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Relativity of Simultaneity, the Global Positioning System and Some Questions About Gravity

Richard Benish
Eugene, Oregon rjbenish@teleport.com
We rarely stop to think how much our conceptions of the physical world are colored by the theories we use to describe it. In the minds of most physicists the idea of rotating and falling clocks communicating with one another by light signals will immediately evoke features of Newton’s Theory of Gravitation and Einstein’s Special and General Theories of Relativity. While acknowledging the incalculable value of these theories, we here reconsider the phenomena mentioned above, trying at every step to distinguish physical facts from theoretical facts; i.e., to distinguish concrete reality from abstraction based on reality; and to distinguish what we know from what we assume. As of course must be the case, the resulting new perspective agrees with the theories mentioned throughout the range over which they have been tested. But it also turns up a couple blindspots where feasible experiments would reveal which perspective is closer to the truth.

1 Introduction
Leaving most gravity issues for later, we begin with the roots of Einstein’s Special Theory of Relativity (SR). [1] A curious and well known fact about SR is that the equations at its core, the Lorentz transformations, were ﬁrst derived by Lorentz [2] – whose conceptions of space and time diﬀered from those of Einstein. Lorentz assumed the existence of a “luminiferous ether.” Einstein did not. Yet the respective theories of these authors yield the same predictions for physical phenomena. There are good reasons that Einstein’s interpretation has turned out to prevail in the course of time. And I do not intend to revive Lorentz’s views of the ether. Nevertheless, it is worthwhile to reconsider some of the ideas that motivated Einstein to give the Lorentz transformations the interpretation that he did. In particular, we shall look critically at what Einstein called “the most important, and also the most controversial theorem” of SR: the relativity of simultaneity. [3]
It is essential that we remain alert to the thoughtcoloring eﬀect of theoretical knowledge. A way to do this is to be perpetually engaged in assessing the distinction between raw physical facts and the abstractions that remain after distillation and mixing by theory. Since this is a recurring theme of this paper, in §2 I present an example to clarify my intent.
In §3 we set up a base inertial system much like Einstein’s “stationary system.” In §4 we add a second inertial system and address the signiﬁcance of the speed of light in any attempt to relate the one system to the other. With the beneﬁt of hindsight, we here also consider how the ideas of Reichenbach [4] (from the mid 1920’s) bear on the way Einstein set up SR. Reichenbach emphasized the interdependence between the relativity of simultaneity and one-way light propagation assumptions. Einstein, of course, assumed that every inertial

observer has the right to regard the one-way speed of light as being equal to the two-way speed of light (second postulate). Thus, one of Einstein’s recurring themes was the idea that everybody has a right to think of themselves as being at rest; that this is the most satisfactory point of view from which to sort out the “laws of nature.” In §5 we take a look at how far Einstein took the idea of “self rest.”
In §6 we introduce a uniformly rotating frame of reference centered on our original base system. Thus we come to the Sagnac eﬀect and comments by various authors concerning how rotation aﬀects the problem of clock synchronization.
If our rotating frame were a sphere, in §7 we temporarily suspend its rotation, but instead endow it with mass. Finally, we consider the sphere having both rotation and mass, and discuss how the combination of these eﬀects applies to the problem of clock synchronization in the Global Positioning System (GPS).
Having up to this point not strayed too far beyond the bounds of standard ideas, §8 is where we begin to suggest some new ones. In the interest of improving palatability, these new ideas are presented as being the hypotheses that a team of imaginary space explorers might come up with to explain their ﬁrst encounter with a large gravitating body.
Coming back to the real world in §9, the highest priority idea that the space explorers conceive to test their hypotheses is discussed in more detail, because it is the most clear-cut and most feasible way to decide between them. The experiment would be a test of gravitational interior solutions. In this case the prediction of our space explorers deviates not only from General Relativity, [5] but from Newton’s Theory of Gravitation. [6] Since their ﬁrst experience with gravity is entirely diﬀerent from Newton’s and Einstein’s experiences, we should not be too surprised to ﬁnd that our space explorers

c Richard Benish 2008

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would devise a profoundly diﬀerent hypothesis. In §10 additional hypothesis-testing ideas are discussed in
terms of experiments and observations that have already been carried out from Earth: the Shapiro time-delay test [7] and the Vessot-Levine falling clock experiment. [8] It is predicted that a new variation of the latter experiment might yield deviations from the predictions of General Relativity (GR).
In §11 we discuss the falling clock experiment as being conducted in a frame of reference undergoing uniform linear acceleration to show that the new prediction actually agrees more exactly with Einstein’s original statement of his Equivalence Principle.
Though some of the predictions of the new hypothesis deviate from Newton’s and Einstein’s theories, in §12 it is shown that it is nevertheless consistent with the operation of the GPS and other well known consequences of GR.
In §13 we point out further implications of the new hypothesis. We conclude by emphasizing that, even if physicists trained in the Newton-Einstein tradition have reasons to doubt the new hypothesis, the experiments proposed to test it are still worthwhile. Their results would still contribute signiﬁcantly to our empirical knowledge of gravity.
2 Facts and Abstractions
There are diﬀerent kinds of facts. Some are direct reﬂections of physical reality, what may be called raw physical facts. And some are derived from theories about physical reality. It is often the case that physical facts are perceptible by children or animals. Whereas theoretical facts require a more sophisticated level of thought to invent or comprehend. Since theories about physical reality are a step or more removed from physical reality itself, the latter kinds of facts are clearly abstractions. Of course there is nothing wrong with abstractions; they are quite necessary. But they are inevitably colored with preconceptions and tacit assumptions. An abstraction can and should be argued about; it is open to debate. Whereas, unless one subscribes to a school of philosophy in which physical reality itself is doubted, raw physical facts are not open to debate. An example follows.
Suppose we have, in the far reaches of outer space, a long hollow cylinder being propelled along its axial direction by a rocket engine. Inside the cylinder a test object is released from the top (leading end). Observations reveal the raw physical fact: the distance between the released object and the bottom (ﬂoor) of the cylinder diminishes at an accelerated rate. (Except to acknowledge that they exist, this paper is not the least bit concerned with the opinions of those who would debate this kind of fact.) Since accelerometers attached to the rocket give positive readings that correspond exactly to the observed motion, and an accelerometer attached to the falling object gives a zero reading, we may conclude further that the reason the distance between object and ﬂoor diminishes is

because the rocket accelerates while the object moves only uniformly. The explanation for why the distance diminishes is so clearly correlated with observable physical facts that this phenomenon can reasonably be claimed to be understood.
Now suppose we have a similar cylinder oriented vertically on Earth’s surface but there is no rocket. The distance between the released object and the ﬂoor diminishes at an accelerated rate. Accelerometers attached to the cylinder give positive readings which correspond exactly to the observed motion, and an accelerometer attached to the released object gives a zero reading. The physical facts are nearly the same as before. But in this case the reasons are not so obvious. The usual description of this circumstance is that the released object accelerates downward; the usual description of Earth and its surrounding space is that they are essentially static things. The accelerometer readings are anti-correlated with these descriptions. Therefore the facts do not yield so clearcut an explanation as in the previous scenario. What are we missing?
In our attempts to correlate the facts, the cylinder on Earth scenario is commonly discussed in terms of gravitational attraction, gravitational ﬁelds, the warpage of spacetime, geodesic vs non-geodesic motion, the Equivalence Principle, force mediation by gravitons, and so on. Each of these concepts is part of one little story or another whose purpose it is to make sense of the facts. Though useful as “talking points,” the explanations these ideas provide for the facts, being uncertain and incomplete, are best regarded as tentative. Depending on which idea is involved, we will even ﬁnd diﬀerent assessments as to whether or not the accelerometers are reliable indicators of motion. More experiments and perhaps more ideas are needed to decide among the possibilities.
To conclude then, in the cylinder on Earth scenario, either we are misinterpreting some relevant facts, we are missing some relevant facts, or both. We know the distance between the released object and the ﬂoor diminishes and that accelerometers behave as described above, but we don’t know exactly why. We cannot reasonably claim to understand this phenomenon as we do the rocket in space phenomenon. So we acknowledge what few physical facts we have and assiduously refrain from giving our abstractions the same status.
3 Base Reference Frame
Our base system is an inertial reference frame (IRF), which we will sometimes also refer to as an inertial system, or simply, reference frame. It should be emphasized at the outset that an IRF is rather artiﬁcial. There is no good reason to think a strictly inertial system of ﬁnite extent would ever be found in Nature. Our conception and construction of such a thing is guided by simplicity. Many physical problems are clearly simpliﬁed by assuming that phenomena are being observed from the vantage point of an idealized inertial system.
Imagine in empty space a massless grid. Suppose the

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3 Base Reference Frame

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Inertial System
K

z y

x
r
Fig. 1: Base reference system. Clocks are preset to time r=c and start ticking upon receipt of ﬂash from center.

such as anisotropic light propagation. Yes, we “stipulated” that light propagation would be isotropic; but it was surely a natural stipulation, since our construction was motivated primarily by simplicity. The question is, is it justiﬁable to think we can have a plurality of IRF’s constructed this way when each one has a unique uniform velocity with respect to all the others? Einstein recognized that a positive answer to this question required a reconciliation between two seemingly incompatible postulates, and that this reconciliation was to be achieved by replacing the Galilean transformation equations with the Lorentz transformations. (See Box 1.) The two postulates are:
1. The laws by which the states of physical systems undergo change are not aﬀected, whether these changes of state be referred to the one or the other of two systems of coordinates in uniform translatory motion.

lengths between intersections are known. Identically constructed clocks at the intersections are synchronized by a ﬂash of light from the origin. (See Figure 1.) The idea is essentially the same as that used by Taylor and Wheeler, in their
abook, Spacetime Physics. [9] Each clock is preset to the time
t r=c, where r is the radial distance and c is the speed of light. The clocks are all programmed to begin ticking at the precise moment of receiving the signal. At the moment
a Hthe ﬂash is emitted, the clock at the origin, whose preset time , also begins ticking. Since we make no pretense of representing a physically realistic system (absence of asymmetries and inhomogeneities) we are justiﬁed to assume that the speed of light is exactly c in all directions. Therefore, it is reasonable to assert that, by the above procedure, all clocks in this system have been synchronized with one another. Let’s call this system, K.
Since this synchronization procedure is equivalent to the various procedures prescribed by Einstein we will refer to it (them) as the Einstein or standard procedure, whose result is Einstein or standard “synchrony.”

2. Any ray of light moves in the “stationary” system of co-ordinates with the determined velocity c, whether the ray be emitted by a stationary or by a moving body. [10]

Without these postulates, or perhaps more accurately, with-

out the Lorentz transformations, the speed v itself would al-

ready be problematical because each frame needs its own set

of synchronized clocks to measure it. That’s why Einstein

was careful to build his theory by ﬁrst deﬁning the concepts

of time and simultaneity before discussing the relative veloc-

iKtyH

of to

inertial systems. In measure the speed v

order for as having

observers in both K the same magnitude

and they

also have to “measure” the speed of light as having the same

magnitude. The only way to arrange this is to agree by “stip-

ulation” that clocks will be set so as to make it appear true.

Clocks “synchronized” so as to make it appear true in one

frame will then appear to be “desynchronized” when viewed

from the other frame. Speciﬁcally, in order to postulates for both frames, it turns out that clocks

isnatKisfH ywbooutlhd

have to appear desynchronized for observers in K, and vice

versa.

4 Uniform Velocity

4.1 Two Postulates

Static emptiness is easy to conceive and problem-free. Things

suddenly adding a

bseeccoonmderceofmerpenlicceatseydsatenmd ,cosnaytroKveH,rstihaal,t

however, by moves uni-

formly with the speed v with respect to K. As is well known,

the root of the problem is the speed of light and its importance

for synchronizing clocks in a given frame. Within our base

system K the problem of “distant simultaneity” is simple and

not really a problem at all because in the course of building

it up we had not yet any reason to even imagine asymmetries

4.2 Desynchronization and Anisotropy

The physical reason for the desynchronization is that, from

the point of in the other

svyisetwemof(KonH)e

system, e.g., K is not isotropic.

, the speed of light Desynchronization

and light speed anisotropy are interdependent and inextrica-

ble. They are both always attributable to all inertial systems

except one’s own. This might be called the “back side” of

the Relativity coin. The “front side,” which gets most of the

attention, asserts the isotropy of light for all inertial systems.

The cross-frame desynchronization resulting from Einstein’s

synchronization procedure is typically thought of as a result

of this “isotropy for everyone” idea, rather than as a result of

the exclusive “isotropy only for me” idea that I’ve presented.

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UNIFORM VELOCITY TRANSFORMATIONS (for motion along x-axis)

Step in Einstein’s derivation of the Lorentz transformations
t

POIINNITTIOAFLk RLAIYGHT

Galilean Transformations

x´= x –vt y´= y z´= z t´= t

The above transformations cannot be generally valid because they fail to account for the finite speed of light and the effect of velocity on length and time standards.

ct

vt

(c – v)t

x´ x´ x´ x´

0

t

=

(c

x´ – v)

x´= (c – v)t

distance

Lorentz Transformations

x´= x –vt

1–

v

2
c

2

y´= y z´= z

Fig. 2: When viewed from K the speed of light in k is not c. In the direction of increasing x along the x-axis, it is (c v). Isotropic light propagation is only assumed for one’s own frame of reference; which means it has to appear as being anisotropic for all other frames of reference.

t´ =

t

–

xv c

2

1

–

v

2
c

2

These transformations satisfactorily account for the finite speed of light and they accommodate (predict) length contraction and time dilation. To Lorentz the time t represents the “true time” of clocks at rest in the “preferred” ether frame; and x represents the corresponding “true” uncontracted lengths. To Einstein, no frame is “truer” than any other.
Box 1.

There’s no argument here about observable physical facts; but there is a diﬀerence in perspective, in philosophical outlook, if you will. Isotropy for everyone perhaps sounds more democratic, and it gives rise to the pleasant sounding expression, “relativity of simultaneity.” But it is not often enough pointed out, in my opinion, that satisfying Einstein’s postulates by adjusting clocks this way only “works” because, as viewed from one’s own reference frame the speed of light with respect to moving bodies does not equal c. You can’t have the speed of light being isotropic in one’s own frame unless you regard it, at least sometimes, as being anisotropic in other frames.
Obvious though this may be, I hesitate to call it trivial. So let’s look at a few examples. In the course of deriving the Lorentz transformations, Einstein describes the progression

of a light ray in a stationary system (K) with respect to a point

in the moving system (k). (See Figure 2.) Time as measured

a H in K
when

is t

referred , and

xtoH

as t; is the

the ray is emitted from the origin distance the ray travels in the time

t. Einstein writes: “The ray moves relatively to the initial

@ A a point of
velocity

k, c

wvh,esnomtheaatsuxrHe=dcin

the v

stationary t.” [11]

system,

with

the

Minkowski diagrams that depict a pair of IRF’s having
a a relative speed v are a more common context showing

the anisotropy of light speed for other inertial systems. Each

system, of course regards itself as being just like our initial

system K, in which light speed is patently isotropic. In the

frame that is typically reserved for ourselves, the space and

RS tsipmeeedaixsesreaprreesveinstiebdlybyperpenldigichut lcaornteos.eaBcuht

other with

and light respect to

¦ the tilted worldline of the other IRF the speed of light is c v;

the magnitude of tilt corresponds to the magnitude of v. Fig-

ure 3 is a Minkowski diagram showing two reference frames
a H T with a relative speed v : c. For this large a velocity the ef-

fects of desynchronization are quite pronounced. By studying

the space and time data marked on the ﬁgure one can deduce

the diagram’s symmetry and reciprocity. Both systems regard

the other as being out of sync and as having slowed clocks

and shortened rods. It would of course be possible to draw

this diagram from the point of view of the “moving” frame,

as a mirror image in which the latter frame is now straight up

and perpendicular to the time line and the asymmetries apply

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4 Uniform Velocity

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WH to the other frame. But such
Although the frame that gets

tahree-drawaxeissns’et ermeasllsyunpeecrﬁescsiaalrlyy.

to be “preferred,” strictly speaking, both frames in this view

are already quite “equivalent.”

We have equations and a diagram. Now let’s consider

a simple physical example. In my left hand I have a laser

pointer that I shine on my outstretched right hand. Suppose

the speed of the beam with respect to my left hand is c. Now

I move my right hand toward the pointer at the speed v. Shall

we insist that even when my right hand is so moving, the

speed of light with respect to it hasn’t changed, that it is still

c? It’s possible to set up a Minkowski diagram representing

my left and right hand. But why should I bother invoking a

new set of clocks conforming to it? So that I can “ﬁnd” the

speed of light to be c with respect to these clocks? Democratic

though it may be, the symmetry of the Minkowski diagram

strikes one as being rather more stipulated and contrived than

physical. SR is all about arranging (by stipulation and con-

trivance) a set of clocks that make the second postulate appear

true. As we will see momentarily, even Einstein endorsed the

C view that it is no less correct to say that the speed of light with
respect to my right hand may indeed be regarded as c v. If

it’s no less correct, then why doesn’t this approach get “equal

time”?

RELATIVE SIMULTANEITY: Stipulated isotropy in both frames

t

t´

3.75 3.00

2.50 3.125

c = 1

(LIGHT PATH)

1.25

v = 0.6

x t

=

x´ t´

=

c

2.50

2.50 2.00

1.60 1.45 1.30 1.15 1.00 .85 2.00 2.00 2.00 2.00 2.00 2.00

0

0

1.25

.80 .65 .50 .35 .20 .05

0

1.00

1.00 1.00 1.00 1.00 1.00 1.00

0 0 0

t´ =

t

–

xv c2

1

–

v

2
c

2

0 -.15 -.30 -.45 -.60 -.75 -.90 -1.05 -1.20 -1.35 -1.50 -1.65 -1.80 -1.95

4.3 Lorentz and Reichenbach
Although most relativists don’t think of it quite this way, it is certainly true that Lorentz did. Lorentz knew that his equations accommodated the appearance of symmetry, while also allowing the idea of a preferred ether frame with respect to which light propagation is really (not just stipulated as being) isotropic. To Lorentz, his transformation equations were an expression of how Nature camouﬂages the existence of the ether. It is worthwhile to consider one of his last statements on the matter (from 1927):
A physicist of the old school says, “I prefer the time that is measured by a clock that is stationary in the ether, and I consider this as the true time, though I admit that I cannot make out which of the two times is the right one, that of A or that of B.” The relativist, however, maintains that there cannot be the least question of one time being better than the other.
Of course this is a subject that we might discuss for a long time. Let me say only this: All our theories help us to form pictures, or images, of the world around us, and we try to do this in such a way that the phenomena may be coordinated as well as possible, and that we may see clearly the way in which they are connected. Now in forming these images we can use the notions of space and time that have always been

00000000000000

0

.50

1.0

1.5

2.0

2.5

3.0

x´

x

0

.50

1.0

1.5

2.0

2.5

Fig. 3: Symmetry and reciprocity of Special Relativity. By comparing clocks at key events, we see that the speed of light = c in both frames, so that all relativistic eﬀects will be attributed to the other frame, no matter which one we think of as being “at rest.”

familiar to us, and which I, for my part, consider as perfectly clear and, moreover, as distinct from one another. My notion of time is so deﬁnite that I clearly distinguish in my picture what is simultaneous and what is not. [12]
Remember, we are not advocating a return to the “old school.” Rather, the idea is to point out that the path taken by the “new school” may not be the only viable one. Indeed, it may be that, by insisting on the theoretical value of the relativity of simultaneity, other alternatives are being overlooked.
In this spirit, note that in the mid 1920’s H. Reichenbach [13] proposed a method whereby any of Lorentz’s “untrue” frames can be explicitly represented as such, by introducing a new “synchrony parameter,” having a magnitude
H I <  < . The idea is a generalization of Einstein’s clock
synchronization prescription, as follows. Suppose A and B are two ﬁxed points in K, tA is the time of emission of a sig-

c Richard Benish 2008

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nal from A, tB is the time at which the signal is received by B and is immediately reﬂected back to A, and tA0 is the time
at A when the reﬂected signal returns:

a @ A tB  tA0 tA :

(1)

Einstein did not include or discuss the range of possible -

values; in place of , Einstein simply stipulated that its value

is

1 2

.

This is the standard approach.

If observers in every

inertial system adopt this convention, they can justify the as-

sumption that light speed is isotropic for each observer with

respect to himself. The obvious advantage of this approach

is its stark simplicity. If it’s simple and it works, one can

understand why many practitioners would want to just leave

it at that. Neither Einstein nor most relativists ever concern

themselves with non-standard -values. Besides Reichenbach

and a few others, a recent exception is Selleri, [14] who has

introduced a “synchronization gauge,” e1, whose purpose is essentially the same as . Since we are here concerned more

with the idea than with its rigorous application, we simplify

the discussion by referring to the idea in terms of  only.
Suppose that the uniformly moving reference frame KH

is the one to which  ourselves as being “at

res12t”aipnplKiesH,.

We may thus think so that with respect

of to

ourselves light propagates anisotropically. Adopting a non-

standard value for  as applying to our own IRF is tantamount

to singling out some other reference frame (e.g., K) in which

the

value

is

1 2

and

light

propagation

is

isotropic.

This

possi-

bility is depicted in the “-synchrony” spacetime diagram in

a H T Figure 4, which adopts the same velocity (v : c) as that

used in Figure 3, and so represents exactly the same physical

circumstance.

Although the clocks in the diﬀerent (purple and green)

a a H cfrharmoneiszheadvaetdtiﬀeretnH t

rates, ;

we the

can see that time given

they are by both

not desynclock sets

RS iasreinndoetpaetnden.t

of distance. Even though

Also we see that the this scheme exhibits

light paths anisotropic

light propagation (for the purple frame) and non-reciprocal

velocities, all clock comparisons made from the origin of the

isotropic frame (world line with blue boxes) and all back and

forth light speed measurements agree with SR. Such a scheme

is thus equally in accord with all empirical evidence. Of

course, Solar System inhabitants have practical reasons for

not adopting it. Many of us instead take for granted that the

a

1 2

scheme

is

the

“universally”

most

appropriate

one.

But

perhaps there is a circumstance somewhere in the Universe in

which this

1 2

scheme

actually

does

make

more

sense.

4.4 Geometrical Approach
The emphasis on light speed anisotropy in the above is likely to be frowned upon by most relativists. To a particular “school” of relativists even slow clocks and shortened rulers

REICHENBACH-GRÜNBAUM ε-SYNCHRONY: Isotropy in only one frame

t´

t

2.50

2.50

3.125

2.40

3.125

3.00

• Light paths ≠ 45°

2.00

• One-way speeds ≠ c.

2.50

• In both frames back and

1.60

forth speed of light = c.

1.60

2.00

• Velocities not reciprocal

2.00

.80

t´= t

1

–

v

2
c

2

1.00

x´= x 1– vc2 2

.80 1.00

.80 1.00

00000000000000 00000000000000

-2.0

-1.5

-1.0

-.50

0

.50

1.0

x´

x

-1.5

-1.0

-.50

0

.50

1.0

Fig. 4: Spacetime diagram whose physical circumstances are exactly the same as in Figure 3 (v=c = 0:6). Even though the -synchrony scheme is not usually practical, “it could not be called false.”

are antithetical concepts. It is appropriate to acknowledge this here.
Within the context of standard physics, as Denker has pointed out, “There are two ways of formulating the essential ideas of special relativity.” [15] Denker laments that the most popular approach – especially as found in most introductory texts – is what he calls the “contraction / dilation approach that alleges that rulers contract and clocks run more slowly when moving relative to the observer.” This is the approach by which SR was born; it is the approach of physicists whom I will refer to as “Einsteinians.” Since I shall also argue that the Einsteinian approach may itself be excessively abstract, the alternative approach mentioned by Denker will be seen as being even more so. In fact, I think it is a perfect example of how the “Einsteinian style” of reasoning might be expected to evolve; it is the natural outcome of what Einstein started. The latter “evolved” approach emphasizes the abstract geometrical relationships of relativistic (Minkowski) spacetime, and is espoused, to one degree or another, by authors such as

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4 Uniform Velocity

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Cross Frame Projection: GEOMETRICAL PROPERTY

Joe Moe

3.00 2.40

2.40

2.80 2.20

2.60 2.40

2.00

1.80 2.20

2.20 2.00 1.80

2.00

1.60

1.60

1.80 1.40 1.40

1.60 1.20
1.40

1.00 1.20
0

1.00

.80

0

.80

.60

.60 .40

.40

0

.20

.20

0

0

0

v c

= 0.6

1 γ

=

1

–

v

2
c

2

= 0.8

0000000000

¡ ¡ t=  merely describes the projection from
Moe’s frame onto Joe’s frame. ...The ratio
¡ ¡ t=  is a property of the projection, not a
property of anybody’s clock.
Figure 5A shows one way of projecting Joe’s time onto Moe’s. Figure 5B shows what looks to me like a physical clock property. Notice that the hands of the clocks do not line up. Moe’s clock shows a smaller time than Joe’s clock. (Are we really not to think of this as a property of the clocks?) Denker argues that this does not mean that Moe’s clock ran more slowly.
After Moe’s journey the side-by-side clock comparison indicates that “Moe took a less time-consuming path, not that his clock ran slower.” To Denker this argument is not only acceptable, it is purported to be “far simpler, more powerful, elegant and consistent.” The time diﬀerence is a “property of the projection, not a property of anybody’s clock.” A properly functioning clock returns from a journey with a smaller elapsed time than the stay at home clock. Denker purports to have provided a simpler explanation than that the moving clock ran slower. Will it make more sense if I repeat it again? My repetition is borne of incredulity. I’m sorry, but I seem to have been deprived of the side of the brain needed to appreciate this style of reasoning. Unfortunately, we will encounter it again.

Immediate Perception: PHYSICAL PROPERTY

4.5 Conventions and Stipulations

Upon Moe’s Return:

Joe

Moe

Fig. 5: Due to their relative velocity, a projection of Joe’s time onto Moe’s is oblique. Symmetry is disrupted when Moe’s velocity changes; the projections no longer line up.

Wheeler and Cuiﬀolini, [16] Ohanian and Ruﬃni, [17] Carroll, [18] Hartle, et al, [19] and of course, by Denker himself.
It will be useful to discuss one example. Denker presents the classic twin problem, starring Joe, who stays at home and Moe, who takes a high-speed journey and returns. Denker insists,
Let’s be clear: Yes, at the end of the journey, Moe is younger than Joe (assuming they started out with equal ages). No, this does not prove that Moe’s clock ran slower. A far simpler explanation is that Moe took a less time-consuming path.
¡ ¡ If t is the ﬁnal time on Joe’s clock and  is the ﬁnal time
on Moe’s clock, then, Denker concludes:

Assuming that the anisotropy and corresponding metric ef-

fects (length contraction and time dilation) apply in one’s

own frame of reference entails many complications. The as-

sumption implies the need to identify the “preferred” frame in

which clocks have maximum rates and rods have maximum

lengths. This would in turn imply the need to keep track of

any pertinent third frames wherein the metric eﬀects are also

due to motion with respect to the preferred frame. Therefore

reciprocity would be lost; symmetry would obtain only with

respect to the preferred frame. These characteristics can be

clearly seen in terms of Figures 3 and 4. The desynchronized

clocks of Figure 3 provide the simplifying advantage of mak-

ing the speed relative. Both frames ﬁnd v to have the same

magnitude. Whereas Figure 4 shows that, if a moving ob-

server uses the same operations to measure his own velocity,

the magnitude comes out greater than that measured by the

resting observer. Therefore (by design) the

1 2

scheme

is

such that one’s “true” speed is the speed as observed from the

“true” rest frame.

Given the obvious complexity of the above approach, it

is understandable that one may sense some superiority to the

scheme resulting from adoption of Einstein’s postulates,

whereby symmetry, reciprocity and simplicity are built into

the foundation. This “simplicity argument” is often invoked

a (if not explicitly, then tacitly) for adopting the standard

1 2

c Richard Benish 2008

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and agreeing to let all inertial observers do the same. But the

simplicity argument does not mean the

1 2

scheme

is

in-

correct. At the same time he proposed his -synchrony idea,

Reichenbach argued that it “could not be called false.” [20]

Within a year of its publication, Reichenbach’s analysis was

endorsed by Einstein: “special care has been taken to ferret

out clearly what in the relativistic deﬁnition of simultaneity

is a logically arbitrary decree and what in it is a hypothesis,

i.e., an assumption about the constitution of nature.” [21] In

spite of Einstein’s endorsement, very little attention was paid

to Reichenbach’s work till Gru¨nbaum [22] revived it in the

mid 1950’s.

Since then, the relativity of simultaneity has been a sub-

ject of lively debate. Einstein’s assertion decades earlier that

this was the most controversial aspect of SR seems no less

true today. This is not the place to hash out the many argu-

ments that have been put forth. For that, consult Max Jam-

mer’s 2006 book, Concepts of Simultaneity [23]. I should

mention, however, a few key contributions. The debate is

divided, roughly, into so called non-conventionalists and con-

ventionalists. The former contend, essentially, that the tempo-

ral and light propagation relationships between IRF’s accord-

ing to SR are ultimately more than conventions, deﬁnitions,

stipulations and assumptions. Corresponding to this view is

the inclination to think of Einstein’s second postulate as being

physically true, and that experiments purporting to be tests of

light speed isotropy are as claimed. The “geometrists” men-

tioned in §4.4 would evidently be classiﬁed as especially hard

core non-conventionalists.

The conventionalists, by contrast, contend that the tem-

poral and light propagation relationships between IRF’s are

indeed only convenient deﬁnitions, stipulations and assump-

tions. However useful the conventions may be, convention-

alists claim it is impossible to prove that one IRF should

be preferred over others or to prove in which direction the

anisotropy axis with respect to any particular frame actually

lies. But this view clearly allows that some such anisotropy

axis may exist, or even that some such anisotropy axis in-

evitably exists for most reference frames, however “dis-

guised” it may be. According to this view (which is similar

to that of Lorentz) experiments that purport to establish the

“isotropy of space” are actually illusions which are implicitly

based on non-trivial conventions.

4.6 Central Logical Unclarity?
SR was published in 1905. Over a century later the status of the core of SR is still a subject of debate. In 1970 Winnie showed that transformation equations could be derived based on Reichenbach’s -synchrony idea; and that these
-Lorentz transformations, for arbitrary choices of , are simply one-one translatable into consequences of the standard Lorentz transformations,

by taking initial clock-settings into account. This result thus demonstrates that diﬀerent choices of  result in kinematically equivalent versions of the Special Theory, and this is precisely the claim made by the thesis of the conventionality of simultineity. [24]

In 1977 Salmon analyzed various performed and proposed

experiments whose purpose was to measure the one-way

speed of light. He showed that they all involve non-trivial

conventional elements and so failed (or would fail) to measure

light’s actual one-way speed. Thus Winnie’s results were fur-

ther reinforced. One of the consequences stressed in

Salmon’s work is a point raised in §4.1; i.e., that reciprocity

of any one-way velocity hinges on (or goes with) reciprocity

of light velocity. In other words, unless clocks are desyn-

chronized exactly as in Figure 3 (i.e., under the assumption

a that

1 2

)

the

speed

of

light

and

the

speed

of

any

material

body will not be symmetrical (will not exhibit reciprocity) as

between one frame and another (as shown in Figure 4). Since

this reciprocity has not the status of a physical fact, but is a

matter of convention, Salmon ponders where that leaves us:

The status of the one-way speed of light...lies at the heart of our analysis of the concept of any one-way velocity whatever. How deeply can we claim to understand the physical world if this concept suﬀers central logical unclarity? [25]

One of the most inﬂuential recent arguments against the

conventionality thesis is a paper by Malament [26]. Later

commentators have acknowledged Malament’s main argu-

ment that only standard synchrony singles out the simplest

Minkowski space (perpendicular simultaneity surfaces) but

many are not convinced that this means the physical world

should necessarily conform to it. Janis [27] for example, has

argued that Malament’s analysis boils down to a restatement

of the simplicity argument: However true the analysis may

be, it by no means proves that space is isotropic in one’s own

frame or that the

1 2

scheme

is

incorrect.

My impression is that most critical physicists and philoso-

phers acknowledge that the various attempts, with thought

or real experiments, to prove that space is actually isotropic,

contain non-trivial conventions. Curiously, both sides of the

argument have appealed to Einstein’s work to defend their

positions. The non-conventionalists cite the second postulate

itself and the many instances of Einstein’s stating or defend-

ing it. And the conventionalists cite Einstein’s need to deﬁne

or stipulate the meaning of simultaneity as conforming to the

light postulate by convention. Doing so results in a simpler

theory. But it does not change the fact that to measure the

one-way speed of light a set of spatially separated synchro-

nized clocks is needed. You can’t know for a physical fact

that your clocks are actually synchronized unless you know

the one-way speed of light. If one of these things is physically

unascertainable, then so is the other. This is the case. So in

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a roundabout way Einstein’s choice to stipulate his way out of the logical circle is a concession that a measurement of the one-way speed of light in inertial systems is impossible; only the two-way speed is measureable. (And this speed comes out equal to c for every permissible value of .)
Einstein’s position on the matter is not as clear or consistent as it could have been. As late as 1949 Einstein writes of his light speed postulate as being “based on experience.” [28] In that context he did not point out that this included only the average two-way speed or point out the conventions behind this “experience.” To my knowledge, Einstein never explicitly expressed the possibility, inherent in his own work and as brought out by Reichenbach, that light propagation might actually be anisotropic with respect to oneself. So one can see how both conventionalists and non-conventionalists might try to adopt Einstein as one of their own. After weighing the evidence on this question, Max Jammer wrote: “Summing up, it seems that, as far as the concept of distant simultaneity is concerned, Einstein can be classiﬁed as a conventionalist, who however sometimes made statements not wholly consistent with the position.” [29]
“How deeply can we claim to understand the physical world if [the concept of one-way velocity] suﬀers central logical unclarity?” Most of the “conventionalism” debate has been carried out by “philosophers” of science or by the more philosophically-minded physicists. One still ﬁnds little discussion of the matter in text books; and one still ﬁnds alleged “Tests of the isotropy of the speed of light...” [30] by competent experimental physicists. Many physicists, no doubt, think such tests have established “space isotropy” to a high degree of accuracy. Meanwhile, conventionalists call these tests “illusions.” [31] If he were alive would Einstein help to settle the debate? Perhaps. But all we have to go on is what he left us with. I think the record shows that Einstein’s deep interest in simplifying and symmetricalizing theories overrode his interest in being critical of his relativity of simultaneity. In one of his last public talks, Einstein was quoted as saying,

“The laws of physics should be simple.” Someone in the audience asked, “But what if they are not simple?” [to which Einstein answered] “Then I would not be interested in them.” [32]

Theoretical and mathematical simplicity are recurring themes

in Einstein’s career. Writing in 1933, he asserted that, “our

experience hitherto justiﬁes us in believing that nature is the

realization of the simplest conceivable mathematical

ideas.” [33] Given the priority Einstein put on mathematical

simplicity, it is not at all surprising that he seems always to

a have assumed

1 2

.

Any scheme that explicitly attaches

importance to anisotropic light propagation is clearly at odds

with Einstein’s view of the world, and is even less congenial

toward the geometrists who have taken Einstein’s quest for

abstract simplicity to even greater extremes.

4.7 Foreshadowing the Importance of Non-Inertial Reference Frames
The relativity of simultaneity and light speed anisotropy are inextricably linked. But notice that they have a crucial difference in character. Light speed anisotropy is an inevitable physical fact with respect to most everybody. Light speed anisotropy an inevitable physical fact with respect to most everybody? To back up this assertion I appeal to the real physical world. As a brief preview of the subject of §6, we have the fact that light signals sent in opposite directions around a rotating body (e.g., Earth) take unequal times to return to their starting point; i.e., the speed of light with respect to rotating observers is not isotropic. This fact can be ascertained without use of a clock and therefore does not depend on any non-trivial stipulations or conventions. If one objects that facts gleaned from non-inertial systems are inadmissible in the context of SR, I would point out that we are not, I hope, as government-sanctioned lawyers and judges abiding by the formalities of a court of law. Our investigation is not to be mind-cuﬀed by legalities. Rather, we are as private detectives whose mission is to ﬁgure out what’s really going on. If in the general case (non-inertial) system we ﬁnd anisotropy to be a fact, then, by what logic should this anisotropy be overruled by an abstract contrivance whose purpose is to make it appear to go away in every special case (inertial) system? Relativistic physics is the only context I know of in which special, local “laws” have the status to squelch global physical reality.
It may not be possible to determine the axis (or axes) of anisotropy. But isn’t it more reasonable to accept this uncertainty than to insist that light is really physically isotropic with respect to, for example, each of the thousands of changing inertial systems attached to every bee in a hovering swarm? Strict adherence to SR, of course, permits (or even requires) one to imagine this multitude of “equivalent” frames. This is a mathematically consistent theoretical fact. On the other hand, relativity of simultaneity is not a physical fact. It is an abstract idea imprinted (or not) on one’s conception of reality. You could argue that the relativity of simultaneity derives from light speed anisotropy; light speed anisotropy explains why clocks in other IRF’s are desynchronized. But turning the argument around makes no sense: desynchronized clocks do not explain why light propagation is anisotropic. As noted above, this is demonstrated in the case of rotation, where the anisotropy can be observed as a physical fact without use of any clocks. Physical facts do not derive from abstract ideas.
One must nevertheless acknowledge the value of the simplicity argument. Both theoretical and experimental physics have beneﬁtted from Einstein’s theory. I don’t mean to suggest that we should stop assuming that light speed is isotropic when adopting this assumption harmlessly simpliﬁes our work. But I would suggest that we should strive to become more conscious of the diﬀerence between physical reality and

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abstract theory; and to be more candid and straightforward about what is a physical fact and what is not. The abstract law is that the speed of light equals c for everybody. The physical fact is that it does not. The “central logical unclarity” concerning velocity is not going to go away if we keep obscuring the complicated physical world with our simple laws.

are at rest relative to the disc, he regards as the eﬀect of a gravitational ﬁeld. Nevertheless, the space-distribution of the gravitational ﬁeld is of the kind that would not be possible on Newton’s theory of gravitation. But since the observer believes in the general theory of relativity, this does not disturb him. [35]

5 Did Einstein Ever Move?

The title of this section may seem like a silly question having an obvious answer. If we take Einstein at his word, however, it would appear that Einstein himself may well have answered, “no.” As we have seen, for Einstein, simplicity in the “laws of nature” was a very high priority. Alongside simplicity is the concept of symmetry, which is not far from the concept of relativity, the impossibility of ascertaining absolute motion and, ultimately, perpetual “self rest.”
Perhaps the best source for examples is Einstein’s book, Relativity, the Special and General Theory. In our ﬁrst example, Einstein refers to the experience of a passenger on a moving train when the brakes are applied. He writes:

It is certainly true that the observer in the railway carriage experiences a jerk forwards as a result of the application of the brake, and that he recognizes in this the non-uniformity of motion (retardation) of the carriage. But he is compelled by nobody to refer this jerk to a “real” acceleration (retardation) of the carriage. He might also interpret his experience thus: “My body of reference (the carriage) remains permanently at rest. With reference to it, however, there exists (during the period of application of the brakes) a gravitational ﬁeld which is directed forwards and which is variable with respect to time. Under the inﬂuence of this ﬁeld, the embankment together with the earth moves non-uniformly in such a manner that their original velocity in the backwards direction is continuously reduced.” [34]

Proceeding immediately to our second example,

. . . we shall imagine KH to be in the form of a

plane circular disc, which rotates uniformly in

its own plane about its center. An is sitting eccentrically on the disk

KobHsiesrvseenr swibhloe

of a force which acts outwards in a radial direc-

tion, and which would be interpreted as an eﬀect

of inertia (centrifugal force) by an observer who

was at rest with respect to the original reference-

body K. But the observer on the disc may regard

his disc as a reference-body which is “at rest”;

on the basis of the general principle of relativ-

ity he is justiﬁed in doing this. The force acting

on himself, and in fact on all other bodies which

Thirty years later (in 1953) Einstein defended the point of view presented in our second example in the course of explaining the Equivalence Principle to a correspondent. Einstein writes: “It [i.e., the relationship between acceleration and gravitation] is the same in the case of the rotation of the coordinate system: there is de facto no reason to trace centrifugal eﬀects back to a ‘real’ rotation.” [36]
Einstein’s motivation for these pronouncements was to either justify or apply what he called the General Principle of Relativity. In his Nobel Award lecture, Einstein stated this principle as follows:
The conclusion is obvious that any arbitrarily moved frame of reference is equivalent to any other for the formulation of the laws of Nature, that there are thus no physically preferred states of motion at all in respect of regions of ﬁnite extension (general relativity principle). [37]
One of the contemporary objections to Einstein’s approach will be mentioned momentarily. First, however, note that these and similar remarks found in Einstein’s work represent not only the meaning that he attached to General Relativity, but key parts of his strategy for building a wider theory. Thus, in his 1949 Autobiographical Notes, Einstein spoke highly of “the plane of insight achieved by the general principle of relativity,” as a route to not only GR but as a worthy guide for going beyond GR. Subsequently he stated the criteria he hoped to meet by his exploration: “It is clear that in general one will judge a theory to be the more nearly perfect the simpler a ‘structure’ it postulates and the broader the group is concerning which the ﬁeld equations are invariant.” [38] Einstein’s intent was thus to keep building on his theories using the same basic strategy, in hopes that he might “succeed in expanding the group once more, analogous to the step which led from special relativity to general relativity.” [39]
It is almost a cliche´ that Einstein persisted for decades, yet failed to reach his goal. His path was solitary and divorced from the developments taking place in quantum theory. Those who worked on GR (whose numbers were not very great prior to the 1960’s) for the most part focussed on Einstein’s ﬁeld equations and had little use for the general principle of relativity. In more recent years Ohanian and Ruﬃni, for example, have explicitly rejected the principle, writing: “tidal eﬀects allow us to measure an absolute diﬀerence between the gravitational force and the pseudo-force found in accelerating reference frames. It is therefore false to speak

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5 Did Einstein Ever Move?

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of a general relativity of motion.” [40] A more thorough critique of the general principle of relativity and its relation to Einstein’s Equivalence Principle is given by Friedman. [41] (Note also that one can ﬁnd arguments more sympathetic with the Einsteinian view. See, for example, Brown [42].)
I don’t suppose many people would ascribe the eﬀects of slamming their brakes or executing pirouettes to transient gravitational ﬁelds and insist that all the while they remained at rest. Einstein failed to reap any additional fruit by adhering to his cherished symmetry / simplicity principles. But he seems to have succeeded in reinforcing the idea that the degree of symmetry in a theory is an indication of how “perfect” it is.
What about the opposite approach? [43] Perhaps the result of reinforcing the above strategy has been in some ways detrimental to physics. Gravitation might be a case in which theoretical symmetry / simplicity principles are being adhered to when physical reality is providing clues that it would be better to let go. Maybe gravity is a more asymmetrical phenomenon than our favorite theories would have us believe. I would advise, for what follows and in general, to remain alert to this possibility.

answers given by the prevailing view appear once again to some authors as being needlessly “awkward.” (This is not the word used by these authors. But I think they would agree with my characterization. Perhaps the best source for an extended discussion of the problem is the book, Relativity in Rotating Frames. [45])
Much ink and paper have been consumed in the course of discussing these issues. Here we will of course strive for brevity, but the questions are of such importance that I hope to also make the facts of the matter quite clear. This will sometimes involve the opinions and the points of view of workers in the ﬁeld. (See also Appendices A and B.)
One of the reasons rotation has been so extensively studied is that it is relatively easy to do so. Not only do we live on a conveniently rotating body and have the need to travel and communicate around it, since its invention, the wheel has evolved in many ways to facilitate probing. And wheels are usually easy enough to ﬁt in a typical physics laboratory. In the latter case, we have the advantage of being observers at rest with respect to the rotation axis.
6.2 Basic Facts

6 Uniformly Rotating Reference Frame
I have yet to see any problem, however complicated, which, when you looked at it in the right way, did not become still more complicated.
– Poul Anderson [44]
6.1 Background
It is well known that Einstein appealed to certain theoretical and physical properties of rotating bodies as a kind of stepping stone from SR to GR. In particular, Einstein inferred that the eﬀect of rotation on uniformly rotating length standards suggests the need for non-Euclidean geometry to properly account for the phenomenon. The idea was then extended to gravitational ﬁelds. But uniform rotation also involves consequences that certainly do not require non-Euclidean geometry, consequences that have been a source of controversy since the early days of Relativity.
The controversy arises for two key reasons: 1) The speed of light is demonstrably anisotropic with respect to rotating observers. The prevailing view is that any such observers may regard themselves as being instantaneously at rest in local inertial frames in which light speed is once again isotropic. But a few authors have emphasized that this view is only rather awkwardly reconcilable with the non-local physical anisotropy. 2) Because of the anisotropy mentioned in (1) and because of the range of speeds found on a rotating body, there arises the question as to how exactly, or whether it’s possible at all, to synchronize clocks on the rotating body. The

Suppose now that our laboratory surrounds the origin of our original system K and that we’ll be observing a rotating body whose axis passes through that origin. To be more concrete, imagine a thin material disk of radius r rotating uniformly with an angular velocity !. Suppose the axis of rotation corresponds to the z-axis. Now imagine that a continuous concave cylindrical mirror is mounted on the disk’s rim so as to facilitate light propagation around the circumference.
Light signals are sent around the disk, one with and one against the direction of rotation. As judged by a clock in K, the time taken for the signal sent in the direction of rotation to return to the source is

T4

a

Pr
c r!

a

c

L v

;

(2)

and for the signal sent against the direction of rotation the time to return to the source is

T5

a

Pr c C r!

a

L
cCv

:

(3)

The diﬀerence in time is

¡T a T4

R P r2!

Lv

a a @I A T5 c2 r2!2 c2

: v2=c2

(4)

A clock located at the emission / reception point on the disk has a lower frequency, fr than a clock in K by the factor

a I a I a I r

r

fr fK

r2!2 c2

v2 c2

:

(5)

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So the reception times as judged by disk-observers using such a clock will be correspondingly smaller. Above and in Figure 6 we have used T and L to denote times and lengths as measured with rods and clocks in K. Using t to denote times measured with clocks in the rotating frame, we thus have

Circumferential light signals: Timed by axis (rest) clocks

a Ip

L

v2=c2

t4

cv ;

(6)

∆T↑

T↑

=

2πr c – rω

=

L c – v

p

I L

v2=c2

a C t5

cv ;

(7)

∆T↑ =

L c

–

L c – v

and

¡t a t4

a IP t5

Lv
p

:

c2

v2=c2

(8)

(We will address the question of lengths measured by rods

in the rotating frame in §6.4.) The ﬁrst order times and the

time diﬀerence given by Equations 6-8 have been observed.

This experimental work began about 100 years ago [46]. One

of the earliest experimenters was Sagnac [47], whose name is

often associated with the observed eﬀect. Sagnac measured

the time diﬀerence (the ﬁrst order part of Equation 4) as a

phase shift in counter-rotating light beams using a rotating

interferometer. Since then, his observations have often been

corroborated in laboratories and around our planet Earth.

I I The second order clock slowping eﬀect given by Equation

5 and implied by the factor =

v2=c2 in Equation 8, is

often neglected in the context of the Sagnac eﬀect because its

contribution to the result is typically very small. Neverthe-

less, various independent experiments and the Global Posi-

tioning System have proven its reality and importance.

∆T↓

T↓ =

2πr c + rω

=

L c + v

∆T↓ =

L c

–

L c + v

( ) RETURN TIME DIFFERENCE

∆T = L
c – v

–

L c + v

= c2

2Lv

1

–

v

2
c

2

Fig. 6: Anisotropic light propagation with respect to rotating disk; basis for the Sagnac eﬀect.
clock to measure the duration of the trip, because the signal returns to its starting point.

6.3 Rotation Plus Translation
The disk as a whole may also have a linear velocity (e.g., in a direction along its rotation plane). We can thus imagine that the rotating body is involved in two kinds of motion, each with its own consequences for the setting of clocks and timing of light signals. But the purely linear motion can be neglected because any consequences of this motion, as we saw in §4, are impossible to determine. They are camouﬂaged as per the Lorentzian interpretation of his transformation equations.
a H With respect to an observer at the center (what we may call
the “lab frame”) where r! , it would therefore be most natural to adopt the simplest possibility by convention; i.e., to assume light propagation is isotropic.
The eﬀects of rotational motion, on the other hand, are not camouﬂaged this way. No convention could make its consequences disappear. What makes it possible to see the eﬀect on light propagation in this case is that a light signal sent in one direction (with or against the rotation) closes back on itself while maintaining the “withness” or “againstness” the whole time. This makes it possible for one observer with only one

6.4 Interferometer Within an Interferometer
Now consider the time average of signals sent in both directions:

hT

i45

a

I P

Pr
c r!

C

Pr ! c C r!

a

L c

@I

I A v2=c2 :

(9)

This is the time as measured in K. We’ve already mentioned
Itphat for disk observers this time would be smaller by the factor v2=c2 due to the slowing of their clocks. With respect to rotating clocks we thus have

hti45

a

L c

pI

I : v2=c2

(10)

We must now address the question whether the motion has any eﬀect on length measurements. The rotating body as a whole can be regarded as a kind of “interferometer,” whose “phase shifts” could be so large, in principle, that they could be timed by clocks or even witnessed by human observers

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SHORTENED RODS INDICATE LENGTHENED CIRCUMFERENCE

Earth. [49] [50]. The results were null. This evidently means that the back and forth path lengths in the perpendicular directions are the same. Just as with the original MichelsonMorely experiment, the most reasonable explanation for this result (ﬁrst proposed by Fitzgerald and Lorentz in 1892 [51]) may be that what we spoke of as “rigid” rods are not really rigid. The lengths of physical bodies, such as the Michelson
Iipnterferometer, are evidently contracted by the factor v2=c2 in the direction of motion. (See Figure 7.)

MICHELSON INTERFEROMETER CONTRACTS IN DIRECTION OF MOTION

LOOSE RODS
LAID ENDTO-END
CONTRACT IN DIRECTION
OF MOTION

Fig. 7: Uniformly rotating bodies undergo changes in length in the direction of the velocity.

who discern a diﬀerence in arrival times of signals emitted

in opposite directions. Now a diﬀerent kind of instrument,

known as a Michelson interferometer, whose dimensions may

be quite small compared to the rotating body, can provide in-

formation on the length question. This device is very sensitive

to changes in length of light paths in perpendicular directions.

As is well known, if light propagation is anisotropic, the path

of a light ray going back and forth, reﬂected from the ends of a

I I rigid rod, say, inpthe direction of the anisotropy (“ether wind”)

is longer by =

v2=c2 than it is when the rod is oriented

in the perpendicular direction. The light path would be longer

because the ray spends more time traveling in the direction in

which it needs to catch up to its destination (c v) than it

does traveling in the opposite direction, in which case the ray

C and its destination are moving toward each otherp(c v). The

I I average time for these two legs is greater by =

v2=c2

than the time needed to travel along the same rigid rod when

it is oriented in the perpendicular direction.

Since rotating observers have already proven that light speed is anisotropic relative to themselves using a Sagnac interferometer, they might also expect the eﬀect to be detectable using a Michelson interferometer. As emphasized by Klauber [48], two independent experiments using Michelsontype instruments have recently been conducted. In both cases the interferometers were sensitive enough to detect such a difference in path length if it were caused by the rotation of the

6.5 Local, Global Agreement

It is important to make a distinction between a device like the

Michelson interferometer and the rotating body as a whole.

The length contraction implied by the null result may clearly

aﬀect small bodies loosely attached to the rotating circumfer-

ence. But these eﬀects are independent of what happens to the

circumference as a whole. The latter question has also been

a lively matter of debate. Aside from any “centrifugal” ef-

fects caused by the rotational acceleration, the rotating body’s

outer portions will also be subject to “relativistic” stresses due

solely to velocity. We need not enter into that debate. It suf-

ﬁces for our purposes to regard the Michelson interferome-

ter much as a pair of small crossed measuring rods laid out

along the circumference. We are concerned with measure-

Pments of the circumference made by observers in K, which
is r. And with measurements of the circumference made

by observers wpho rotate. Evidently, they will measure it to be

I I lponger by =

v2=c2 because their rods are shorter by

I v2=c2.

With this in mind, consider again light rays completing

circuits around the disk in opposite directions. Or better, con-

sider a single ray that completes one circuit in one direction

and is then reﬂected back to return to its source in the opposite

direction. We can rearange Equation 10 (whose time refers to

a rotating clock) to give the average speed of the light ray:

L
hti45

a

c pI

v2=c2 :

(11)

But L is the circumference measured by lab rods. The evi-

dence discussed above indicates that the circumference

measured with rotating rods comes out longer than that mea-

sured by lab rods:

L

al p

:

I v2=c2

(12)

Therefore, rotating observers would ﬁnd the average back and forth speed

l
hti45

a

c45

a

c:

(13)

The Michelson interferometer doesn’t tell us the speed of light. It tells us that the length / time ratio is the same no mat-

ter which direction the apparatus is oriented. That’s the nature

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of a “null” result. By contrast, Equation 13 refers to actual speed measurements, made with rulers and a clock. Furthermore, it need not be a back and forth average, but yields also deﬁnite one-way speeds. Bearing in mind the relationship between L, T , l, and t, the average one-way speeds corresponding to the back and forth average are given by

a I l

c v

t4

v2=c2

nd

l
t5

a

I

c

Cv
v2=c2

:

(14)

Time in a Rotating Frame: in Sync or out of Sync?

Einstein

Center-Flash

“Synchronization” Synchronization

(impractical) (navigator-friendly)

The speeds are faster than those measured in K because the

path lengths are judged to be longer and the travel times

shorter.

These speed measurements break down to a length mea-

surement and a time measurement. The length is measured

either with many rods laid end to end or with one rod placed

3

multiple times from one position to the next. Whereas the time (between emission and reception) is measured with only

2

one clock at one location. It would clearly be useful for a resi-

dent of the rotating body to synchronize all clocks around the

rim so that any given pair of adjacent clocks would yield the

same results. This is simply achieved by synchronizing all

clocks by a ﬂash of light from the rotation axis. In this way

the local (clock-pair) speed measurements become consistent

with the global (one-clock) speed measurements.

5 4 3 2 1

6.6 Time Gap
This is where we ﬁnd some controversy. Although the synchronization procedure just mentioned is so reasonable and practical that it is, in eﬀect, the same as that actually used by the GPS, opinions vary as to how to “regard” the end result. Many authors prefer to think of the clocks synchronized this way as actually being desynchronized. Why? Short answer: Einstein and his second postulate again. Long answer:
The Einstein method of synchronization is supposed to work for any two closely spaced clocks that are at rest with respect to each other. If the speed of light with respect to such clocks is actually equal to c in both directions, then the method does of course work. What’s more, even if the speed of light is actually anisotropic you can get away with supposing (or stipulating) that the method works if the light path doesn’t close on itself. In other words, the method works if you send a synchronization signal and merely assume that the time at the receiving clock is r=c as in our original grid. What happens if we adopt the same procedure for a circumferential segment of our rotating body? It becomes a problem when you ﬁnd that, as you work your way around from one segment to the next, upon returning to the starting clock, the times don’t agree. For a straight line light path you can never ﬁnd out if the time for the synchronization signal to arrive is really l=c or not. But on a rotating body you can’t avoid discovering that when the signal returns to its starting point
% it’s oﬀ by lv=c2. The last clock in the chain is behind or

Fig. 8: Tubular spacetime diagram for rotation: World line is green helix. Using Einstein synchronization, the line of simultaneity is also a helix; it does not close on itself and so yields two diﬀerent times for a point in spacetime that should have just one time. Centerﬂash synchronization does not have such problems.
ahead by about this much, depending on whether the procedure is conducted with or against the direction of the rotation. Taking the average, as judged by rotating observers using rotating clocks and rods, the diﬀerence is exactly lv=c2. Using lab clocks and rods the average diﬀerence is one half of Equa-
@I A tion 4: Lv=c2 v2=c2 . This is known as the “time gap”
(or time lag). (See Figure 8.)
Since a synchronization time gap is totally impractical and it’s so easy to eliminate it by the center-ﬂash synchronization method, why isn’t it simply accepted that this latter method is the way to synchronize clocks on a rotating body? The “problem” is that by using center-ﬂash synchronized clocks, the speed of light comes out being anisotropic and not equal to c. Some people don’t like that, or at least they seem to not like it, because they call the clocks desynchronized and insist that the speed of light is still locally equal to c. (See Appendix A.) This approach strikes me as making the postulated constancy and isotropy of light speed such lofty inviolable absolutes, that any real world measurement that fails to maintain their sanctity implicitly becomes less worthy, no matter how practical it may be.

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6.7 Digression: Map vs. Territory
Before following up on this point, it will be useful to reconsider the “fact vs abstraction” discussion of §2. For reasons that need not presently concern us, I was once given the following advice by a well known astrophysicist: “You must believe Newton’s equations because your life depends on them every time you cross a bridge or ﬂy in an airplane.” Is it really so? With a little reﬂection one sees that the answer is no. A logical error like the one committed here is sometimes called, the fallacy of misplaced concreteness. Since powered ﬂight (birds) and bridges were around long before Newton was born, we see that the confusion is between the physical properties that make these things possible and the abstract laws used to express their interrelationships. Just as we don’t eat restaurant menus, we don’t ﬂy in or tread over the equations. Due to repeated use and a reputation for reliability, the equations have come to be regarded as being real (concrete) as concrete, real as aluminum or feathers or muscle. But they’re not. When the law becomes as real as the physical reality, insensibly we forget that our job is to twist theories to suit facts; not to twist facts to suit theories. It bears repeating: The abstract law is that the speed of light is c for everybody; the physical fact is that it is not.
6.8 Dramatization by Ampliﬁcation
To more dramatically illustrate the reality of the Sagnac effect, imagine that the disk is much larger and rotates much faster than the Earth. Suppose then, at a given moment, we send a green beam in the direction of rotation and an orange beam against the direction of the rotation. It is not hard to imagine that further details of the setup could be arranged so that anybody – even a child – could see the time gap with their own eyes and with their own sense of time. The orange beam returns to the observer before the green beam. In our actual experience the eﬀects require sensitive instruments to detect. In this scaled-up example we have only, so to speak, ampliﬁed the eﬀects to the level of gross human perception. The anisotropic light speed and the time gap then become starkly evident physical properties.
6.9 Einstein and Others on Rotation
Now let’s return to our disk, whose physical and abstract properties, we’ve begun to see, have sometimes been confused. Going back to the source helps to clarify our perspective. Einstein wrote:
Thus on our circular disk, or to make the case more general, in every gravitational ﬁeld, a clock will go more quickly or less quickly, according to the position in which the clock is situated (at rest). For this reason it is not possible to obtain a reasonable deﬁnition of time with the aid of

clocks which are arranged at rest with respect to the body of reference. A similar diﬃculty presents itself when we attempt to apply our earlier deﬁnition of simultaneity in such a case, but I do not wish to go any further into this question. [52]
One might quibble with exactly what Einstein meant by “not possible,” “reasonable,” and “diﬃculty.” But the truth is that physicists, engineers and technicians have succeeded in doing the “not possible.” They have built a system around our massive rotating Earth in which time and position are quite reasonably deﬁned. The end product provides a physically robust spacetime grid on and near Earth’s surface that is accurate down to an extremely ﬁne scale. Note further that improvements to the system are limited only by our technological ability, not by the abstract theoretical “diﬃculties” Einstein was referring to.
And yet, as late as the 1960-70’s one could still ﬁnd in the literature statements to the eﬀect that synchronizing clocks on a rotating body is “impossible.” The most extreme “impossibility statement” I know of is from Zeldovich and Novikov, who wrote not only that “it is impossible to synchronize time on a rotating body,” but that “the very concept of synchronization does not exist.” [53] (Some stupendously misplaced concrete here, we observe.)
Other authors who have at least allowed rotating observers to conceive the idea of synchronizing their clocks include Born [54], Landau & Lifschitz [55] and Rosser [56]. Still these authors, following Einstein, have claimed it is not possible to actually do so. The “problems” for these authors are essentially the same as those mentioned by Einstein: the rate diﬀerences due to radial distance diﬀerences and the time gap resulting from the Einstein synchronization method. It turns out that the rate diﬀerence problem is actually quite easy to solve. As we will see in the next section, the solution has been implemented in practice in the Global Positioning System. The time gap “problem” is even more of an imagined problem than a real problem. It arises if one insists on using Einstein’s synchronization method so as to maintain that the speed of light will equal c with respect to an adjacent pair of clocks on the circumference.
Note that the time gap has two diﬀerent manifestations: As stated above, the gap can be measured using one clock to ﬁnd the diﬀerence in times taken for co-rotating or counterrotating light beams to return to their source. This doesn’t cause any real problems and it is consistent with the centerﬂash synchronization method for any two adjacent clocks. The problematic manifestation is a result of Einstein synchronization in which case the gap would be the diﬀerence in time setting that one clock would need to have with itself.
I think it is safe to assume that one of the reasons Einstein did “not wish to go any further into this question,” is that he was loathe to contemplate a scheme in which light speed did

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not equal c, and yet he, too, found the time gap to be objectionable (awkward?). Many authors nevertheless advocate Einstein-synchrony because if one ignores the need to synchronize all the way around the circumference, it allows one to maintain that light speed is always locally equal to c. This is sometimes thought of in terms of imaginary “Locally Comoving Inertial Frames” (LCIF’s). These are essentially the inertial frames attached to clocks released from rotation onto inﬁnitesimal tangents. At the moment of release, one such clock suddenly ﬁnds itself in a bona ﬁde inertial frame so that observers in it are supposed to synchronize their clocks according to the Einstein prescription, and thereby arrange it so that light speed suddenly becomes isotropically equal to c. One of the questionable aspects of this approach is the sheer multitude of frames that must be invoked in the course of working one’s way around the rotating disk.
Superﬁcially, this sidesteps the time gap problem, but it is much more complicated than a simple center-ﬂash procedure and it denies rotating observers a global set of synchronized clocks. Is it really so important to have the speed of light “locally” equal to c? Does the possibility of implementing this procedure mean that the speed of light actually does locally equal c? Or are the stipulators further enabling the fallacy of misplaced concreteness? That is, are Einstein’s postulates and his interpretation of the Lorentz transformations taking undue precedence over the physical fact that, at least globally, the speed of light with respect to rotating observers clearly does not equal c? If the speed does not equal c in the global (general, non-inertial) case, then why insist that it does equal c in the local (special, inertial) case? The soundness of the standard logic is not at all obvious.
Having similar misgivings as my own concerning this “traditional (LCIF) approach” to the Sagnac eﬀect, R. D. Klauber has commented:
The plethora of possible settings for the same clock in a rotating frame results from insisting on “desynchronization” of clocks in order to keep the (one way) speed of light locally c everywhere. And thus, one is in the position of choosing whichever value for time one needs in a given experiment in order to get the answer one insists one must have (i.e., invariant, isotropic local light speed). One can only then ask if this is really physics or not. [57]
It appears we have three choices: 1) You can arrange to have isotropic and constant light speed with respect to adjacent clocks if you’re willing to accept an objectionable time gap. 2) You can “avoid” the time gap and arrange to have isotropic and constant light speed with respect to adjacent clocks by invoking a multitude of local inertial frames, which denies the possibility of a set of globally synchronized clocks. Or 3) you can simply accept the measurement made using one clock indicating anisotropic non-c light speed with no time

gap – which is the same as measurements made locally when clocks are center-ﬂash synchronized.
Clocks in the GPS are synchronized by the latter approach. It is practical and reasonable. What is not practical nor reasonable is the set of desynchronized clocks that you get by following Einstein’s prescription – whether the ambiguous local only variation or the unambiguous but troublesome time gap variation. The “impossible to synchronize” stance is thus clearly not that of a sailor or pilot or detective who would persist at solving the problem till he found a scheme that works. It would seem rather to be that of a stubborn “philosopher” who would prefer not to disrupt his foundational precepts.

6.10 Interlude and Assessment
In Appendix A and B you’ll ﬁnd further examples from the literature that reinforce the above impressions on the Sagnac eﬀect and the problem of rotation in SR. In the next subsection an imagined scenario is presented whose purpose is to illustrate one ﬁnal point concerning all this. The point is the limited nature of SR, especially with regard to its applicability to the problems raised by the phenomenon of rotation. Ultimately, a more comprehensive theory is needed to satisfactorily answer the questions that rotation draws attention to. (On this point, Einstein and I agree.) I would recommend reading §6.11 and §6.12 so as to fully absorb this point. But readers eager to see where I’m going with all this may wish to skip ahead to Sections 7 and 8.
In any case, in the interest of keeping our bearings straight, the reader is reminded here that we are not out to “refute” SR. The theory is mathematically consistent and I make no pretense of falsifying it with my arguments. The main purpose has been rather to show that the empirical facts that are often associated with SR need not be looked at through orthodox Special Relativistic glasses. It is never too late to question the cornerstones of the prevailing theoretical ediﬁce – especially if those cornerstones are more abstract than physical. And especially if their apparent validity depends on them keeping to their local corners.

6.11 Hypothetical Dilemma

Imagine an astronomically huge circular ring that rotates with

respect to the axis perpendicular to its center with the speed

a r!

1 2

c.

Suppose

the

ring

is

inhabited

by

intelligent

beings

who have synchronized their clocks by a ﬂash from the center.

P Q IH % ¢ The ring’s radius is r :

15 meters, which makes its

centripetal acceleration an Earthlike 9.8 m / s2. Now suppose

the inhabitants need to take an expedition to quell a distant

emergency – a problem that requires not only urgency, but a

lot of personnel.

I The plan
line directed

is to

to launch 360 ships onto the the distant trouble. At every

saimnteertvanalgeanlotinagl

the outer periphery, a ship is prepared with crew and two

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clocks, whose purpose will be speciﬁed momentarily. The

ﬁrst ship to be launched is designated #1, the last, #360, with

corresponding numbers in between.

For many many years events on the ring have been co-

ordinated and recorded on the basis of the center-ﬂash clock

synchronization scheme. For this is the only way the inhab-

itants could keep a continuous (no time gap) set of clocks

which both individually and in coordination with one or more

additional clocks, would reﬂect the physical fact that the av-

erage speed of light signals relayed around the circumference
@ C A @ A % % is c r! against the direction of rotation and c r!

with the direction of rotation. The proportions of the system

are so extreme, that, in the fast direction, a signal would re-
% I turn to its starting point in year and in the slow direction Q% the signal would return in years.

Note: Although length contraction and time dilation

would be important eﬀects to include for certain contexts, for

ours they would play a small role. In the interest of simplic-

ity, we therefore sacriﬁce some accuracy by omitting these

I secoTnhdeord esrpeaﬀtieacl tisnftreorvmaltshecoforrlelospwoinndg

discussion. to propagation

times

I Q % % day or days, depending on direction. Following the

initial launch of ship #1, over the course of three years, each

ship is released onto the tangent.

As to the reason for having two clocks on each ship: One

of the hoped for scientiﬁc byproducts of the voyage is that it

might help determine which of two hypotheses is more ac-

curate with regard to light propagation speeds. As per the

recommendation of Dr. Sellauber, one clock on each ship is

to remain continuous with the time it kept while back on

the ring. Using these unadjusted original clocks, Sellauber

has argued that it may be best to maintain the impression of

anisotropic light propagation from one ship to the next. These

times would thus still be measured as

t4 % c L v

nd

t5

%

c

L
C

v

;

(15)

P where L % % r, v r!, t4 is the time light would take to

go in the forward direction, from ship #360 to ship #1, and t5

is the time light would take to go in the backward direction,

from ship #1 to ship #360.

As per the recommendation of Dr. Ashstein, on the other

hand, each ship should also be equipped with a clock that re-

ﬂects the fact that the tangential train of ships now comprises

an inertial system. Being in uniform motion, the ﬂeet should

regard light propagation as being isotropic. Using ship #1 as

the Master clock, the second clock on all of the trailing ships

is accordingly reset. Only on ship #1 do both clocks show the

same time, t1. The propagation times can be expressed as in Eq. (15), except that the velocity, v drops out, so the time is

supposed to be the same in either direction.

If n is the number of ships along the line of communica-

tion and l is the distance between any two adjacent ships, then

the propagation time (in either direction) would be given by

t(n

a 1)

l@n
c

IA :

(16)

If n is thought of as the number of ships connecting a given ship with ship #1, the initial synchronization signal propagat-
C @ IA ing backwards from ship #1 yields the time t1 l n =c.
Each second clock (except for the one on ship #1) is thus reset to show this time and so abide by the idea of stipulated isotropic light propagation. As we should expect, this procedure results in a discrepancy with the original clocks:

¡t(n

% @lv n

1)

c2

IA

(17)

This amounts to about 18 hours between any two adjacent clocks along the train. So for ship #360, the diﬀerence will
Q R be about = of a year.
Now here’s the problem: An engineer on board ship #180 has discovered quite conclusively that a design ﬂaw in the Freem Thruster Containment Module (FTCM) will cause a fatal malfunction in every ship unless it is corrected. The correction would only take a few minutes; the problem is informing everyone about it. To make matters worse, the transmission antenna of ship #180, which needs to be focussed in one direction at a time in order to maximize signal strength, has been badly damaged by cosmic debris collisions. It is only good for one more transmission. Unfortunately, for budgetary reasons, only ships #1, #180 and #360 were equipped with transmitters powerful enough to reach the whole train. Signals from other ships’ transmitters dissipate too rapidly to remain coherent to receivers beyond the nearest adjacent ships. This rules out the possibility of having ship #181, e.g., transmit a signal forward to ship #179 for a relay up to ship #1. The time of FTCM failure was predicted with 99.99% conﬁ-
¦S dence to be 180.5 days ( minutes) as given by the ﬂeet’s
original clocks. Given the above equipment limitations, it is therefore impossible to warn the whole ﬂeet in time. Damage control will require some hard decisions.
Recall that the mission is of some urgency in the ﬁrst place, but even more important is that enough personnel arrive to accomplish the goal. If more than half the ﬂeet were to perish, the mission would become increasingly diﬃcult with each additional lost ship. If the speed of light is now really isotropic, as suggested by the second set of clocks, then, whether a signal is sent forward or backward, it would reach only 2/3 of the ships in that direction (i.e., 1/3 of the total ﬂeet) before the rest were obliterated by Freem radiation. Sending warning signals forward would thus save some time, but leave alive at most 1/3 of the ﬂeet.
According to Sellauber the situation might not be quite so dire. According to his hypothesis, light propagation is not likely to be any more isotropic now than it was back on the ring. Resetting the clocks didn’t change that. If he is right, then sending the signal forward would probably only save 1/3 of the forward ships (i.e., only 1/6 of the ﬂeet). But sending

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the signal backward would probably save the whole rear half. The mission would be delayed a bit, but they’d have ample personnel to get the job done. Sellauber is compelled to point out a caveat (implied by the use of words, “likely,” “might” and “probably,” above): There is no guarantee that light propagation was actually isotropic with respect to their original rotation axis. (That’s why the one-way speeds given above were stated as averages.) If light propagation was anisotropic even with respect to their original rotation axis, the present dilemma could possibly be diminished or exacerbated, depending on the direction. But since both the magnitude and the direction of this possible anisotropy is unknown, its eﬀect on the present situation would be random and so should not inﬂuence their decision.
Though not conclusive, another clue is pointed out by Sellauber, which nevertheless implies that the anisotropy found as they rotated on their ring would persist in the same sense along their tangential train. While attached to the rotating ring, the Cosmic Background Radiation (CBR) revealed a spectral red / blue dipole whose magnitude remained constant, but whose direction changed so that the blueshift was always on a tangent in their direction of rotation. Correspondingly, with respect to the axis of rotation, there was no dipole at all. After being launched onto one of these tangents, the dipole has remained constant (blue) in the direction of their motion.
Since we are contemplating this whole scenario, essentially, from the point of view of rest with respect to the original rotation axis (from which) the red / blue shifts reveal no dipole) and we have followed every important detail, it should be obvious that sending the signal backward is the right choice. From the point of view of the crew of ship #180, the decision is similarly based on this remembrance of the motional history of the ﬂeet and the consistency with the CBR red / blue shifts. But suppose all record of this history, in their minds and data records, was somehow wiped out so that they don’t even know they are “heading towards” their troubled destination. Suppose further that their view of the microwave background radiation has been obscured. They would thus ﬁnd themselves in a string of ships whose direction in indiscernible. Having no clues as to their state of motion (aside from that it is uniform) they reasonably assume that they are at rest. In this case they would certainly be forgiven for setting their clocks based on the assumption that light propagates isotropically.
If the bad news from the engineer on ship #180 was discovered under these circumstances, they could hardly do better than to ﬂip a coin to decide which direction to transmit the warning. But with an unobscured view of the cosmos and a non-amnesiatic consciousness that considers history and any other possible clue, a better than random decision can be made. With so many lives at stake, it would surely be an act of horrible negligence to decide by ﬂipping a coin or to deliberately send the signal forward, when experience and awareness

of one’s surroundings provide such strong evidence that more lives would be saved by sending the signal backward. This scenario thus implies not only that light propagation cannot in general really be isotropic with respect to uniformly moving bodies, but that it may be possible, in some circumstances to make a reasoned guess as to the direction of the anisotropy. Dogmatically insisting that light propagation must be isotropic with respect to oneself or other uniformly moving bodies (IRF’s) just because one such frame is “related” to all others by the Lorentz transformations, could cause fatal damage.
6.12 What Does it All Mean?
Any statement claiming or implying that light propagates with a constant speed equal to c, if it does not at the same time point out that this is just an average back and forth speed, is plainly false in most every case. If correctly determining the direction of light’s anisotropic axis becomes a matter of life and death, any right-thinking physicist will surely consider all clues, not only in accelerating reference frames, but in uniformly moving frames as well.
If, after being released onto a tangent, you choose to deﬁne away the anisotropy, if, while moving uniformly you ignore any clues gathered from your history and your global, non-local surroundings, if you still choose to deﬁne light propagation as being isotropic, it may cost your life. Does light go faster toward the east or west; north or south; up or down? How does the CBR ﬁt into the picture? How might huge concentrations of matter aﬀect light propagation? One had better think clearly and thoroughly examine all the evidence because the Universe cares not a whit about petty human stipulations nor “elegant” geometrical principles.
Similarly, any claim or implication that the relativity of simultaneity provides some kind of profound insight into the nature of space and time which neglects to clearly point out that this abstract idea derives from the physical fact of anisotropic light propagation is also sorely lacking in truth. The purpose of the scenario in the previous subsection and this section as a whole, is to distinguish as clearly as possible what we know from what we don’t know, and to emphasize that stipulations are a poor substitute for knowledge – especially if we tend to forget the diﬀerence.
The domain of SR is not broad enough to be of any use for solving the above dilemma. Without bringing in clues from cosmology, no certainty can be attached to the ﬂeet crew’s decision to send the signal backward. One needs only to point out that the tangent-traveling ﬂeet could well be at rest with respect to the rotation axis of another huge ring, a mirror image of their own, we might say, upon which clocks were also synchronized by the center-ﬂash method. Launching a tangential ﬂeet from this second ring back “toward” the original ring would put these tangentially launched (Ring #2) ships at rest with respect to the original ring. Is light propagation isotropic with respect to this train of ships or with respect to

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the original train heading in the opposite direction? If it’s possible to ﬁnd out, it would only be because our Universe is much more complicated than Minkowski space.
It is an intriguing theoretical fact that the phenomenon of rotation appears as a sort of stepping stone or guide post. Rotation seems to deﬁne a zone beyond which SR does not really go without causing suspicious awkwardness or raising questions whose answers require a more comprehensive theory. It is well known that Einstein had a similar regard for rotation, as it helped him to conceive GR. As we recall from §5, Einstein’s conception of gravity, which included the idea of static gravitational ﬁelds, inspired him to regard any observer, even those in non-inertial systems, as being in a state of rest. In later sections we will take the opposite approach. We will adopt the rotation analogy as inspiration for a model of gravity in which motion plays the more signiﬁcant role. For now, the most important thing to bear in mind is that the simpliﬁcation of theoretical principles brought about by accepting the symmetric, isotropic conceptions of SR comes at the cost of obscuring asymmetries and anisotropies that are likely to exist – and in the case of rotation have been proven to exist – in the real physical world.
7 Simpliﬁed GPS
The conﬂict between Einstein synchronization and the desire to have a continuous deﬁnition of time around the circumference is just one of the reasons given (or implied) for the “impossibility” of ﬁnding a “reasonable deﬁnition of time” on a rotating body. Another reason is that the frequency of a rotating clock depends on its distance, r, from the axis. Even though a given axially concentric circle of clocks can be synchronized by the center-ﬂash method, they will not tick at the same rate as other circles of clocks and so would quickly grow ever more out of synch with clocks at diﬀerent radii.
The solution to this problem is called “syntonization” – adjustment which makes the frequency of one oscillator the same as another. Knowing how rotation, with its varying rdependent speeds, aﬀects the rates of identical clocks, it becomes a simple matter, in principle (and these days not even too diﬃcult in practice) to adjust clocks’ frequencies so they are all the same – even though their “natural” frequencies would be diﬀerent.
In the GPS syntonization comes into play not only for the eﬀect of orbital motion and rotation, but also for the eﬀect of gravity. Instead of immediately combining these eﬀects, let’s ﬁrst consider gravity by itself. Accordingly, let’s imagine a massive non-rotating sphere centered on the grid we started with in K. Imagine as well, that the sphere has an array of very tall vertical poles planted on the surface. By “very tall” we mean extending to some astronomical distance. At each one of many regular intervals along the poles we have mounted an accelerometer, a clock, a trio of measuring rods

INSTRUMENT STATION
0
ACCELERATION

Fig. 9: The essential basics.

– mounted as cartesian axes – as well as devices to send and receive light signals. We will refer to such places as “Instrument Stations,” (IS’s). (See Figure 9.) In practice, of course, such assemblies of instruments are carried on board satellites. But orbital motion adds a level of complexity which for the moment we’ll neglect. The GPS comprises 24 satellites orbit-
a R ing at a distance of about four Earth surface radii r R. So a R let’s imagine 24 poles whose IS’s include one each at r R.
As predicted by GR and as data from our poles would conﬁrm, even clocks that appear not to be moving, have rates that depend on distance from the center of the sphere. Following is a scheme for completing the setup of a simpliﬁed GPS on our idealized sphere. First we’ll syntonize, then synchronize, then syntonize again.
Syntonization has been achieved in the real GPS by either equipping the satellite clocks with frequency synthesizers that can be controlled from the ground, or by pre-programming them to run at a rate that would compensate for the eﬀects of gravity and orbital motion. The rate that would need to be compensated for gravity alone would be

r
fr a I

PGM ;

fI

rc2

(18)

where fI is the rate of a clock at inﬁnity, G is Newton’s
constant and M is the sphere’s mass.
Since most inhabitants of the sphere live on or near its surface, it is deemed most practical to have the rates of all other clocks adjusted to match those located on the surface. This idea comes directly from the real GPS design. Due to deviations from sphericity, slightly eccentric satellite orbits, etc., implementing the idea is not as simple in practice as it is for our idealized sphere. But the principle is the same. Let us therefore imagine that the clocks on our tall poles – especially

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a R those at r R – have been pre-programmed to tick at the

rate which matches the surface clocks. This completes the

ﬁrst syntonization.

Synchronizing the GPS clocks, in practice, involves calcu-

lating and compensating for rotation and orbital motion. But

the idea is to end up with an array of clocks whose settings are

the same as what you’d get by the center-ﬂash method. As we

recall from §3, this is essentially the same as the method de-

scribed by Taylor and Wheeler. [9] Imagine that this is

achieved for our sphere with an actual center-ﬂash, as though
a H there were no matter obstructing the signal from r . As-

suming that the radial distance r (as that appearing in Eq. 18)

is an accurate measure of distance, we’d want all clocks to be
a pre-set to the time t r=c and to start ticking as soon as the

ﬂash arrives.

Various factors make the real world syntonization / synch-

ronization scheme a little more complicated than the above.

Only one of these factors is of fundamental importance for

our purpose. That is the fact that the r in Eq. 18 and that used

to compute the propagation time, refers to a “coordinate” dis-

tance, not a “proper” distance. The importance of the distinc-

tion and the magnitude of the diﬀerence will be discussed in

more detail later. For now, note that in the Earth’s GPS the

diﬀerence is so small as to involve a ranging error of only a

few millimeters.

The last major part of the System is the user, i.e., receiver-

bearing observers who want to know where they are, what

time it is and how fast (and in what direction) they’re going.

For details on how all these parts function so as to give the

observers what they want, consult [58]. Here it will be suf-

ﬁcient to say, ﬁrst, that the clocks built into most receivers

are less accurate than the clocks on the satellites (poles). The

System’s accuracy depends on the fact that every location on

the sphere is in view of at least four IS’s (satellites). With tim-

ing and ranging signals from four diﬀerent directions, by us-

ing a sophisticated calculation algorithm, a receiver can have

its clock adjusted to the correct GPS-time and have its three

dimensional location determined to within a few meters, cen-

timeters or millimeters, depending (crudely speaking) on how

expensive the unit is.

Assuming that the syntonization needed for gravity alone

has been achieved, having begun with a non-rotating sphere,

we see that it is simple, in principle, to set up a system of

synchronized clocks. Replicating the actual GPS requires one

more step: to set everything in motion. After the center-ﬂash

signal has been received Synchronization would require no

further adjustment. But extra syntonization factors will be

needed to account for the rate change arising from change in

motional state. So let’s imagine that the extra factors needed

for rotation and orbiting are designed to kick in at a given

moment when the “planet” begins to rotate and all the IS’s

a R at r

R are launched onto circular orbits. The result

would be, in essence, what we’ve got on and around our

planet Earth.

So as to make the sequel logically consistent, let us now go back to the state of our sphere-with-poles prior to rotation and prior to syntonization. In other words, we’re back to our non-rotating sphere whose array of Instrument Stations contains clocks that now all tick at their natural frequencies.
8 Gravity Virgins
Does the totality of the observable interactions compel us to adopt [the] standard interpretation, or might the same pattern of experiences be explainable within some other, possibly quite different, conceptual framework?
– Kevin Brown [59]
The foregoing contains many clues. Those who are satisﬁed with the status quo may well ask, clues to what? What’s the mystery? But those who feel some discontent with our present state of knowledge of space, time, matter and gravity, will be eager to try looking at it all from a new angle. To make it easier for the mind to accommodate that angle, the idea is presented as the experience of “space explorers” who come to discover the eﬀects of gravity for the ﬁrst time. How else could we be assured of suﬃciently minimizing our accumulated metaphysical baggage so as to get a truly fresh look at the problem?
8.1 A Diﬀerent World
Imagine a civilization advanced enough for space travel, yet ignorant of gravity. This civilization has evolved far from any large masses – perhaps as an experiment conducted by an even more advanced civilization – living inside a huge selfsustaining Rotating Cylindrical space station. Let’s call members of this civilization, RC’s. The RC’s “ground” is the inner wall of the Cylinder. They have conducted many experiments involving light, clocks and motion. Motion is “sacred” to the RC’s because they realize that if their Cylinder did not rotate they would not survive.
The two devices that have most beneﬁtted the RC’s concepts of motion are accelerometers and clocks. While spending most of their time happily pressed against the Cylinder’s inner wall, they have also thoroughly explored the Cylinder’s immense bracing, which spans across the rotation axis, where the acceleration and velocity are both zero. The RC’s have conducted Sagnac-like experiments with light and understand that clocks on the inner wall are slowed while those at rest with respect to the axis have a maximum rate. They understand that it’s not acceleration, but velocity that causes clocks to slow down.

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Coordinate time - Proper time Projection: Minkowski Spacetime Epstein “Speedometer”
EPSTEIN
MINKOWSKI
Movement through space, movement through time: Pythagorean breakdown
.8 .5
.6 .866 c
Fig. 10: Space-propertime diagrams, adapted from L. C. Epstein, Relativity Visualized. 8.2 Odometer Having full access to the rotation axis, both inside and (its imaginary extension) outside their cylinder, the RC’s have even developed a “mileage” tracking system based on the effect velocity has on clock rate. The rotation axis has proved itself to be a most reliable standard of rest. Thus any movement with respect to a stationary point on the axis results in a slowed clock and a correspondingly reduced elapsed time.
The RC’s basic odometer relation is

is perfectly reliable for simple situations, such as any point

attached to the Cylinder’s bracing or wall. For more compli-

cated motion paths the RC’s have developed more sophisti-

cated equations and algorithms that are built into all vehicles.

With very sensitive gyroscopes, accelerometers and clocks,

the RC’s have devised a tracking system giving them accurate

measurements of distance travelled with respect to a point on

the axis based entirely on the elapsed time of a moving clock.

The RC’s and their vehicles usually carry a second clock that

can be reset to match the local time. But the factory installed

original clock is left untouched so that “total mileage” is a

readily available datum.

How diﬀerent is this world from ours, with its clearly de-

ﬁned standard of rest, where accelerometers and clocks are

absolutely reliable indicators of motion! To the RC’s it is a

fact of everyday life that motion is absolute. For example,

they have no reason to doubt the implications borne of their

Sagnac-like experiments, that the speed of light is isotropic

only with respect to the rotation axis. This is the basis of

their odometer relation and their general reckoning. Their

kinematic accounting system is thus quite like the -Lorentz

a transformations discussed in §4. For the RC’s

1 2

only

for

the rotation axis and for all moving bodies the

1 2

values

depend on velocity, seemingly absolute velocity with respect

to the axis. Therefore, when they take an excursion from the

Cylinder, either by tangential launch or by rockets, it would

never occur to the RC’s to “resynchronize” their clocks for

the purpose of making the speed of light equal c. For uni-

form motion their excursions can be charted essentially as in

Figure 4. For excursions involving more complicated maneu-

vers, the eﬀects can be charted as a series of similar ﬁgures,

each one having the appropriate  value. The RC’s know well

that the back and forth speed of light – when measured with

their co-moving rods and clocks – is always equal to c and

that it is therefore a meaningful universal constant. But the

one-way speed is generally equal to something else.

Traveling in the neighborhood of their Cylinder, the RC’s

not only measure the speed of light to be anisotropic, they also

ﬁnd lengths in the rest frame to be “expanded” and clocks in

the rest frame to be ticking faster than their own. This is all

to be expected and is perfectly consistent with empirical re-

sults of Einsteinian SR. The superﬁcial diﬀerences are only

to do with the diﬀerent synchrony parameter values. Though

the RC’s may be able to appreciate the computational con-

a venience of Einstein’s preferred “

1 2

for

everybody”

ap-

proach, it is so far removed from their physical experience

that the idea is only rarely discussed – such discussions being

found mostly in obscure philosophical journals.

q
a L c T2 T 2 ;

(19) 8.3 Speedometer

where L is the distance travelled, T is the time given by an The RC’s basic odometer – the model used to represent mo-
axis clock and T is the time given by a clock that moves dur- tion of the various parts of their rotating cylinder – may be
ing the same T-time with a constant speed. This equation simply depicted graphically. Not surprisingly, the odometer

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Schematic of “Odometer Principle” for Uniformly Rotating Body

scheme (Figure 11):
1. All curved paths drawn on the tubes are as “speedometer needles” whose identical lengths represent the same distance through coordinate time: one half turn around the Cylinder.

Fig. 11: Identical length worldlines, showing that diﬀerent radial distances in the same rotating system travel proportionally diﬀerent distances through space and time. Compared to the time given by a clock on the axis, the smaller the elapsed proper time, the greater the distance traveled through space. (Equation 19.)

WH is directly linked
Epstein calls his

to

a

“speedometer.” In any case, that’s what projection of the usual Minkowski dia-

gram. (He also calls it a “space-propertime diagram, as dis-

tinct from the usual Minkowskian space-coordinatetime dia-

gram.) [60] Figure 10A shows how the two diﬀerent projec-

RS tions of the diagram are related to each other.
this diagram a light ray is represented not by a

Aclcionredbinugt

to by

the space-axis (abscissa). And the ordinate represents not co-

ordinate time but proper time. The range of the speedometer

“needle” is thus a quarter circle whose height is a coordinate

time unit (e.g., second) and whose width is the coordinate

distance traveled in that time.

In terms of this diagram, a light ray thus travels entirely through space (zero proper time). Whereas an observer at rest (the RC’s axis) travels entirely through time. For this latter extreme, coordinate time equals proper time. Whereas, for all speeds in between, the Pythagorean Theorem gives an easy decomposition in which the proper time is always less than the coordinate time when some portion of the total “speed” is diverted through space. Figure 10B shows two representative uniform speeds.

For the cyclic motion of the RC’s world, the whole diagram is most intuitively depicted by rolling the individual diagrams corresponding to particular speeds into a set of nested tubes (Figure 11). As we see, the variation in “speed through time” is clearly related to the variation in “speed through space.” Let’s itemize a few other key points arising in this

2. Though not explicitly shown in the ﬁgure, it nevertheless follows from the RC’s practical center-ﬂash synchronization method, that all clocks on a given circle (tube) are synchronized with one another. Quite possibly the RC’s would want to have a second set of clocks on each circle whose rates would all be syntonized with clocks on one particular circle – perhaps that of the cylinder wall, where they usually live, or perhaps that of the axis, or perhaps all three. In any case, they would certainly want to have a range of clock sets that reﬂect the natural unsyntonized rate diﬀerences from axis to wall.
3. Finally, note that the tubes in the ﬁgure represent a passive backdrop; the blank map of empty spacetime. As far as the RC’s are aware, there is a rather sharp discontinuity where empty space meets rotating (or even non-rotating) gross matter. Whether we depict only one or many clocks per circular cross-section, the corresponding helical lines represent movement of these clocks
a through spacetime. The extreme cases, r! H a and r! c correspond to the maximum or
zero passage of proper time and zero or maximum speed through space, respectively. But in both cases, it is either a material object (clock) or a light ray (non-clock) that moves. It is not spacetime that moves. Motion is conceived as the change of position of something with respect to the passive backdrop of space, whose coordinates are typically based on one designated material body (e.g., the Rotating Cylinder) or a plurality of bodies (e.g., the “ﬁxed” stars).
Item number 3 above might have been abbreviated by stating simply that tubular motion models are conducive to illustrating the diﬀerence between motion through space as opposed to motion of space. Motion of space is not part of standard physics; it is mentioned here as a hint of an idea that the RC’s will make use of later. As noted above, the nested tubular “odometer” of Figure 11 is directly applicable only to the motion of diﬀerent radii of their Cylinder. The spacetime paths and corresponding odometer data representing independently powered, steerable vehicles would require more complicated deviations from this base helical system.

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Linear propulsion sources coupled to produce high-speed rotation

Fig. 12: The limit to linear speed applies also to the angular speed of a rotating body. If the acceleration is uniform, as the speed with respect to the axis approaches that of light, Equation 20 will be obeyed instead of the linear Newtonian relation. (This has been empirically demonstrated in particle accelerators, where the propulsion is produced by electromagnets, not rockets.)

8.4 Acceleration and Universal Speed Limit

We need to understand a few more kinematical principles that the RC’s have adopted. They have arisen not only in the course of their natural technological development and their Space Exploration Program, but in their conjecture as to what got their Cylinder rotating in the ﬁrst place.
In the tubular odometer of Figure 11, we see that the rotation speed increases linearly with tube diameter. The RC’s have long recognized that, for a given angular velocity, !, there is a corresponding size limit, because the product r! cannot exceed the light speed constant c. Rather than investigate the limit by adding material to increase the radius, the RC’s apply what they have learned about linear acceleration. It has long been known that uniform proper acceleration (as by a rocket) will increase one’s speed with respect to the
a Cylinder’s axis, not linearly by the simple formula, v at,
but by a modiﬁcation which takes the limit into account:

av

at
p

:

I C a2t2=c2

(20)

Even for small accelerations a, as t becomes very large v approaches, but never reaches c. The origin of the RC’s world and, in particular, how it acquired its rotation, is an unsettled question. The most widely accepted hypothesis is that a large ﬂeet of powerful rockets were attached all around the circumference, each pair being “coupled” by a huge “spoke”

(inner bracing) spanning the Cylinder’s diameter. (See Figure

12.) Given an endless supply of fuel the Cylinder’s angular

velocity could ideally be increased so that the circumferential

speed would approach ever closer to c. Well before it attained

such a huge speed, they surmise, the rockets were turned oﬀ

and removed, leaving the Cylinder to “forever” maintain its

state of uniform rotation. So the story goes.

The pertinent kinematic principles can be starkly illus-

P Q H VTT trated by imagining that the Cylinder’s circumference has

p % been brought up to the speed c=

: c. This is the

speed at which the rates of clocks and the lengths of tangen-

tially oriented measuring rods at the circumference will be

SH7 of their QH gram we see

rest that

rates and lengths. On this means the needle

twheillspdeipedtoometerfrdoima-

the space-axis.

Now imagine that a section of the Cylinder has been

nested inside a huge non-rotating “ring dial” that nearly

touches the Cylinder’s outer wall. This portion of the outer

wall and the ring dial both have numerous equally spaced

clocks arrayed all along their edges. The ring dial is basically

one continuous circular “ruler” having tick marks along its in-

ner edge. Whereas, the outer wall of the Cylinder is equipped

with many small measuring rods – each one being attached

by a single “point” fastener to allow the rod to expand or con-

tract. Referring to Figure 13A, we show the position of a

rotating measuring rod and rotating clocks as they move with

respect to the non-moving ring dial. All clocks have been

set (by a center-ﬂash synchronization signal) to read zero at

this instant. Figure 13B shows the same object one “smallo-

second” later by the ring dial clocks. With respect to the ring

dial instruments, as expected, this reveals the rotation speed
v a r! a H:VTTc.

These circumstances could also be modelled with a space-

time diagram as in Figure 4. But showing how the clocks

and gauges might actually be arranged in physical space lends

some concreteness to the idea that could not be so easily con-

veyed in Figure 4. Also we have troubled to illustrate these

details so as to make it abundantly clear how un-relativistic

this reckoning system is.

Consider, for example, how the rotation speed could be

interpreted by an observer in the rotating system if such an

observer were to consider herself as being at rest. Suppose

this observer counts the tick marks on her own measuring rod

I I a passed by the point P of the ring dial in the timpe elapsed be-

tween Figures 13A and 13B. She then ﬁnds =

v2=c2

twice as many marks as what she would ﬁnd by counting the

number of ring dial tick marks that pass one point of her rod

I a in the same time. Fpurthermore, this is the length that moves

past the point P in

v2=c2 one half “smallo-second”

a P Q Q RTR according to her own clock. Using these values, she could

p % calculate that her speed is length / time

: c.

Having been born and raised in the Rotating Cylinder,

however, she knows that she is not really moving that fast;

she knows that her speed reckoning should always be referred

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Rotating Cylinder Kinematics EXTREME CASE: v = 0.866 c
P
RING DIAL (at rest)
P
RING DIAL (at rest)
Fig. 13: Using center-ﬂash synchronization means diﬀerent velocities could be measured, depending on whether the data are referred to rotating rods and clocks or resting (Ring Dial) rods and clocks. Note that if Einstein synchronization were “locally” applied to the adjacent rotating clocks, they would appear here to be desynchronized exactly so as to make the velocity appear to have the same magnitude in both frames. But this results in a most extreme and impractical time gap; it implies a “relativity” to motion which contradicts the RC’s physical experience.
back to the axis, which is the same thing as the ring dial. Her absolute speed is the number of distance ticks that a point
H VTT on her rod passes on the ring dial in ring dial time. That is,
: c. Our observer recalls having found in the RC’s obscure
philosophical journals discussions about a conceivable alternative called “Minkeinstein spacetime.” According to Minkeinstein, a rotating observer may begin with a series of closely spaced rotating clocks that are “locally” “synchronized” by

a linear method which assumes that the speed of light is the

same in both directions. She could then “locally” “measure”

that the number of tick marks on her rods passed by a point on

the ring dial in the time given by her (Minkeinstein synchro-
a H VTT nized) clocks yields the speed v : c. Although appre-

ciating the imagination needed to conceive it, this scheme –

wherein the speed of light is artiﬁcially forced to be isotropic

so that all velocities will exhibit reciprocity – seems funny

to her. She alternately chuckles and winces to think any-

one might actually employ it because the scheme obviously

breaks down globally. Light signals sent around the circum-
¦ ference back to herself reveal speeds (c v) regardless of

how she might synchronize her clocks. So (again) the most

reasonable synchronization scheme is that which is consis-

tent with this absolute light speed anisotropy for rotating ob-

servers; i.e., which honors the axis as the true reckoning rest

frame. The RC’s have never entertained any abstract “rela-

tivity principle” as having any special virtue, so the idea of

velocity reciprocity means little to them. What’s much more

important is having clocks synchronized so as to reﬂect the

empirical fact of light speed anisotropy in opposite directions

around their Cylinder. That this makes virtually all velocities

asymmetrical, non-reciprocal and absolute does not trouble

them. On the contrary, it serves to simplify their reckoning

by ﬁxing their bearings.

Our observer and her world are rotating through space;

the axis and the ring dial are at rest in space. Acknowledging

this axis as the “preferred” rest frame amounts to the same

thing as the Reichenbach-inspired -synchrony scheme, ac-

a cording to which, for this frame,

1 2

,

and

the

non-preferred


1 2

frames

are

all

determined

correspondingly.

Recalling

Reichenbach, not only is it true that this scheme “could not be

called false,” given their experience, this is clearly the most

reasonable kinematic scheme the RC’s are likely to come up

with.

8.5 First Encounter
During their entire evolution the RC’s were deprived of a solar system; no orbiting planets; no orbiting moon; no large dense bodies massive enough to arouse any inkling of gravity. They understand electromagnetism well enough; they have a pretty sophisticated understanding of motion; but the RC’s have no conception of gravitational attraction. Of course, they have often contemplated the apparent vastness of space and the mysterious points of light scattered across the otherwise black sky. After many years of practicing their excursions beyond their womb-like cylinder, gradually increasing the distance and the time away from home, they’ve just ﬁnished preparations for a much longer voyage.
A handful of RC’s set out in two rocket ships (1 and 2) to explore the Universe. Many years pass and eventually they happen to ﬁnd the end of one of our tall poles. They are delighted to ﬁnd the accelerometer and the clock, etc.

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RC’s discover top of pole
0
ACCELERATION
Fig. 14: After years of exploring in the void of deep space, the RC’s happen upon a sign of intelligence: Instrument Station #1.
Having stopped to inspect these things, they notice that the accelerometer reading is extremely close to, but not exactly zero. Of course they realize that the rates of clocks are trickier to read and interpret. For example, they need at least a pair of clocks to get a meaningful assessment. The best they can do is to ascertain that while “co-moving,” the instruments on the pole behave the same as those on the spaceship. In any case, the RC’s recognize this station of instruments as a sign of intelligence – motion indicators have the same meaning anywhere in the Universe.
The RC’s decide to leave their engines oﬀ and drift with respect to this pole. After a long while, they begin to visually discern a slight motion. After a very long time they ﬁnd the ﬁrst Instrument Station (IS-1) very slowly receding from view. Eventually, they are pleased to see another IS (IS-2) coming into view further along the pole. Since the pole is moving so very slowly, the RC’s take readings and carefully inspect IS-2 without having to ﬁre their rocket.
They are in no hurry, so the RC’s continue this process for many more IS’s, and a pattern begins to emerge. The accelerometer readings are slowly increasing and the speed of the pole past them is increasing correspondingly. They think, “whatever is at the other end of this pole, it is ever so gradually accelerating and ever so gradually increasing its acceleration.” It’s too early to draw further conclusions; they need more data.
As the RC’s continue in this manner, after many more IS’s have accelerated past them, it becomes necessary to scan the passing IS’s with electronic instruments, as the pole’s speed is now too great for casual visual inspection. The RC’s curiosity grows as the speed increases, so a strategy is devised for

unravelling the mystery and making a more thorough investigation. Since they have two ships, they decide to send Ship 2 toward the propelling end of the pole. After blasting onward for a while, Ship 2 eventually discerns not only a “spot” in the direction of the pole, but faint evidence of other poles that seem to be emanating from the same spot. Rather than continue directly toward the spot, it is decided that Ship 2 will veer out a ways beyond the pole to get a diﬀerent view of the thing and take a long way around to the other side. Maybe the source of propulsion is on the other side of the spot. Or, if there’s also a pole on the back side, then the plan is to see if the propulsion source is at its far end – or at least go as far out along it so as to be symmetrically positioned with respect to Ship 1, with the spot midway between them.
Ship 2 approaches the destination near the far side pole and the RC’s are puzzled to discover that, in order to progress along it, they need to exceed a minimum acceleration away from the spot. If their acceleration is less than this minimum, the pole appears to accelerate in the direction opposite to that of the initial near side pole (which Ship 1 is still exploring). This is most baﬄing; the RC’s have never before seen anything like it.
It’s as though the spot is propelling both poles in opposite directions. Ship 2 manages to attain a symmetrical relationship between itself, the spot and Ship 1, which means rockets are oﬀ, yet the poles are accelerating past both ships in opposite directions. Could it be that the poles are being built up from “below,” as though growing like trees planted on the spot? Not an unreasonable hypothesis, except that the spot now too appears to be getting ever larger. With their “neutrino-beam” radio, the ships remain in communication with each other even as the spot ominously begins to ﬁll the view along the line of sight between them. Before long the RC’s become totally awed by the size of the spot, which they now realize is a huge spherical chunk of matter. Another bewildering clue is that the signal propagation delays have grown shorter and shorter, as though the distance between ships were decreasing. How could that be if (as is the case) they have never registered any acceleration toward each other?
8.6 Puzzled Landing
There’s no sign that the outward acceleration of the Sphere and its poles is going to let up; on the contrary, it continues to increase. The idea of maintaining a safe distance from the Sphere by veering oﬀ into elliptical orbits, perhaps understandably, does not occur to the RC’s. So, it begins to dawn on them that the only way to avoid fatal collisions is to blast their rockets, to accelerate radially away from the Sphere, and to match its surface speed and acceleration just before contact. In the nick of time the feat is accomplished. The ships “land” without mishap.

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What are they to make of this? It’s still too early to say.

But an important clue observed by both ships is that the accel-

eration of the poles – as indicated by the pole’s accelerometer

readings as well as their motion – increased as the Sphere got

closer according to a deﬁnite pattern. The acceleration varies

as the inverse square of the distance from the Sphere’s cen-

ter. Having avoided any local maneuvering (except for the

landing) Ship 1, especially, was able to compute that the ve-

locity of the pole at every point along the way was exactly

as if the acceleration shown by the accelerometers were con-

tinually applied to it. Having access to the data signals from

the IS’s, they know that this does not mean all points of the

pole shared this increase. Rather (remarkably) the acceler-

ation shown by every accelerometer remains constant over

time. The seemingly rigid pole somehow maintains a range

of constant accelerations – varying as the inverse square of

the distance. Surely this is a key piece of the puzzle.

Now that they have managed soft landings, they are able

to conﬁrm another piece of the puzzle, which is also provided

by the IS’s continual data signals. It turns out that the fre-

quencies of the IS’s clocks are diminished exactly as they’d

expect from the measured velocities. For example, for three

a IT a R particular distances, r

R, r R and at the surface,

ar R, the corresponding clock rates are in the ratio

q

I f16R a I fR

q

GM 8rc2 2GM rc2

nd

q

I f4R a I fR

q

GM
2rc2 ;
2GM rc2

(21)

where G and M are unknown constants. As these equations imply, the RC’s have already deduced that the eﬀects have something to do with the Sphere’s mass. But the mind boggling revelation is that the mass evidently combines with an entirely new constant, which they surmise they’ve overlooked because of their prior lack of experience with any mass so large as the Sphere. A poignant clue from these equations,
P of course, is that GM=r is the square of a velocity, which
indeed corresponds to the velocity of the poles as they accelerated past ships (1) and (2). So some things add up, even as other things remain most puzzling and uncertain.
After a period of intense contemplation, the RC’s emerge with two hypotheses. Note that, although previously ignorant of gravity, the RC’s are mathematically sophisticated. They had long ago developed geometries involving non-Euclidean ideas; they are fully capable of thinking abstractly in terms of curved (or ﬂat) spaces of three and more dimensions. It blows their minds to now be thinking that some of these ideas may be needed to describe the real world. Their experience, with all the data they’ve obtained, leaves them no choice. The
QCI conception of a static ( )-dimensional Euclidean space has
been convincingly shattered. But it’s not yet obvious which of their hypotheses, which we’ll call Hypothesis 1 (H1) and Hypothesis 2 (H2), would become the appropriate replacement.

Remember, the RC’s have never before had even the slightest reason to doubt the truthfulness of their motion sensing devices: accelerometers have always reliably told them how fast they were accelerating; and clocks could always be trusted to determine a velocity with respect to their rotation axis and provide a record of distance traveled (according to the elapsed time-odometer connection). So the vast majority of RC’s naturally lean toward that hypothesis which honors the heretofore utter reliability of these instruments.
8.7 Accelerometers and Clocks: Betrayal?
The main reason most RC’s dislike H1 is because it would involve the conclusion that their motion sensing instruments are not necessarily reliable. This would be the case if there were some kind of strange mechanism emanating from this big chunk of matter, that latched onto every nook and cranny of every other chunk of matter, producing a kind of “pulling at itself.” If this were really happening and the “strength of the interaction” happened to be extremely “ﬁne tuned,” (that is, exactly the same for every kind of matter) then an accelerometer that was being so pulled, would yield a zero reading, even though it was accelerating.
The redeeming quality of this view is that it maintains the impression that the RC’s had had for thousands of years, that material things did not show any obvious sign of “moving by themselves.” On the scale of familiar objects in their experience, matter was seen as “inert.” Either rotation or some kind of propulsion was needed to make a body move. Some kind of propulsion was needed to get a body to start rotating in the ﬁrst place. And space was pretty much just a vast passive emptiness within which matter could be found to move or not. H1 involved only a relatively subtle modiﬁcation of this ancient world view.
In their prior experience, however, forces always pointed in the same direction as positive accelerometer readings. Most RC’s found it troubling that the mutual pulling idea would mean that this would no longer generally be true. On a large massive body, in fact, an accelerometer reading would appear to mean the opposite of what it says. If the massive body is really an unmoving (static) thing, in spite of the accelerometer readings, then a positive reading would indicate a state of rest. That is not only bizarre, it cuts against every deepest instinct the RC’s have concerning motion.
A similar conclusion follows with regard to clock rates. Clocks attached to the Sphere have rates that would “ordinarily” indicate that they possess a substantial velocity. So now it has been proposed that these clocks may actually be in a state of “rest”? Most RC’s didn’t like it, but they could not yet prove otherwise. In trying to put as positive a spin on this hypothesis as they could, its proponents cast it as a “geometrical” account of space, time and matter in which this mutual pulling of static chunks of matter was a matter of course.

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H1 proponents even found it ﬁtting to “bend” the common meaning of words in their eﬀort to explicate their geometrical model to others and themselves. For example, they invented the expression, “acceleration of a particle at rest,” which refers to an object attached to the Sphere. Asking, well, is the particle accelerating or resting, one comes back to the fact that, according to H1, the Sphere is an utterly static thing. Therefore, the “deep down” answer is evidently that H1 proponents really think of it as being at rest.
They use the word “acceleration” to refer to a state of rest because of the positive accelerometer readings and because this is the state of motion with respect to geodesics, i.e., trajectories of “falling” objects that are thought of as “really” accelerating, but which have some properties of not accelerating. As we might expect, H1 proponents have an uphill battle trying to convince H2 proponents – who are simply inclined to a straightforward interpretation of their motion sensing instruments and aren’t impressed by counter-intuitive geometrical word games.
At this stage it must be admitted, however, that the facts do not yet warrant a decision. It just might suﬃce to posit that space and time in the vicinity of a suﬃciently massive body are perceptibly neither Euclidean nor Minkeinsteinian, but pseudo-Riemannian.
What could possibly make it so?

perpendicular to this axis. Rotation, of course, means that the stationary velocity with respect to the axis, is similarly perpendicular; all velocities are on tangents of the circular crosssections. Note that the symmetry of the circle thus also has a handedness. With respect to the Cylinder’s circumference, clockwise motion diﬀers from counterclockwise motion in its eﬀects on rulers, clocks and the speed of light. The Rotating Cylinder thus manifests the combination of stationary inward acceleration and stationary tangential velocity.
The massive Spherical world also exhibits a pattern of accelerometer readings and clock rates that persists without change. This evokes the possibility that it too should be regarded as a manifestation of stationary motion. In this case the pattern has a “convex-spherical” character. The acceleration is directed outwardly; beyond the surface the acceleration decreases with distance; and a circularly symmetric crosssection can be found on planes cutting the center along any axis. Since the acceleration is directed outwardly, and given the above symmetry properties, it follows that the velocity is also directed outwardly. Of course, this is also dramatically consistent with the RC’s recent experience. Thus, H2 involves the concepts of stationary outward acceleration and stationary outward velocity, by analogy with the cylindrical, rotation-induced stationary motions.

8.8 Space, Matter and Time in Light of a Faithful Interpretation of Accelerometer Readings and Clock Rates
8.8.1. Stationary Motion. Even as it allows them to retain the sanctity of their motion sensing devices, H2 is in some ways even more radical. Before getting to the radicalness, however, we should emphasize that the new matter-induced motions bear a profound similarity to the RC’s experience with their Rotating Cylinder back home. In the Cylinder an array of accelerometers and clocks manifests readings and rates indicating a pattern of stationary motion. The bracing and the cylinder are practically rigid, yet they reveal a range of accelerations and velocities that do not change over time and that leave the structure intact.
These latter characteristics are exactly what they have found to exist with respect to the huge massive sphere. The RC’s clearly perceive the analogy and seize it as a basis for building and comparing hypotheses. A key question, of course, is how far does the analogy extend? At what point would new ideas be needed to maintain the validity of the analogy and what exactly would those ideas be? The RC’s organize their thoughts as follows.
In the Cylinder, the stationary motion pattern has a “concave-planar” character. The magnitude of the acceleration increases with distance from the axis, it is directed inwardly and because there is only one linear axis, a circularly symmetric cross-section can be found only on a plane that is

8.8.2. Unaccelerated Test Objects. One of the most obvious diﬀerences between these two circumstances has to do with the behavior of small bodies that are not ﬁrmly attached to the Cylinder or the Sphere. (For the sake of argument we presently allow, of course, that in spite of its large size, the mass of the hollow Cylinder is so small that its gravitational eﬀects have gone unnoticed by the RC’s.) In the case of the Cylinder, both inside and outside, it is possible to place objects at ﬁxed locations with respect to the axis and their spatial relationships will not (“noticeably”) change.
This is most unlike what they ﬁnd even at large distances from the surface of the massive Sphere. The motional eﬀects of the hollow Rotating Cylinder have a negligible eﬀect on the spatial relationships in the surrounding space. Whereas the motional eﬀect of the solid massive Sphere very noticeably alters the spatial relationships of objects similarly placed (“falling”) in the surrounding space.
8.8.3. Dimensionality: Motion Through or Motion Of Space. This diﬀerence in the behavior of unaccelerated test objects is clearly related to the characterization of the rotational motion as “concave-planar” and the matter-induced motion as “convex-spherical.” Rotational motion has a lower dimensionality. It is conﬁned to the thing that rotates and has no noticeable eﬀect on the surrounding volume of space. It can be conceived as motion through this three-dimensional volume of space [(3+1)-dimensional spacetime].

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Whereas matter-induced motion evidently requires one more space dimension because the pattern is not of motion through space but appears to be more accurately characterized as motion of space. Radical may be too mild a word for the implied shift in perspective. But it makes sense because of (among other things) the inverse-square acceleration pattern.
If the accelerometers are telling the truth about their states of motion, it would not make sense to think of it as happening in (3+1)-dimensional space because the coherence of the sphere and its poles would not be possible if all the accelerations were naively thought of as so many diﬀerent rates of “material expansion” in 3D space. But if matter is the driving agent of the (4+1)-dimensional motion of space, the inversesquare acceleration is exactly the pattern one should expect. Matter would then be seen as the source of space; matter is perpetually generating space.
The RC’s regard this not only as a reasonable interpretation of their experience, but as being directly related to the physical dimensions of the new constant appearing in Equation 21. In terms of Length, Time and Matter, the dimensions of the constant, G are L3=T 2M . In words, G may thus be expressed as “acceleration of volume per mass.” Multiplying by M=r2 gives the linear acceleration corresponding to how the generation of space spreads out over distance. At any two given distances from a massive sphere, in a given small increment of time, shells of equal volume are accelerated outwardly. The radial thickness of both shells is given by the inverse square law.
So the idea is that matter propels itself and its surrounding space ever outward. Fortunately, there are several ways to continue building on the rotation analogy and ways of comparing the (3+1) vs (4+1)-dimensional schemes so as to facilitate visualization and improve palatability of this, in some ways disturbingly new idea. Ultimately, the ideas need to be tested with experiments, of course. But this is all so new to the RC’s that they are still mentally digesting the hypotheses – which process will naturally lead to the most eﬀective experimental ideas in due course.
8.8.4. Embedding Space. An essential concept for understanding the relationship between a given dimensionality and the next higher or lower dimensionality is that of embedding. The RC’s recognize that H2’s appeal to a higher a dimensional space is analogous to a more familiar circumstance involving lower dimensional spaces.
Consider the surface of a sphere. Imagine that the surface is inhabited by beings having no extension perpendicular to the surface. They extend only along the east-west and north-south directions in the surface. Virtually of necessity, the description they would devise of their world would be two-dimensional; it would involve only two coordinates (e.g., longitude and latitude). This is known as the intrinsic description, which is suﬃcient because it’s all that’s needed to

specify any location on the surface. It is also suﬃcient to reveal the essential properties of the surface, that it closes back on itself and that the sum of angles of intersecting straight lines (great circles) depends on the distances between vertices, etc. The two-dimensional geometry of the sphere is thereby shown to be quite diﬀerent from the two-dimensional geometry of a plane or a torus, for example.
Beings who extend in three space dimensions should well appreciate the suﬃciency of the intrinsic description. But an obvious alternative for them is to include the third dimension, to add a coordinate which facilitates understanding that the spherical surface is but a cross-section of their much larger world view. The 3D beings reside beyond, above and below the surface, so their three-dimensional description is called extrinsic. The 2D sphere (and its ﬂat inhabitants) is embedded in the higher space of three dimensions. The third dimension is the embedding space and it is, for mapping purposes, superﬂuous. For other purposes, however, the third dimension is essential.
Since the extrinsic description is an obvious possibility for 3D beings, it is at least conceivable that, were the 2D beings suﬃciently imaginative, they too could conceive a higher dimensional embedding space. It is even conceivable that they could devise ways of testing whether their world was really physically so embedded. For example, suppose the sphere were subject to various deformations due to rain, spherequakes, meteor impacts, heat, etc. Nothing on the surface itself helps to explain these things. Perhaps the inhabitants note the regularity of some of these phenomena and have ﬁgured out how to predict the time of their recurrence. If they insist that the whole world consists of their 2D surface, these phenomena could only be ascribed to magical gods or “forces of nature.” Whereas, if they allowed that the world extended into another dimension, the inﬂuences impinging on their surface could be seen, or at least imagined, as originating in the higher dimension and ultimately having an explanation that did not require magic.
The point of discussing the relationship between the second and third dimensions, of course, is to establish an analogy for the relationship between the third and the fourth. Though only two dimensions are needed to map a sphere, for the sphere to exist in physical reality it requires the embedding space of the next higher, third dimension. If our analogy does in fact extend to the relationship between the third and fourth dimensions of space, it implies that, for a seemingly threedimensional spherical mass to exist in physical reality, it requires the embedding space of the next higher, fourth dimension.
Before exploring this idea further, note that a similar train of thought was entertained by R. Swinburne in 1968. In his chapter on “The Dimensions of Space,” after setting up the analogy much as we have, with two-dimensional beings embedded in the surface of a sphere, Swinburne presents various possible consequences and implications. His arguments

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led him to conclude that three space dimensions are all we get, that the idea of a fourth physical space dimension is not logical. In the chapter’s last paragraph, however, Swinburne perspicaciously admits:
But, it may be objected, how do we know that we are not in the same situation as the inhabitants of the purported two-dimensional world described earlier? It seemed to them that only two lines could be mutually perpendicular. They made this mistake because it was not physically possible for the ‘objects’ with which they were familiar to move outside their surface. Might not we be making a similar mistake in supposing that our space is three-dimensional because it is not physically possible for the objects with which we are familiar to move outside the threedimensional hyperplane? It must be admitted that we might be making just this mistake. [61]
That Swinburne did not further explore this objection is not surprising, as his whole chapter rests on the tacit assumption that an understanding of the motion of objects on and around a large massive body should scarcely be aﬀected by speculations about extra dimensions. Having Newton’s and Einstein’s theories of gravity already deeply ingrained in his mind, he was not alert to the possibility that his discussion – especially his ﬁnal paragraph – points to a possible profound connection. Perhaps this connection could lead to an understanding of why Newton’s and Einstein’s theories appear to work so well. Could it be that “moving outside the three-dimensional hyperplane” is the same thing as the outward generation of space by matter?
We’ll come back to this question. First, let’s widen our perspective to see how the RC’s might come to it without hesitation, because their minds are not ingrained with any gravity theories at all.
8.8.5. Lore of Hyperdimensionality. The RC’s mathematicians have a long history of hyperdimensional and non-Euclidean research. To some extent the mathematician’s results have been adopted by theoretical physicists and have spilled over into popular culture. Since the RC’s are now faced with a real world need to discover which part (if any) of this research rings true, they ﬁnd it worthwhile to freshly contemplate some of the key ideas.
By deﬁnition, increasing the number of space dimensions means introducing a new perpendicular direction along which motion can occur. Though aware of this deﬁnition for many years, it had not previously occurred to the RC’s that one of the underlying concepts in the deﬁnition was unwittingly being slighted: motion. The extra dimension had been conceived, rather, more as an abstract device for developing theories or accounting systems for phenomena or data that were

so complex that three dimensions, or coordinates, were not suﬃcient. So the perpendicular “motion” was generally not in physical space, but “parameter space.”
On occasions when the RC’s physicists guessed they might beneﬁt from having more than three space dimensions to describe physical reality (which had especially lately become common) they were inclined to invoke a lot more than just one. But the purpose was always to do with simply (or complicatedly) widening the stage, so to speak, to provide more coordinates with which to describe physical phenomena. This did not change the primal view, mentioned above, that gross matter and space were essentially static things. So even the physicists’ invocation of a fourth space dimension slighted the concept of motion – at least any motion that has a perceptible connection to the “ﬁrst three” space dimensions.
The appearance of the fourth spatial dimension in RC pop culture typically shared this lack of connection to physical motion. A 4D creature supposedly would be capable of materializing and dematerializing at will; and could do magical things like remove the contents of locked boxes. But matterinduced motion played no part in such ideas. The fourth dimension also enjoyed a phase amongst certain “spiritual” RC’s and continues as a recurring theme in their science ﬁction genre.
A key aspect of the use of the concept by some of the RC’s academic physicists is the idea that a given space dimension may have a characteristic size, like a meter, a millimeter, the wavelength of a subatomic particle, or some unimaginably tiny length scale. In this context, the unobservability of the extra dimension would be explained as being due to its being curled up, “compactiﬁed,” into these extremely small sizes.
Most of these ideas stand in sharp contrast to the present one (H2), whereby the fourth space dimension doesn’t have a “size,” is not compactiﬁed or invisible, and it doesn’t merely add an extra static coordinate, because it represents the motional extension of every “lower” dimension. Based on their recent experience, the RC’s now conceive this “motional extension” as the perpetual generation of space by matter.
A common graphic image presented in discussions about the fourth space dimension is very suggestive of this space generation idea. It’s the so-called “hypercube” shown at the upper right of Figure 15. Also depicted in Figure 15 is the “hyperspherical” counterpart. Considering one alleged proof of the three-dimensionality of space, the historian of science, Max Jammer, described a progression much like the ﬁrst three steps of Figure 15. In conclusion, he wrote that this kind of
...deduction comes to an end apparently because the conception of a motion of a three-dimensional space as a whole lacks all intuition based on experience. [62] (My emphasis.)
The implication that motion should play a fundamental role in the hierarchy of dimensions was hit on by the inventorphilosopher, Arthur Young, who said: “In terms of dimen-

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Fig. 15: Hierarchy of dimensions: Linear, rotational and omnidirectional projection.
sions, the line is extension and the birth of time.” [63] Considering the profound and pronounced (motional) eﬀects a large body of matter has on accelerometers and clocks, considering this image of the hierarchy of dimensions and the comments by philosophers, mathematicians and historians about what hyperdimensionality entails, isn’t it ironic that the interconnections are not perceived as an obvious possibility? Just imagine a lot of little accelerometers attached to Figure 15’s “hyper-objects.” If the extension of a (zero-dimensional) point to create a line is the “birth of time,” how much more may time be “birthed” by the (hyper) extension of a massive three-dimensional volume?
With the kind of intuition the RC’s had about motion after thousands of years of evolution in their Rotating Cylinder, topped by their recent experience and present situation, it’s easy to imagine that they would proceed, at least tentatively, beyond the third dimension. As explorers, they are rightly inspired even to name their hypothetical new territory. Their map now includes: height, width, depth and gravity.
The cubical culmination of Figure 15 had been recognized by the RC’s geometers as the simplest “head on” view or “central projection” of a four-dimensional hypercube. [64] But before encountering the large massive Sphere, it had not occurred to them that the “projection” aspect might be due to a process that was constantly happening in time, or that its physical counterpart necessarily involved matter. Having now discovered that matter has a motional eﬀect on space, that space appears to be constantly, outwardly projected by matter, in H2 the RC’s conceive that space is not just a preexisting background. Rather, space would not exist, there would be no dimensions at all were they not all being perpetually generated by matter.
8.8.5. Curvature and Hyperdimensional Motion. The diﬀerence between H1 and H2 is crucially exempliﬁed by the signiﬁcance of curvature in the respective hypotheses. In H1 cur-

vature is a given property of spacetime, rather like the myste-

riously impinging phenomena in the case of the 2D spherical

surface. In other words, the curvature is regarded as an in-

@Q C IA trinsic feature of

-dimensional spacetime, which has

no extrinsic reality because an extrinsic reality would require

a higher dimensional embedding space. H1 proponents ex-

plicitly deny that an embedding space is necessary or helpful.

This point of view makes sense if (big if) matter and space are

static things. Thus, in H1, aside from that it is somehow re-

lated to mass, nobody knows what causes the curvature. The

RC’s draw a blank to the question of what mass must be do-

ing to make spacetime curve. The motional implications of

the readings of accelerometers attached to large massive bod-

ies go ignored and unrecognized like a tyrannosaurus on the

couch. (H2 proponents don’t understand how the H1 propo-

nents can be so oblivious!) Yet the existence of static curva-

ture and the way it varies from place to place are presumed to

be known in ﬁne detail, based on the equations of the hypoth-

esis.

Now recall that the RC’s had made observations in their

Cylinder of the contraction of lengths due to motion. Length

contraction always occurs in the same magnitude as the slow-

ing of clocks. Although the RC’s had noted that this combi-

nation of eﬀects could be described in terms of non-Euclidean

geometry, it was not suspected that this might have any par-

ticularly far-reaching implications. For all practical purposes,

the length contraction and clock slowing were accounted for

in the RC’s kinematical scheme. Non-Euclidean geometry

could well be used to describe the metric relationships on a

rotating disk, as the lengths of rods placed on its surface (and

the rates of clocks) varied with location. But since this had

no noticeable eﬀect on the behavior of objects not attached

to this surface, for the most part, it was of no more than aca-

demic interest.

In their present circumstances, the RC’s focus on the most

important consequence of this prior research: it’s that the cur-

vature was entirely caused by motion. So one of the most

questionable things about H1 is that not only does it lack an

explanation for how curvature is caused, it then regards static

curvature as a cause of motion. By this view the RC’s ex-

perience in the Cylinder and now on the Sphere would most

properly be regarded as anti-analogs, because the cause-eﬀect

relationship is reversed. This idea (H1) thus strikes most RC’s

as being utterly backwards. They see the “proper” analogy as

being to ascribe to the same eﬀects, as far as possible, the

same causes. Rotational motion causes spacetime curvature.

Uniform rotation is widely regarded as being an instance of

absolute stationary motion. Therefore, evidence of spacetime

curvature implies the existence of absolute stationary motion.

(Possibly, it does not require the existence of absolute sta-

tionary motion; but it’s not a bad guess that it does.) Con-

cerning the evidence the RC’s have found on and around the

Sphere, the signiﬁcance of this connection is far more than

academic because the length and time diﬀerences, rather than

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being conﬁned to a 2D surface, would pervade all of volumetric space. This diﬀerence again suggests the existence of a higher space dimension, as it underscores the diﬀerence between motion through space and the movement of space (concave-planar symmetry vs convex spherical symmetry).
In and around the Cylinder it made sense to regard motion as being through space. But now, if the motion attributed to the mass of the Sphere was conceived as being through space, it would require the (H1-motivated) backwards interpretation of the relationship between motion and curvature, which corresponds to the equally backwards interpretation of accelerometer readings and the rates of clocks. Another graphic construction helps to convey these features and may then be adapted to convey the possibly more “forwards” interpretation embodied by H2.
8.8.6. Odometer-Speedometer for Matter. The nested tubes of Figure 11 represent Epsteinian space-propertime diagrams that are rolled up so that time goes up the axial direction and space goes angularly around the tube. This is well-suited to represent the range of speeds corresponding to diﬀerent distances from the home Cylinder’s rotation axis. To represent the eﬀect of matter on space and time, Epstein has rolled his diagram in the perpendicular direction, so that time goes around the tube and space is represented by the tube’s axis. [65] We will ﬁrst describe how Epstein’s tube corresponds to H1, and then we’ll describe an adaptation that is better suited to represent H2.
Before considering how a tube’s shape may be dictated by matter, let’s ﬁrst apply this rolled up space-propertime diagram to zero mass inertial frames. In this case a rest frame corresponds to zero motion through space and so to just looping around the tube in endless circles along the time coordinate (or perhaps “inﬁnitesimally thin” rolled up layers). Uniform motion with respect to the rest frame then corresponds to looping around in a uniformly pitched helix. The maximum speed of a light ray corresponds to motion parallel to the axis with no turning (zero passage of proper time). The diameter of the tube is arbitrary; it may be chosen, for example, so that one turn corresponds to one second of time, or to the distance light travels in one second of time.
Now let’s consider the eﬀect of matter. We imagine that the center of a very long tube corresponds to the center of a massive sphere. In this case, whatever diameter we may have chosen for pure inertial frames far from matter, it will change as distance to the sphere decreases so as to reﬂect the change in clock rate. Speciﬁcally, the diameter of the tube then corresponds inversely to the rate of a clock attached to the Sphere. This causes the tube to ﬂare open, getting wider as distance to the sphere decreases. The angle of this ﬂaring with respect to the axis then corresponds to the degree of space curvature (length contraction). The tube itself is static. The motion of objects or observers attached or unattached to the Sphere is

GR’s Static Schwarzschild Field Modelled as Epsteinian Spacetime Tube
MASSIVE SPHERE
A
Tube Centered on Mass

Continuous envelope with geodesic trajectory (see B)

Segmented conical slices with rest lines (see C)

B Straight Line Trajectory
INDICATES: • Geodesic motion (falling) • Zero accelerometer reading • Significant motion through space

UNFURLED (Laid flat) CONICAL
SLICES

C Curved Trajectory
INDICATES: • REST in STATIC field • Positive accelerometer reading • Motion only through time

UNFURLED (Laid flat) CONICAL
SLICES

Fig. 16: Key characteristics of Epstein’s tube model for rest and motion in a gravitational ﬁeld. Rest in the ﬁeld (through time only) is indicated by lack of motion along tube’s axis; i.e., straight up and around the tube (green lines). Motion in the ﬁeld (through space) is indicated by deviation from straight up the time axis (red line). Rest corresponds to positive accelerometer readings; motion (falling) corresponds to zero accelerometer readings. Our motion-sensing devices seem to be telling us that this scheme is backwards.
represented by lines traced on the tube. Objects that are ﬁrmly attached to the sphere move only through time, so their paths (“speedometer needle”) are straight up and around the tube. Since the tube ﬂares, these lines are not geodesics, which means that if the approximately conical “slice” of tube containing the time line were split and laid out ﬂat, the line would curve. This indicates an acceleration such as would be measured by an accelerometer. On the other hand, the lines corresponding to objects that are unattached to the Sphere trace out a helix along the tube. They are geodesics, which means the corresponding cone slices, when laid out ﬂat, would reveal the lines to be straight. And this corresponds to a zero accelerometer reading. (See Figure 16.)
The slowness of clocks at rest near the surface corresponds to the smaller angular travel of the “speedometer needle” around the tube; a needle of the same length far away

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from the Sphere travels a further angular distance around the

tube in the same increment of coordinate time. The staticness

of the tube is explicitly represented by the “straightupness” of

the time lines of objects attached to it. Since the tube is static,

even though lines traced on it have the correct character corre-

sponding to geodesic vs non-geodesic or zero vs positive ac-

celerometer reading, the question remains as to how the lines

come to be drawn at all. The curvature correctly accommo-

dates the diﬀerent kinds of linear motion through spacetime;

but it sheds no light on how either the curvature or the motion

come to be in the ﬁrst place.

The inadequacy of H1 to answer such questions is re-

vealed by Epstein himself, who, in the course of presenting

his model, at one point asserts that, “gravity slows the speed

of time.” Then, a few paragraphs later, he asserts that the dif-

ference in “the speed of time...is suﬃcient to cause gravity

itself!” [66] Circular reasoning leaves you where you started.

The shape of Epstein’s tube reﬂects the variable quantity

@I P A known in H1 as the metric coeﬃcient,

GM=rc2 . H2

also involves a similar quantity, which will be referred to as

the curvature coeﬃcient. Since this is a second order quan-

tity (with a “velocity squared” in the argument) whose mag-

nitude typically deviates from unity by a very small amount,

Epsteinian tubes representing typical stars and planets have

only a very slight ﬂare. The extreme is reached, not at the

surface but at the center. If the sphere is uniformly dense and

its surface radius is called R, then the temporal metric coeﬃ-

@I Q A cient corresponding to the center becomes

GM=Rc2 .

a I (Where the corresponding spatial coeﬃcient .)

Now let’s consider the H2-motivated Space Generation

tube model. Its key features are: 1) The outer envelope rep-

resents not the (second order) curvature, but the ﬁrst order

P speed, i.e., the stationary oputward velocity. Its magnitude is
(approximately) given by GM=r. This is the speed that

the pole appeared to have acquired with respect to the RC’s

spaceships and the speed that would then cause clocks to slow

and rods to contract. 2) The tube turns. Every body of matter

is evidently a source of motion and a source of space. This

motion may be represented so that the graph of every body

turns at the same angular rate. And 3) The interior of the

tube is a coordinate space where additional motion data are

plotted. (Unlike the Epsteinian tube, whose interior contains

no information.) The whole volume within the tube moves

and the motion causes curvature whose magnitude is plotted

in the interior space. (See Figure 17.) Both of these features

are reﬂections of the fourth dimension of space, whereby the

(moving) graph now represents motion of space rather than

motion through a non-moving graphical surface. It’s not just

lines on the surface, nor even just the surface itself, but rather

the entire tubular volume that is in a state of, and therefore

cogently represents, stationary motion – motion into (or out-

from) the fourth dimension of space.

Every cross section of the tube contains the curvature

data, in the form of space-propertime diagrams, where the

speed is proportional to turning radius. As we recall, this correlates to the angle of the speedometer needle. The outer envelope represents the stationary outward velocity of objects attached to the Sphere and so has a maximum deviation from straight up the time coordinate. Whereas the tube’s axis represents the trajectories of objects fallen from inﬁnity whose clock rates are a maximum and so have no deviation from straight up the time axis. Proponents of H2 suspect that their experience of watching the entire length of pole accelerate and speed past them means their own (“falling”) trajectory corresponds to this center-of-the-tube spacetime path. Being the extreme case which represents the trajectory of a clock having a maximum rate, the RC’s call it a maximal geodesic. What may appear as accelerated motion of an object falling radially from inﬁnity (maximal geodesic) is therefore, according to this scheme, more accurately regarded as the closest thing known to a state of rest. A maximal geodesic is the purest “tracer” of the motion of space past it, which is most rapid near the generating mass’s surface (assuming uniform or nearly uniform density beneath the surface).
This latter parenthetical remark draws attention to one of the most obvious diﬀerences between these models: The H1 tube has a central bulge; the H2 tube converges to a point. This then also reveals where the models diverge most dramatically with regard to empirically testable predictions. It didn’t take long for the RC’s to realize that conclusive proof revealing which hypothesis was closer to the truth would have been at hand if, instead of landing on the Sphere’s surface, the spaceships “fell” into a hole that spans a diameter. According to H1, the mutual pulling of matter ensures that the speed of objects pulled into the center would increase until the center was reached. Therefore, in this case the spaceships would have experienced a high speed collision. (If only one ship were swallowed by the hole, it would have gone up and out the other side to its original radial distance.) The central bulge of this model thus corresponds to the minimum clock rate. The decrease in clock rate toward the center is supposed to cause objects to reach their maximum speed there.
According to H2, on the other hand, matter does not pull objects inward through a pre-existing space; it moves new space outward. Recall the RC’s bewilderment when their neutrino beam radio communications indicated that the distance between Ships 1 and 2 was decreasing, when they knew they had never accelerated toward each other. We now see how the RC’s conceive this: As stated, they never did accelerate toward each other. Rather the space between them was continually moved past them. Near the surface the speed of space is a maximum; to keep up with it requires maintaining the constant acceleration revealed by accelerometer readings on the surface. But at the center the speed and acceleration go to zero. So the rockets would appear to asymptotically approach each other as the amount of space between them similarly diminished. The convergence to a point thus also corresponds to a maximum clock rate – the same rate as clocks

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Generation of Space Inside and Outside Massive Sphere
MASSIVE SPHERE

CROSS-SECTION

Vs
2GM / rγ GM / rγ

0

MAGNIFIED PROFILE (from r =0 to +4R)

R

rγ

Fig. 17: COARSE PROFILE and MAGNIFIED PROFILE: Movement of space is represented by rotation of whole volumetric tube. Helical trajectory is a projection (“tracer”) of unaccelerated object released at inﬁnity (maximal geodesic). The outer envelope is undergoing stationary outward motion. Maximal geodesics, being always unaccelerated, have maximum clock rates and so are represented by the axial line of zero velocity. Trajectories branching oﬀ from outer envelope represent objects released from r = R, r = 2R and r = 4R. CROSS-SECTION: Space-propertime diagrams (“speedometers”) at every point within the tube indicate the degree of clock slowing and length contraction applicable to objects released from various distances. In every case both metric eﬀects are a maximum prior to release. At the extremes are the stationary outward velocity (Vs) of outer envelope and the standard of rest (maximal geodesic) of the central axis.

at or falling from inﬁnity. At the center all causes of motion are cancelled by symmetry, so a maximum clock rate is intuitively quite reasonable. (Contrast this with the central bulge in the mutual pulling hypothesis, where nobody has any idea what could cause a clock at the center to run slow.)

8.8.7. Horizons, Singularities or Well-Behaved Limits? As

the RC’s often found to be true, considering an extreme case

proved most illuminating: What happens when the stationary

outward velocity (by H2) becomes so large as to approach or

exceed the speed of light? Or (by H1) when the curvature

becomes so great as to freeze light and prevent its outward

motion? First, H1: Consider again the metric or curvature

@I P A coeﬃcient

GM=rc2 . As noted earlier, the RC’s sur-

mise that G is a Universal constant and that its magnitude

is rather small. (That’s why they failed to notice its eﬀect

back home in their Cylinder.) The question then amounts

P 3 to what happens when the ratio M=r c2=G? Or could PM=r even equal or exceed c2=G so that the coeﬃcient be-
comes zero or negative? These questions motivated the RC’s H1-sympathetic theorists to invent diﬀerent ways of expressing the problem so that the answers were not so objection-
P a able when M=r c2=G (where they ﬁnd a horizon). They
failed, however, to mitigate or eliminate the problem for the
a H case that r (where they ﬁnd a singularity). An industry
could perhaps be built on the perplexities surrounding these consequences – if H1 were found to prevail.
But what of H2? Rather than jump immediately to this extreme, the RC’s perceive the beneﬁt of starting from the opposite extreme. They begin by considering a very small body of ordinary density – perhaps a grain of sand or a speck of dust. Having discovered quite by accident how the acceleration and velocity produced by a massive body varies with distance outside the body’s surface, the RC’s reason that with each additional shell of matter of the same density, these ac-

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celerations and speeds will increase in like proportion. For example, if the radius were doubled by a layer of matter of the same initial density (which would thus multiply the mass by eight) the speed and acceleration at the surface would also both be doubled.
The ﬁrst beneﬁt of this approach, as we’ll see, is that it lends credence to the RC’s prediction concerning falling into the center of a massive body. The inverse square law entails the consequence that spherical shells of matter should produce motional eﬀects at or beyond their outer surfaces, but not within their inner surfaces where the eﬀects are cancelled by symmetry. Therefore, the only thing that can aﬀect the speed at a given distance within the sphere as a whole is the matter within that given distance. By adding successive layers of matter, one arrives at a new and larger maximum velocity at the surface of each new layer. But this cannot increase the velocity within that layer. Which means both acceleration and velocity vary directly as the distance.
Although this is certainly a reasonable (and testable) ﬁrst approximation for small bodies of matter, the RC’s see that the process of adding shells of matter indeﬁnitely superﬁcially leads to problems similar to those of H1. Consider that by H1, if all the mass of a body is contained within the dis-
a P tance r and r GM=c2, then the spherical suface at r con-
stitutes a “light front,” a surface at which light is frozen. Perhaps if one accepts the mutual pulling idea (or if it proves to be true by experiment, of course) extremes of curvature where this could happen may not seem so objectionable. But by the Space Generation idea, if the body’s mass were increased so
P that the distance GM=c2 corresponded to its surface, that
would mean the matter there was making space move at the speed of light. Also, clocks at the surface would stop ticking. The RC’s surmise that both of these things are impossible. It would be quite like trying to accelerate a body of matter to the limiting speed c with some other means of propulsion. Happily, the RC’s have already dealt with this circumstance (Equation 20, repeated here for convenience) in the context of uniform linear acceleration (through space):

av

at
p

:

I C a2t2=c2

(22)

VELOCITY and DENSITY near and within large masses

( ) 1 –

2GM rc2

≤ 0.

c

0.9

0.9

0.8

V =

2GM r

HORIZON

0.8

0.7

0.7

0.6

0.6

0.5 0.4

Vs =

2GM rγ

c

0.5 0.4

0.3

0.3

0.2

0.2

0.1

0.1

r

Fig. 18: Mutual pulling (static ﬁeld) approach: As mass is added, when 2GM=c2 comes to equal the coordinate radius r, light is frozen and clocks stop (horizon). Since clocks are supposed to get even slower toward the center, there is an unavoidable singularity at r = 0. Space generation approach: As mass is added, it’s not the coordinate distance, r, by itself that matters. Rather, it’s the larger distance r
 = r(1 + 2GM=rc2). No matter how much mass is added, r
 increases so that 2GM=r
 only approaches c2. The variation in proper density
 thus approaches a 1=r
2 law. There is no horizon and, since all metric eﬀects go to zero at the center (as the mass within r
 goes to zero) there is no singularity.

The RC’s surmise that a similar limiting relationship should exist for volumetric acceleration (of space). By analogy, the expression for the velocity caused by this acceleration would be

Vs2

a

r

PGM C PGM=c2

;

(24)

Vs a

Pp GM=r
pI C PGM=rc2 ;

(23)

where Vs is the stationary outward velocity and r is the usual coordinate distance; i.e., the distance as gauged by an observer at inﬁnity. This equation implies the need for a new distance coordinate. Squaring Equation 23 gives

whose denominator is the new distance coordinate. The RC’s call it

r
 a r C PGM=c2 :

(25)

This is the whole distance of relevance for gravitational phenomena. The curvature coeﬃcient may then be expressed as

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ACCELERATION for large M / r ratios

ACCELERATION

.50 .45 .40 .35 .30 .25 .20 .15 .10 .05
0 0

2GM c2

G = M = c = 1

rγ = r (1+2GM rc 2)

(H1): GM

1

r 2 1– 2GM rc 2

(Newton): GM r2

(H2):

GM

1

r 2 (1+2GM

) rc 2 2

1 c2

2

2

r2 γ

1

2

3

4

5

6

7

8

9

10

r

Fig. 19: Mass-induced acceleration compared for three models. The magnitude may go to inﬁnity for Mutual Pulling models. The magnitude is prevented from reaching inﬁnity in the Space Generation model.

@
I

PGM

A

a

@
I

C

PGM

A

1
;

r
 c2

rc2

(26)

which is greater than zero for any ﬁnite M , r or r
. Inside the massive body M goes to zero as r goes to zero, so there
is no singularity. Outside the body mass shells may actually
be added indeﬁnitely, but the proper density cannot remain
I constant. In the limit this density would vary as =r
2, as PGM=c2 grows larger than r and as

a Ps

Vs

GM r

3 c:

(27)

(See Figure 18.) We should now compare how the acceleration itself dif-
fers from one hypothesis to another. For this purpose, instead of having the mass vary by adding successive shells of matter, let’s keep the mass constant. In all cases, when the spacetime is close to ﬂat, the acceleration reduces to the simple (Newtonian) approximation GM=r2. As one would expect, the “horizon” appearing in H1 for the velocity calculation appears also in the acceleration:

a I PI gH1

GM r2 p

: GM=rc2

(28)

a P When all the mass is within r and r GM=c2 this “ac-
celeration of a particle at rest” thus becomes inﬁnite (which
a P corresponds to the idea that r GM=c2 represents a “light
front.”)

By contrast, replacing r with r
, as per H2 gives

gH2

a

g

a

GM r
2

a

GM r2

I @I C PGM=rc2A2

:

(29)

The graphs for H1, H2 and the Newtonian approximation are shown in Figure 19. A couple noteworthy features should be pointed out. First, bear in mind the common Newtonian
a P relationship between escape velocity, V 2 GM=r and g:

g

a

GM r2

a

I P

V2 r

:

(30)

Unlike H1, there is no horizon here. With decreasing r the

acceleration increases indeﬁnitely and becomes inﬁnite when
a H r . In H2, on the other hand, r
 cannot become zero as
long as there is mass. So as r
 approaches zero, we get

a a PI IP g

GM r
2

3 Vs2
r

r

c2
3( 0)

:

(31)

This limit is shown in the graph (Figure 19).

8.8.8. Indelible First Impression. Beyond these theoretical ideas, in the forefront of their deliberations to assess the merits of H1 and H2 is the RC’s unforgettable, thrilling experience of “falling” alongside the poles. During the whole trip (minus local maneuvers and ﬁnal landing) the ships’ accelerometers read zero. Whereas accelerometers on the poles were positive. The RC’s thus, in eﬀect, motionlessly watched as the spherical “spot” grew ever larger. It seemed obvious the

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whole time that the Sphere was moving toward them, and not the other way around. Then, ﬁnally, the experience of navigating “soft landings”: The RC’s needed to accelerate away from the Sphere’s center. Remember, “gravitational attraction” is not part of their training. Rather, it is in their blood and their bones to believe their accelerometers perhaps even more than it is in an Earthian astrophysicist’s blood to believe Newton’s equations. Their motion sensing instruments are telling the RC’s that space itself moves. In the process of generating it, matter moves space. While “falling,” the space between their ships and the Sphere was being constantly moved outwardly past them. That’s their best guess and they’ll stick to it unless it leads to an untenable conclusion.

8.9 Further Considerations and Testability

From all of the above we see that, except for their small contingent of H1 proponents, the RC’s prefer space generation over mutual pulling. But they know better than to think they can just take their pick. They must settle the question with conclusive empirical evidence. Their experience so far does not provide that. So they conceive of two crucial tests. One of these tests could be performed using the instruments on the poles and one of their rocket ships. Since the RC’s nearly ran out of fuel in order to make their soft landings, they would rather save what little they have left, if they can. Still it is worthwhile to explain what they had in mind, since it well illustrates a key diﬀerence between the competing hypotheses.
When the RC’s ﬁrst encountered the top of the pole, and upon passing later IS’s, they took note of the clock readings with the intent of analyzing this data later. As it turns out, in the course of all their maneuvering and their general befuddlement, they ended up losing this clock data. If they hadn’t lost it, they could perhaps already safely decide between H1 (mutual pulling) and H2 (space generation). According to H1, since the “ﬁeld” around the sphere is supposed to be a static thing, the rate of a clock that “falls” in the ﬁeld is supposed to depend only on its location and its speed. Its direction is not supposed to matter. Mathematically, this is expressed by adding a velocity squared term to the argument in Equation 18 or 21. Therefore, as a clock falls, its rate decreases, reaching a minimum just before colliding with the surface.
The RC’s have reservations about this picture for the following reason. It seems to cause twice as much clock slowing as it “should.” The spacetime curvature somehow causes or is a manifestation of slow clocks that are not moving at all. Then, with respect to these clocks the curvature also causes motion which has the potential to slow falling clocks’ rates by another factor of that which applies to the static clocks. This is the factor that somehow causes clock slowing due to curvature:

I P p GM=rc2 ;

(32)

and for the case of a clock radially falling from inﬁnity, one is supposed to ﬁnd another factor due to motion:

I P p GM=rc2 :

(33)

As a consequence, within the pulling ﬁeld there is no state of motion corresponding to the rate of a clock at inﬁnity; the only analog for a clock on their cylinder’s rotation axis is a clock at inﬁnity. So the picture is of a static thing that somehow manifests the instrument detectable characteristics of motion. Then this static thing produces motion of other things which manifest even more eﬀects of motion with regard to their clock rates. But these latter (falling) things manifest no eﬀects of motion with regard to their accelerometer readings. For most RC’s the gut reaction is that the whole scheme is scrambled and nonsensical.
Fortunately, as mentioned above, the picture is diﬀerent enough from H2 that it lends itself well to a confrontation by experiment. According to H2, since the “ﬁeld” around the sphere is stationary, i.e., always moving with the same outward acceleration and velocity (at any given location) the rate of a clock that radially falls cannot get any slower, it can only increase. The extreme case would be a clock that falls radially all the way from “inﬁnity.” That is, of course, what the RC’s just did, in essence. Such a clock would retain the maximum rate it had at inﬁnity, because nothing ever changes its velocity.
During the whole trip, motion sensing devices consistently indicate that it’s the pole that does the moving, not the falling clock. As noted eariler, the RC’s have come to call these special case trajectories (falling radially from inﬁnity) maximal geodesics. (See Figure 17.) The rate of any clock in this “space generation ﬁeld,” whether rigidly attached to the central mass (e.g., attached to a pole) or falling in any direction, will depend on its speed with respect to the local maximal geodesics. Maximal geodesics are thus the local “preferred” frames; the local standards of rest.
If the RC’s had not lost track of their clock data, the idea would be to compare the elapsed time of their own clock with the elapsed time that they could calculate would have elapsed if the rate decreased as per H1. According to the latter hypothesis the expression for the rate of a “probe” clock moving in the ﬁeld is

¡ I fH1 fP fG

a a I fG

fG

q q

2GM v2
rP c2 c2
2GM
rGc2

I;

(34)

where G and P denote ground and probe, and v is the probe’s speed. We see that, by adding (subtracting) a squared velocity
P to the matter-induced clock slowing factor, GM=rpc2, the
direction of the moving clock is irrelevant. By H2, on the other hand, the expression is

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8 Gravity Virgins

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¡ I p fH2 a a I fG

fP fG fG

r

¢ 2GM=rP ¦v£2

q

c2

2GM

rGc2

I ; (35)

where we see that the matter-induced factor is treated as a speed (stationary outward velocity). This speed is then added (before squaring) to the apparent speed of a clock moving with respect to the stationary body so as to get the total clock slowing eﬀect. Since the falling velocity is positive or negative depending on direction, the sum results in a signiﬁcant diﬀerence between H1 and H2. For a maximal geodesic the sum is zero. (Note that Equation 35 has been somewhat simpliﬁed by using r instead of r
 for the radial distance.) If the RC’s had enough fuel, they’d launch a clock to a high enough vertical distance so as to make the elapsed times large enough that the diﬀerence would be measurable.
(Some readers may be aware that the prediction of H1 would seem already to have been borne out by the VessotLevine experiment. [8] If true, this would of course rule out H2. It is shown in §10 that the results of this experiment are actually equivocal. When account is taken of the apparatus used to make the measurements, the prediction of H2 actually agrees quite well with the experimental result. What follows here is a simpliﬁed description of a modiﬁcation of the Vessot-Levine experiment whose result would not be equivocal.)
The idea is to take data at three key events: Soon after releasing the launched clock upwardly onto a ballistic trajectory, record the elapsed time at a well-deﬁned height; call this the “perisphere.” Another recording would be needed at the midway (apex) point; call this “aposphere.” And ﬁnally, record the elapsed time at perisphere again on the descent phase. According to H1 the elapsed time for the ascent phase would be the same as that for the descent phase. By this view, the trajectory is symmetrical in all respects. The launched clock goes up and then comes back down. But the direction does not aﬀect clock rate; only speed (and location) matters.
According to H2, by contrast, the elapsed time for the ascent phase should be signiﬁcantly less than for the descent phase. The trajectory is actually asymmetrical. The ballistically launched clock only goes up; the only force ever applied to it was in the upward direction; nothing ever forces it back “down.” Just before being released onto the ballistic trajectory the clock had the maximum upward speed, so it had a minimum rate. After that the clock’s speed with respect to a maximal geodesic only decreases, so its rate increases on both the ascent and the descent phases.
Two comparisons should thus be made: 1) Elapsed time for ascent phase (H1 time > H2 time) and 2) Elapsed time for descent phase (H1 time < H2 time). Both H1 and H2 predict the same total elapsed times for the whole trip; so the key to

a conclusive result is getting accurate determinations for the two distinct phases.
Although the results of this experiment should clearly reveal whether the ﬁeld is stationary or static, an even more ironclad, and fortunately less technologically demanding experiment could be conducted in a laboratory at the sphere’s surface. This will be discussed in the next section, not from the point of view of hypothetical space explorers, but from that of diligent Earthian physicists.
9 In Progress
When it’s actually ﬁnished, §9 will have a diﬀerent name. Sections 9-13 are still in progress. Having recently hammered §8 – the most important section in this essay – into a reasonably complete state, I deemed it appropriate to make this draft available. §9 is also important in that it discusses the most feasible experiment to test the gravity model. But most of §9’s content is already available on the GravitationLab.com home page and in Paper 1: “Laboratory Test of a Class of Gravity Models.”
Reader comments are encouraged: webmaster@gravitationlab.com. Richard Benish, June 26, 2008.
Appendix
A Modern Commentators on the Sagnac Eﬀect and Relativistic Rotation
The practical purposes of positioning and navigating are extremely well served by the GPS. The system’s clocks are synchronized, in eﬀect, by the central-ﬂash method. Accordingly, when used to “measure” the one-way speed of light, they indicate that the orange beam (of our previous example, §6.8) returns to the observer before the green beam because the orange one travels with respect to the observer faster than c, and the green one travels slower than c. The diﬀerence is plus or minus the speed of rotation. This concurs with the conclusion of our “child” observer in our “ampliﬁed” example in which the time-gap is so large as to be discernible using only human eyes and mind. It is a simple physical property of rotating reference frames. Most children, one can be sure, would have a diﬃcult time understanding why so many grownups would make such a big deal of it.
Happily, since the GPS has been up and running, one no longer (to my knowledge) ﬁnds blanket “impossibility” statements. The spirit of Einstein, however, still resides in many theoretical accounts of the System. In spite of how simply the situation can be described, there is still some debate about

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the “proper” way to interpret the Sagnac eﬀect. A key focus in these discussions, as indicated above, is the time gap. It is useful to see exactly what’s been said. Three examples thus follow. First, Rizzi and Seraﬁni:
The deep physical, non-conventional, nature of the time-lag...the dark physical root of the Sagnac eﬀect [is] desynchronization of slowly travelling clocks. ...such a desynchronization is just a variance of the well known twin paradox. [67]
Next, D. Dieks:
Because of the diﬀerence in arrival times of the two light signals, the velocity of light obviously cannot be everywhere the same in the rotating coordinates. This is a consequence of the fact that in the rotating frame events with equal time coordinate t are not standard simultaneous. [68]
N. Ashby:
The Sagnac eﬀect can... be regarded as arising from the relativity of simultaneity in a Lorentz transformation to a sequence of local inertial frames co-moving with points on the rotating earth, or as the diﬀerence between proper times of a slowly moving portable clock and a Master reference clock ﬁxed on earth’s surface. [69]
Before considering the general import of these statements, I’ll begin with some remarks about each one. Rizzi and Seraﬁni refer to another physical operation besides counter-rotating light beams that would result in an identical time diﬀerence. Two very slowly transported clocks, upon returning to their starting point, will have become “desynchronized” by the same amount. Notice how far removed this is from the original empirical fact. The authors may have pointed out some kind of corollary or analogy, but is it true to say that this is the “dark physical root” of the Sagnac eﬀect? No clocks are needed to demonstrate the eﬀect. So how can clocks be the “physical root”?
The t Dieks refers to corresponds to the time of the rotation axis, or the time clocks on the rim would have (to ﬁrst order) if they were synchronized by a ﬂash from the center. He points out that the velocity of light cannot be isotropic with respect to the rotating disk when using this time coordinate. However, one is still struck by Dieks’ characterization of light speed anisotropy (or time lag) as being a consequence of the synchronization scheme. Remember, the lag (anisotropy) occurs with respect to any one clock on the rim. The (physical) eﬀect is patently independent of (abstract) synchronization scheme.
Ashby is recognized as one of the foremost authorities on the GPS. He attributes the Sagnac eﬀect to the relativity of simultaneity or a clock transport operation like the one mentioned above by Rizzi and Seraﬁni. He reiterates these

and other causes in at least three papers. [70] [71] [72] In describing the details of the System, Ashby often refers to the “constancy of c” in the underlying (Earth centered) inertial coordinate system. With regard to rotating observers and the signals they send with or against the rotation around the circumference, he refers to “path-dependent inconsistencies” or “discrepancies”; he refers to diﬀerences in the travel times or diﬀerences in the path lengths of such signals. But, to my knowledge, Ashby never explicitly describes what happens as being due to non-c light speeds with respect to rotating observers. It is amazing to me how consistent he is, at every step avoiding the simplest explanation. Instead, he concludes: “In fact observers in the rotating frame cannot even globally synchronize their own clocks, due to the rotation.” [73]
Navigators and time keepers around the world are surely satisﬁed with how well synchronized the clocks of the GPS appear to be. Yet one of the physicists who helped to build and maintain it regards the clocks as being desynchronized. In the discussion section of the book from which the above quotations were taken, we see that Ashby graciously grants as “admissible” the viewpoint that the rotating clocks are indeed synchronized (with the corresponding anisotropic light propagation which follows). But he evidently prefers not to think of it this way himself. Why not? Evidently, it’s because it would mean that the speed of light with respect to rotating observers would then even locally not be equal to c. While allowing that others may look at it that way, Ashby appears to maintain that, for himself, it is against his principles.
B Classical, Relativistic or Generic Fact?
Although Ashby and others have explicitly referred to the Sagnac eﬀect as a “Special Relativity eﬀect” [70] [74], K. Brown, who has provided an in-depth analysis of the problem, points out that, actually, it is a classical eﬀect:
It’s worth emphasizing that the Sagnac eﬀect is purely a classical, not a relativistic phenomenon, because it’s a “diﬀerential device,” i.e., by running the light rays around the loop in opposite directions and measuring the time diﬀerence, it eﬀectively cancels out the “transverse” eﬀects characteristic of truly relativistic phenomena. [?]
We should add that there’s little doubt that Sagnac himself would have emphatically denied that the eﬀect that he demonstrated is “relativistic.”
Be that as it may, Brown derives the Sagnac “time-gap” four diﬀerent ways and comments: “We’re simply decomposing those absolute intervals into space and time components in diﬀerent ways.” [75] Though Brown regards it as a classical eﬀect, his analysis is clearly couched in Relativistic terminology. While a “classical” physicist (e.g., Sagnac) would not

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B Classical, Relativistic or Generic Fact?

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hesitate to describe light propagation with respect to the rotating body as manifesting a speed anisotropy with respect to the body, for the most part, Brown avoids this point of view. He points out that “Special relativity does not entail invariant or isotropic light speed with respect to non-inertial coordinates.” Yet his “space and time decompositions” are all from the point of view of inertial observers, relative to whom, he repeatedly asserts, the speed of light equals c. Indeed, in the course of defending SR against those who would suggest that the Sagnac eﬀect is somehow inconsistent with SR, Brown emphasizes the special status of inertial frames:

It’s self-evident that since the speed of light is isotropic with respect to at least one particular frame of reference, and since every other frame is related to that frame by a transformation that explicitly preserves light speed, no inconsistency with the invariance of the speed of light can arise.

As we recall from §4, however, we see that Brown is tacitly

denying

the

possibility

of

adopting

a

value

other

than

1 2

for

the synchrony parameter, . What Brown says is true “in the-

ory” if we insist that all uniformly moving observers adopt

a

1 2

.

One

can

“uphold

the

law”

(second

postulate)

by

this

approach. As Brown writes: “The second principle states that

light always propagates at the speed c, assuming we deﬁne

the time intervals...as whatever they must be in order for the

speed of light to be c.” [76] Sounds like stacking the deck,

doesn’t it? This is what we get if everyone agrees that  must

equal

1 2

.

Physics

by consensus?

Judges

and

lawyers would

perhaps be happy with it. But detectives would not.

References
1. A. Einstein, ‘On the Electrodynamics of Moving Bodies,’ (1905) reprinted in The Principle of Relativity, (Dover, New York, 1952) p. 37.
2. H. A. Lorentz, ‘Electromagnetic Phenomena in a System Moving with Any Velocity Less than that of Light,’ (1904) reprinted in The Principle of Relativity, (Dover, New York, 1952) p. 11.
3. A. Einstein, quoted by M. Jammer in Concepts of Simultaneity (Johns Hopkins University Press, Baltimore, 2006) p. 129. The original reference is given by Jammer as: “Vom Relativita¨sPrinzip,” Vossische Zeitung, 26 April 1914 (no. 209), pp. 1-2; The Collected Papers (1996), vol. 6, p. 4; English translation (1997), vol. 6, pp. 3-5.
4. H. Reichenbach, The Philosophy of Space and Time (Dover, New York, 1958) translation of original, published in 1927.
5. A. Einstein, ‘The Foundation of the General Theory of Relativity,’ (1916) reprinted in The Principle of Relativity, (Dover, New York, 1952) p. 111.

6. Sir Isaac Newton, Principia (University of California Press, Berkeley, 1934) translated from original (1686) Latin by A. Motte in 1729, revised by F. Cajori, 1930.
7. Reasenberg R. D., Shapiro I. I., et al ‘Viking Relativity Experiment: Veriﬁcation of Signal Retardation by Solar Gravity.’ Astropysical Journal, v. 234, 15 December 1979, L219-L221.
8. Vessot R. F. C. and Levine M. W. A ‘Test of the Equivalence Principle Using a Space-Borne Clock.’ General Relativity and Gravitation, v. 10, 1979, 181.
9. E. F. Taylor and J. A. Wheeler, Spacetime Physics (W. H. Freeman, New York, 1966) p. 18.
10. A. Einstein, ‘On the Electrodynamics of Moving Bodies,’ (1905) reprinted in The Principle of Relativity, (Dover, New York, 1952) p. 41.
11. A. Einstein, ‘On the Electrodynamics of Moving Bodies,’ (1905) reprinted in The Principle of Relativity, (Dover, New York, 1952) p. 45.
12. H. A. Lorentz, Problems of Modern Physics (Dover, New York, 1967) p. 221. (Original, 1927).
13. H. Reichenbach, The Philosophy of Space and Time (Dover, New York, 1958) p. 127; translation of original, published in 1927.
14. F. Selleri, ‘Sagnac Eﬀect: End of the Mystery,’ and references therein, in Relativity in Rotating Frames (Kluwer, Dordrecht, 2004) pp. 57-77.
15. J. Denker, http://www.av8n.com/physics/odometer.pdf
16. I. Ciufolini and J. A. Wheeler, Gravitation and Inertia (Princeton University Press, Princeton, 1995).
17. H. Ohanian and R. Ruﬃni, Gravitation and Spacetime (W. W. Norton, New York, 1994).
18. S. Carroll, Spacetime and Geometry (Addison Wesley, San Francisco, 2004).
19. J. B. Hartle, Gravity, An Introduction to Einstein’s General Relativity (Addison Wesley, San Francisco, 2003).
20. H. Reichenbach, The Philosophy of Space and Time (Dover, New York, 1958) p. 127; translation of original, published in 1927.
21. A. Einstein, quoted in Jammer in Concepts of Simultaneity (Johns Hopkins University Press, Baltimore, 2006) p. 181. Original reference given as: Review, Deutsche Literaturzeitung, 1928, pp. 19-20.
22. A. Grunbaum, ‘Logical and philosophical foundations of the special theory of relativity,’ American Journal of Physics, 23 (1955) pp. 450-464.
23. M. Jammer, Concepts of Simultaneity (Johns Hopkins University Press, Baltimore, 2006).
24. J. A. Winnie, ‘Special Relativity Without One-Way Velocity Assumptions: Parts I and II,’ Philosophy of Science, 37 (1970) pp. 81-99 and 223-238.
25. W. C. Salmon, ‘The Philosophical Signiﬁcance of the One-Way Speed of Light,’ Nouˆs, 11 (1977) pp. 253-292.
26. D. Malament ‘Causal Theories of Time and the Conventionality of Simultaneity,’ Nouˆs, 11 (1977) pp. 293-300.

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27. A. Janis, ‘Conventionality of Simultaneity,’ Stanford Encyclopedia of Philosophy (online article) http://plato.stanford.edu/entries/spacetime-convensimul (July 18, 2006). Also ‘Simultaneity and Conventionality,’ in eds., R. Cohen and L. Laudan, Physics, Philosophy and Psychoanalysis (Boston Studies in the Phlosophy of Science (D. Reidel, Dordrecht/Boston, 1983) 76, pp. 101-110.
28. A. Einstein, ‘Autobiographical Notes,’ in ed. P. A. Schilpp, Albert Einstein: Philosopher-Scientist (Library of Living Philosophers, Evanston, Illinois, 1949) p. 57.
29. M. Jammer, Concepts of Simultaneity (Johns Hopkins University Press, Baltimore, 2006) p. 137.
30. S. Herrmann, A. Senger, E. Kovakhuk, H. Mu¨ller, and A. Peters, ‘Test of the Isotropy of the Speed of Light Using a Continuously Rotating Optical Resonator,’ Physical Review Letters, 95, 150401 (2005).
31. E. Minguzzi, ‘On the Conventionality of Simultaneity,’ Foundations of Physics Letters, 32, Number 5 (May, 2002) p. 813.
32. A. Einstein, as recalled by A. Komar, in ed., G. E. Tauber, Albert Einstein’s Theory of General Relativity (Crown, New York, 1979) p. 185.
33. A. Einstein, Ideas and Opinions, (Crown, New York, 1982) p. 274.
34. A. Einstein, Relativity, The Special and General Theory (Crown, New York, 1961) pp. 69-70.
35. A. Einstein, Relativity, The Special and General Theory (Crown, New York, 1961) pp. 79-80.
36. A. Einstein, quoted in J. Norton, ‘What was Einstein’s Principle of Equivalence?,’ eds., D. Howard and J. Stachel, Einstein and the History of General Relativity (Birkha¨user, Boston, 1989) p. 21.
37. A. Einstein, ‘Fundamental ideas and problems of the theory of relativity,’ Nobel Lectures, Physics 1901-1921 (Elsevier, Amsterdam, 1967) p. 486. Lecture delivered July 11, 1923.
38. A. Einstein, ‘Autobiographical Notes,’ in ed. P. A. Schilpp, Albert Einstein: Philosopher-Scientist (Library of Living Philosophers, Evanston, Illinois, 1949) p. 77.
39. A. Einstein, ‘Autobiographical Notes,’ in ed. P. A. Schilpp, Albert Einstein: Philosopher-Scientist (Library of Living Philosophers, Evanston, Illinois, 1949) p. 91.
40. H. Ohanian and R. Ruﬃni, Gravitation and Spacetime (W. W. Norton, New York, 1994) p. 377.
41. M. Friedman, Foundations of Spacetime Theories (Princeton University, Princeton,1983) pp. 204ﬀ and pp. 228-229.
42. K. Brown, ‘Controversies Over the Equivalence Principle,’ mathpages, http://www.mathpages.com/home/kmath622 /kmath622.htm
43. M. Faraday wrote: “It is absolutely necessary that we should learn to doubt the conditions we assume, and acknowledge we are uncertain...In the pursuit of physical science, the imagination should be taught to present the subject investigated in all possible and even in impossible views; to search for analogies

of likeness and (if I may say so) of opposition – inverse or contrasted analogies; to present the fundamental idea in every form, proportion, and condition; to clothe it with suppositions and probabilities – that all cases may pass in review, and be touched, if needful, by the Ithuriel spear of experiment.” Experimental Researches in Chemistry and Physics (Taylor, London, 1859) p. 480. Quoted in W. Berkson, Fields of Force (John Wiley and Sons, New York, 1974) p. 57.
44. P. Anderson, original source unknown.
45. G. Rizzi and M. L. Ruggiero, Relativity in Rotating Frames (Kluwer, Dordrecht, 2004).
46. R. Anderson, H. R. Bilger and G. E. Stedman, “‘Sagnac” eﬀect: A century of Earth-rotated interferometers,’ American Journal of Physics, 62, (11) November 1994, p. 975.
47. M. G. Sagnac, ‘The Existence of the Luminiferous Ether Demonstrated by Means of the Eﬀect of a Relative Ether Wind in a Uniformly Rotating Interferometer,’ in eds., G. Rizzi and M. L. Ruggiero, Relativity in Rotating Frames (Kluwer, Dordrecht, 2004) pp. 5-7. (Translated from original: Comptes rendus de l’Acade´mie des sciences, 157, (1913) p. 708.)
48. R. D. Klauber, ‘Relativistic Rotation: A Comparison of Theories,’ Foundations of Physics, 37, No. 2 (February 2007) pp. 202-203.
49. S. Herrmann, A. Senger, E. Kovakhuk, H. Mu¨ller, and A. Peters, ‘Test of the Isotropy of the Speed of Light Using a Continuously Rotating Optical Resonator,’ Physical Review Letters, 95, 150401 (2005).
50. P. Antonini, M. Okhapkin, E. Go¨klu¨, and S. Schiller, ‘Test of constancy of speed of light with rotating cryogenic optical resonators,’ Physical Review A, 71, 050101(R) 2005.
51. H. A. Lorentz, Problems of Modern Physics (Dover, New York, 1967) p. 87. In this collection of lectures delivered in 1922, Lorentz gives two references for the earliest publication of the contraction hypothesis, one of his papers: Abhandlungen u¨ber theoretische Physik, Bd. I, p. 441; and one by Sir Oliver Lodge, who relayed Fitzgerald’s idea in: ‘Aberration Problems,’ Phil. Trans. A, 184 (London, 1893) p. 727.
52. A. Einstein, Relativity, The Special and General Theory (Crown, New York, 1961) p. 81.
53. Y. B. Zeldovich and I. D. Novikov, Stars and Relativity (University of Chicago, Chicago, 1971) p. 11.
54. M. Born, Einstein’s Theory of Relativity (Dover, New York, 1962) p. 320. Born’s remarks closely follow Einstein’s, as he writes of the problem of synchronizing clocks on a rotating body: “It is thus impossible to arrive at a reasonable deﬁnition of time...”
55. L. D. Landau and E. M. Lifschitz, The Classical Theory of Fields (Pergamon, New York, 1971) p. 253. The authors state: “Clocks on the rotating body cannot be uniquely synchronized at all points,” due to the resulting time gap. Funny how the word, “synchronization” has come to mean Einstein synchronization, as though he had a patent on the idea. If it can’t be “uniquely” done without maintaining the second postulate,

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we’ll just say what Einstein said: it’s impossible. Well, anyhow, that was before they got the GPS up and running. 56. W. G. V. Rosser, An Introduction to the Theory of Relativity (Butterworths, London, 1964) p. 460. 57. R. D. Klauber, ‘Relativistic Rotation: A Comparison of Theories,’ Foundations of Physics, 37, No. 2 (February 2007) pp. 236-239. 58. R. C. Daniels and R. H. Huxford, ‘Using Global Positioning Systems (GPS): How it Works, Limitations, and Some Guidelines for Operation,’ (January 16, 2001) http://www.ecy.wa.gov/biblio/0006015.html. 59. K. Brown, Global Interpretations of Local Experience, http://mathpages.com/rr/s7-08/7-08.htm 60. L. C. Epstein, Relativity Visualized (Insight, San Francisco, 1988) p. 205. 61. R. Swinburne, Space and Time (St. Martin’s Press, New York, 1968) p. 155. 62. M. Jammer, Concepts of Space... 63. A. Young, Which Way Out? (Robert Briggs Associates, Berkeley, 1980) p.164. 64. T. F. Banchoﬀ, Beyond the Third Dimension (Scientiﬁc American Library, New York, 1990) pp. 114-115. 65. L. C. Epstein, Relativity Visualized (Insight, San Francisco, 1988) pp. 143ﬀ. 66. L. C. Epstein, Relativity Visualized (Insight, San Francisco, 1988) p. 145. 67. G. Rizzi and A. Seraﬁni, ‘Synchronization and Desynchronization on Rotating Platforms,’ Relativity in Rotating Frames (Kluwer, Dordrecht, 2004) pp. 92, 99. 68. D. Dieks, ‘Space, Time and Coordinates in a Rotating World,’Relativity in Rotating Frames (Kluwer, Dordrecht, 2004) pp. 37-38. 69. N. Ashby, ‘The Sagnac Eﬀect in the Global Positioning System,’ Relativity in Rotating Frames (Kluwer, Dordrecht, 2004) p. 24. 70. D. W. Alan, M. A. Weiss and N. Ashby, ‘Around-the-World Relativistic Sagnac Experiment,’ Science, 228 (5 April 1985) p. 69. 71. N. Ashby, ‘Relativity and the Global Positioning System,’ Physics Today 55, No. 5 (May 2002) p. 41. 72. N. Ashby, Relativity in the Global Positioning System, Living Reviews of Relativity (28 January 2003). http://relativity.livingreviews.org/Articles/Irr-2003-1/ 73. N. Ashby, ‘The Sagnac Eﬀect in the Global Positioning System,’ Relativity in Rotating Frames (Kluwer, Dordrecht, 2004) p. 12. 74. J. F. Pascual-Sa´nchez, A. San Miguel, F Vicente, ‘Isotropy of the Velocity of Light and the Sagnac Eﬀect,’ Relativity in Rotating Frames (Kluwer, Dordrecht, 2004) p. 167. 75. K. Brown, mathpages, http://www.mathpages.com/rr/s2-07/207.htm 76. K. Brown, mathpages, http://www.mathpages.com/rr/s1-06/106.htm

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