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EOVE3IIBBR 1, 193Z

PII75ICc4L RZ VIBTV

VOLUME 4Z

Experimental Establishment of the Relativity of Time
J. By RQY KENNEDY AND EDwARD M. THQRNDIKE
University of TVcshington and Polytechnic Institute of Brooklyn
(Received July 9, 1932)
None of the fundamental experiments on which the restricted principle of relativity is based requires for their explanation that the classical concept of absolute time be modified; the present experiment was devised to test directly whether time satisfies
the requirements of relativity. It depends on the fact that if a pencil of homogeneous
light is split into two components which are made to interfere after traversing paths of different length, their relative phases will depend on the translational velocity of the optical system unless the Lorentz-Einstein transformation equations are valid. Hence, such a system at a point on the earth should give rise to an interference pattern which varies periodically as the velocity of the point changes in consequence of the rotation and revolution of the earth. The effect to be expected for a small velocity is so very small that it has been necessary to devise a special source of light, an interferometer of great stability and a refinement of the technic of measuring displacements in the interference pattern. With the apparatus finally employed, we have shown that there is no effect corresponding to absolute time unless the velocity of the solar system in space is no more than about half that of the earth in its orbit. Using this null result and that of the Michelson-Morley experiment we derive the LorentzEinstein transformations, which are tantamount to the relativity principle.
MONG the several classical experiments which suggested the restricted pri~ncip~ le of relativity there appears to be none in which any question
as to the nature of time is involved. That is, in any of them, time as indicated by an ideal clock moving with the earth might be related in any way to that indicated by a hypothetical fixed clock without at all affecting their results, at least insofar as can be inferred from such theories of the experiments as we are at present able to construct. In experiments such as those of Rayleigh and Brace, of Trouton and Noble, and of Fizeau, all of which yielded null results, there is present the theoretical difficulty that unknown properties of matter are involved. The Michelson-Gale experiment gives a positive result, which is consistent with the concepts of either relative time or absolute time. In fact, it seems that the only experiment heretofore reported that permits of any definite interpretation is that of Michelson and Morley; and the null result of this experiment is completely explained if we suppose that space dimensions in the direction of motion are contracted by an amount depending upon a suitable function of velocity; so here, too, no question as to time is raised. Hence, although such experiments have suggested the relativity theory, they do not form a sufficient basis for the logical derivation of it.
It appears, then, that the theory has needed confirmation, particularly
in its most revolutionary aspect; i.e., its denial of a significance for absolute
time. Such confirmation has been obtained in the work reported in this paper, and by combining our results with those of the Michelson-Morley experiment, we derive the Lorentz-Einstein transformations which are well known to embrace the whole theory.
400

RELATIVITY OIi TIME

The principle on which this experiment is based is the simple proposition

that if a beam of homogeneous light is split at a half-rejecting surface into

two beams which after traversing paths of different lengths are brought

together again, then the relative phases of the superposed beams will depend

upon the velocity of the apparatus unless the frequency of the light depends

upon the velocity in the way required by relativity. Furthermore, the phase-

difference can be made to determine the positions of fringes in an interference

pattern, so that by measuring these positions for various velocities of the

system, the question whether the frequency follows the relativity require-

ment can be decided. The variation of the velocity of the system comes about

because of the motions of rotation and revolution of the earth.

The theory of this experiment requires the following two assumptions:

(a) There exists at least one coordinate system in which Huyghen's principle

is valid and the velocity of light is the same in all directions. This assumption

is unobjectionable from the standpoint either of relativity or of any plausible

hypothesis involving an ether; for relativity, it is true for all uniformly

moving systems, and in the latter case for any system at rest in the ether.

(b) The Michelson-Morley experiment indicates that a system moving with

uniform velocity v with respect tion of motion contracted in the

troatsiuoch[1a—ssy's/tcem']'"

has as

dimensions compared

in the directo dimensions

in the fixed system, while dimensions perpendicular to this direction are un-

changed. This is in part assumption, for although there can be little doubt

that the experiment yields a strictly null result, nevertheless it actually

shows only that dimensions in the direction of and perpendicular to the mo-

tion are in the ratio mentioned; either of these dimensions might be any

function of the velocity so long as that ratio is preserved.

A
Fig. 1.
Let us consider one such system S', and suppose that a system S (attached,
for instance, to the surface of the earth) moves practically uniformly with
velocity v with respect to it. In S is set up an arrangement for producing
interference; i.e., one in which a pencil of homogeneous light is divided as
mentioned above into two pencils which are recombined after traversing paths of different lengths. We can simplify the discussion by treating the general case instead of the particular arrangement used in the experiment, and by adopting a rule regarding expressions for the distances, angles and
times in system S that will be of interest; i.e., the magnitudes of these quan-
tities will be expressed by unprimed letters when they are referred to standards moving with S, and by the same letters primed when referred to stand-
ards fixed in S'. The path of a typical ray with respect to S can be represented schemati-
cally as in Fig. 1 where the ray coming from the left is divided at A. into rays
1 and 2 which recombine at B. The courses of the same rays with respect to

402

R. J. XZEXEDY AND B. 3II. TIIOREDlEE.

S' are evidently determined by the requirement that to each element ds' of
a ray is to be added an elementary vector vdt' where dt' is time required
for light to traverse the element, and c is the velocity of light with respect to
5'. The length of the resulting element is evidently cdt', hence c'(dt')' = (ds')'
+v' (dt')-'+2v ds'dt' cos O'. Hence,

— + dt'

=

c(1

ds
—P')

[P

cos

9'

(1 —P' sin' 9')'(']

where p = v/c and 9' the angle between v and the element ds'.
If for the moment we consider a set of rectangular coordinates in 5 and S' with corresponding axes parallel and x-axes parallel to velocity v, we have

from assumption (b)

d ' = [(d ')' + (d. ')'+ (d")']"' = [(d- )'(1 —P') + (dy)'+ (d ) '1"'

— = ds l. —P'

a- i/z = ds(1. —P' cos' 9)'"

ds

—=— — cos 0' = dx' ds'

"' dx(1 —P')
ds(1. —P' cos'

= 9)'"

cos0

1 P2

1/2

P' cos' g

sin' 9' = sin' 9/(1 —P' cos' 9).

When these expressions are substituted in Eq. (1), it reduces to

+ dt' = [ds/c(1 —P')(('] (1 P cos 9),

(2)

the right side of which equation involves only quantities referred to standards
moving with S. The time for light to traverse the whole ray AB along path 1
is therefore

f + f dt( = 1/c(1 —P')((~ (1 P cos 9)ds

1

1

and a similar expression holds for path 2. Hence difference of time for the two paths is

f' —t' = 1/ (1 —0')'~'[

— + d

~f d

l3

6d — f 9d

1

1

2

The term in brackets multiplied by P vanishes, since in order to interfere the rays must intersect, and therefore their projections on the line joining A and
8 are equal; these projections are the integrals in brackets. Hence
t&' —tz' = — (s& s&)/c(1 —P')"' = As/c(1 —P')'" say

and the number of waves corresponding to this difference of time is
I = v'(tg' —t2') = v'As/c(1 —P')"'

where u' is the frequency of the light employed as measured by an observer
in S . This number n is seen to be independent of orientations, lengths and

RELATIVITY OF TI1IIE

403

dispositions of paths, but to depend upon difference of path-lengths, the
relative velocity of S and S' (through P) and the frequency.
The foregoing treatment is strictly valid. only if the moving system is regarded as not subjected to forces, but is undoubtedly suf6cient for the purpose in the small constant field of gravitation and acceleration at the surface of the earth. Moreover, although the rotation of the apparatus with the earth involves a slight effect on the time difference computed above (whether regarded from the standpoint of relativity or classical theory), it turns out to be altogether negligible in amount. This effect is a function of rotational
velocity, not of orientation of apparatus. We have now to consider the effect of a change in the velocity v on the
number n expressed by Eq. (4). In that equation c is evidently a constant, while the difference As, because it is referred to standards moving with the
system, is constant unless the courses of the rays between the points of separation and recombination are dependent on the velocity; that this is not the
case can be shown by Huyghens' principle. A direct consequence of this principle is that the course of the ray is determined by the condition that the time required for traversing the path is a minimum compared with the time for any neighboring path. Now, Eq. (3) expresses the time in terms of coordinates moving with S, and if minimized in the usual way would yield the equations of the paths. For the present purpose, however, it is unnecessary to carry out this operation. Rewriting (3) we have

It will be observed that although the expression involves the velocity of the

moving system, nevertheless the course of the ray is quite independent of it.

That this is so is evident from the following considerations: the second

integral is equal to the projection of the path on the line joining A and 8,

and being the same therefore for all paths, cannot contribute to thedeter-

mination of the minimizing path. The first integral is expressed in terms of

distances referred to standards moving with the system and so is independent

of the velocity. Hence the actual courses of the rays, which are got by mini-

mizing integrals of this form, are independent of the velocity, and As is a con-

stant. This proof is essentially that of Lorentz extended by the inclusion of

the contraction hypothesis.

The quantity v' in Eq. (4) is the only one whose possible variability with

velocity remains to be considered.
=v(1 —P')'~' where v is the constant

From the standpoint value of the frequency

of relativity, which would

v
be

determined with standards moving with S; this value of v' would evidently

make n a constant. Furthermore, it will be shown later that insofar as the

atom is to be regarded as a typical clock, the Lorentz-Einstein transforma-

tions hand,

can be derived from this relationship
v'Wv(1 —P')'" these transformations

and assumption do not apply

(b). If, on the other and it turns out

that there exists but one system S' satisfying assumption (a); this unique

system would be the absolute reference frame postulated in the classical

404

R. J. KENNEDY AND E. 3II. THORNDIKE

ether theory. In .this case n is evidently a function of the velocity of S with

respect to the absolute reference frame. Evidently, then, the relativity

hypothesis can be tested by determining whether n is constant as v changes

in consequence of the motions of rotation and revolution of the earth,

For the present purpose the total velocity of the apparatus can be got by

adding vectorially a presumably constant velocity vp of the sun, the orbital

velocity v& of the earth and the circumferential velocity v2 due to the rotation of the earth (taking account of latitude). Its square can be reduced to

+ + + — + — + — = v

vp

vg

'v2

2v vy sin (8~ co~) 2vvv, sin (8, &u2) 2v~v~ cos (8~ 8~),

where v is the projection of vp on the orbital plane, vp is projection of vp on

the equatorial plane, co& and co2 are constants related to the direction of vp,

and 0~ and 02 are angles expressing the position of the earth in its orbit and

its orientation on its axis with respect to the fixed-stars. This procedure as-

sumes only that the fixed-star system has no great angular velocity with
respect to the fundamental system S'; there is an unimportant approxima-

tion in the last term.

In order to get an idea of the magnitude of the effect that might be ex-

pne=cAtesd/,X(l1et—uPs ')a'"ss.uEmxepanthdiantg

v'=v
this,

and replace v ignoring terms

by
in

c/X; then (4) becomes P ahorse second degree,

substituting for the velocity from the expression above, and gathering con-

stant terms into one,

e = (As/1)(1+ ,'(v'/c'-) + ) = — + — + (As/Xc') [v v~ sin (8& o&q) vvvv sin (8q &aq) J a constant
= 8n+ np.

Here the variable part of n is represented by 8n and the constant by np and we assume v and vp to be large compared with the orbital and circumferential velocities v~ and v2. Hence 8n should be proportional to the sum of a term with a period of a year and one with a period of a sidereal day.
In performing the experiment, we wish, of course, to make Sn as large as possible. The only factor that can be controlled is the ratio As/X, the largest feasible magnitude of which is a measure of the homogeneity of the light. For various reasons the most suitable light seems to be the mercury line of wave-length 5461. With this, sufficiently clear interference fringes could be got when As was as large as 318 mm (the value finally used) and on substituting this in the expression for n it turns out that the rotation of the earth would produce a daily variation of a thousandths of a fringe for 200 km per
' More specifically, 8& is the angle between the projection of vo on the orbital plane and a
direction in that plane determined by the angle ~1 which depends on the position of the earth in its orbit or the time of year at which 01 is taken as zero. Similarly, 0& is the angle between the projection of vo on the equatorial plane and a direction in that plane determined by the angle cd which depends on the time of day at which 02 is taken as zero. In the reduction of the data, the 8's are taken as zero at the beginning of each run, so the cv's depend on the times of starting runs. In the comparison and final summary of data, the 8's are of course referred to the same sidereal time.

RELATIVITY OIi TI3EE

405

second, while the orbital motion would produce the same variation in six
months for 3 km per second.
Because of the probable minuteness of these effects it was necessary to contrive new ways of detecting them. In the rather complicated method first proposed' the phase variation would show itself in the rotation of the plane of polarization of a beam resulting from the superposition of two oppositely circularly polarized beams. This scheme, although theoretically capable of great precision, was abandoned in favor of the much simpler one finally employed. In the latter, ordinary interference rings were formed and
photographed, and the problem became one o:f measuring very small changes
in the diameters of the rings. It was satisfactorily solved by devising a special
comparator which will be discussed later. Evidently, it was necessary to take every precaution to keep the experi-
mental conditions constant; we were able, in fact, to reduce the average d-ily periodic error in As/X to about two parts in 10"'. This great stability was attained mainly by using interference apparatus made almost entirely of fused quartz and kept in a vacuum at a temperature constant to within about a thousandth of a degree. The apparatus was furthermore (partly accidentally) compensated for temperature to such an extent that one degree change produced a shift of only about a hundredth of a fringe. The vacuum was employed as simplest way to eliminate variations in pressure, which would have caused variations in index of refraction of optical paths, and, by mechanical action, variations in lengths of paths.
Several disturbing factors producing spurious effects had to be dealt with. Perhaps the most troublesome was the variability in density of the photographs due (in the earlier green-sensitive plates) to rapid aging which affected the emulsions in varying degrees. Since the photographic effect of light is not proportional to its intensity, it follows that a spurious displacement of an interference pattern of the type used is to be expected if the density of the photographs is not constant. The methods adopted to eliminate this and other difficulties are discussed elsewhere in the paper.
APPARATUS AND EXPERIMENTAL PROCEDURE
The general arrangement of the experimental apparatus is sketched in
Fig. 2. Light from source S passes through a small circular opening in screen
S&, is rendered approximately plane-parallel by lens L& is dispersed in direct
vision prism P and the green (X5461) image of first opening is focused over a
second one in screen 52 by lens L2. The water-cell C is to absorb stray heat radiation. The green light from second opening is polarized by nicol prism N so that the electric vector is horizontal, and. then enters the vacuum chamber V through a window and is concentrated by lens L3 to the extent required to produce the greatest intensity in interference pattern. The light is then split into two pencils at the half-reflecting mirror M» which is inclined at such an angle (Brewster's angle) that reflection of the polarized light occurs only at its platinized face; the faces of the compensating plate 3ll4. are equally
2 Kennedy, Phys. Rev. 20, 26 (1922).

406

R. J. EEPcVEDY Ale E. 3II. TIIORXDIEE

inclined. Hence no stray (non-interfering) light can be superposed at these faces on the two pencils from 2&I&, these pencils are rejected by mirrors 3II2
I I' and 3553 back to 3EI~, at which one is partially transmitted and the other
partially reflected through lenses 4 and which focus the light as a system of interference rings on a wide horizontal slit just in front of a photographic
plate in the holder II. The slit is 5 or 6 mm wide, so the plate receives a sym-
metrical. central section of the interference pattern of that width. The plate
is held by a spring in the holder lightly against the metal tube T which is sealed against the window S" of the vacuum chamber. Most of the length
of the tube as well as the vacuum chamber is within the tank V containing

Fig. 2.
water at a temperature constant to within less than 0.001'C; hence slight
variations in room temperature cannot affect focusing and thereby diameters of rings. The plate holder, which is kept from contact with the vacuum chamber in order to preclude the possibility of jarring the latter when the holder is operated, is arranged to let the plate slip down two slit-widths automatically every half hour. On each plate six photographs are taken consecutively in this way, and twelve hours after the start of the first series six more are taken in the spaces left vacant during first exposure of the plate; hence the developed plate will contain a series of photographs alternately taken twelve hours apart. The purposes served by this procedure will be explained later. Four such plates are taken during a day's run.
The temperature of water-bath was easily kept nearly constant for many wt:eks in succession. The temperature was chosen only slightly above that of

RELATIVITY OF TIDDLE

407

room, the water was circulated continuously and the mercury-toluene ther-
— mostat was arranged to control the potential of the grid of a vacuum tube
which actuated the relay in the heating circuit in this way only a minute
current is broken at the mercury surface and it does not become contaminated with a 61m of oxide. The optical part of the apparatus was enclosed in a small dark room within a larger one. The temperature of the inner room was kept constant to within a few hundredths of a degree, that of outer room to within about a tenth.
The interference apparatus consisted essentially of a set of four interferometer plates of the best quality obtainable, mounted on a circular fused quartz base 28.5 cm in diameter by 3.8 cm thick. The method of mounting the plates is perhaps worth describing: the support of each plate was cut from a fiat plate of fused quartz, and fused to a tapered plug of the same material which, after being ground to fit a tapered hole in the base, was etched away

ggrror

ggrtz go,se
Fig. 3.
over the whole conical surface except in four spots of two or three square millimeters area, which were therefore the sole points of contact of the plug with the base. This procedure was necessary to insure a definite 6xed position
of plug, since if it were merely ground into the plate it would probably fit the hole only in a region near its middle. The positions of the bearing spots are indicated by the small squares in Fig. 3, the dotted one being on the opposite side of plug from the others. The plugs were held down by light springs as shown in the figure. The end mirrors were circular etalon plates 25 mm in diameter. Their supporting frames were fashioned so as to provide three projections of quartz against which the mirror was held by a light spring opposite each projection. The faces of the projections were ground Hat and so as to be very nearly in a vertical plane when the frame is in place on the
base. Final adjustment of mirrors was made by rotating them about their horizontal axes; it will be evident that in this way (because of slight in-

408

R J KENNED 7 AND E 3II THORNDIKE

clination to each other of the faces of the mirror) a very fine adjustment can
be made. It was sufficient simply to rotate the mirrors with the unaided
fingers, to correct for the departure from the vertical, while viewing the interference rings with a telescope. The reflecting surfaces were of platinum
applied by cathode deposition. It was impossible to use silver for the purpose
because traces of mercury vapor in the vacuum chamber would quickly dis-
solve it. The light lens system which formed the rings on the photographic
plate was attached to the base by means of invar plugs similar to those described above.
The quartz base rested on a piece of uniform velour, the back side of which was cemented to a heavy flat brass plate which was supported in an accurately horizontal position at three points. Each fiber of the nap of the velour thus served as a tiny spring so that the weight of the quartz plate was evenly distributed; this is important, since a fused material of this sort is essentially only semi-solid. The friction between the velour and the rough bottom face of the base sufficed to hold the latter accurately in position.
In order to produce interference under the existing condition of large
difference of paths of the two beams, the image in the half-reflecting mirror of the face of either end-mirror must be nearly parallel to the face of the
other end-mirror; it will be shown that such an adjustment of the mirrors gives rise to a pattern consisting of a series of concentric circular rings. In order that the effective diameter of each ring may be sensibly independent
of accidental variations in distribution of light intensity over the faces of
mirrors and with respect to direction in the beam, it is necessary to make this parallelism very accurate. The accuracy of adjustment could be tested by
the simple procedure of moving a broad slit in various directions across the pencil incident on the half-reflector while the rings were observed in a tele-
scope or photographed; when the diameters of rings were constant for all positions of slit the adjustment was the best obtainable.
The particular spectral line employed in the experiment was chosen on basis of several requirements. As has been pointed out, it must be capable of producing interference with large path-difference; it must also be entirely controllable as to intensity, the intensity must be fairly large, and the line must be easily separable from adjacent ones. On the whole, these conditions seemed best satisfied by the line X5461 of mercury. The homogeneity of any light is roughly proportional to the inverse square root of absolute temperature of source; hence the first source employed was a mater-cooled mercury arc. This produced excellent interference rings, but it was soon noticed that their diameters depended on the part of the arc from which the light was taken; this suggests a Doppler effect due to motions of evaporating molecules from the hot liquid surface where the arc was brightest. Such an effect due to velocities variable by only a few centimeters per second would evidently be objectionable in view of the stability required.
The source finally used was an electrodeless discharge in unsaturated mercury vapor. The tube is sketched in Fig. 4. The inner tube in which the discharge took place was connected to a continuously operating pumping

RELATIVITY OF TIDDLE

409

system through a capillary tube (heated to prevent condensation in it) of such length and diameter as to keep the pressure of the vapor just below that of saturated vapor at the existing temperature. The vapor was supplied by the mercury well at the rear of tube, and the small amount escaping through the capillary would condense and return by way of the other vertical tube. The temperature of the source, and thereby the pressure of vapor, were kept constant by means of carbon-tetrachloride in the jacket surrounding the inner tube; the liquid was maintained at its boiling-point by heat from the discharge, and its vapor was condensed and returned by the water-cooled
condenser connected to top of jacket. It will be evident that with the dis-
charge occurring at some distance from the mercury mell, erst-order Doppler effects would be eliminated since no mercury condenses in the forward part
f;o pumps

ca pi(tory

«ndenser

~meelrcE ury Fig. 4.

of the tube and therefore velocities of vapor molecules are on average same

in all directions. Electrical energy was supplied by a coil of some thirty turns

of wire around the outside of the jacket, in which oscillations of 20 meters

wave-length were produced by a 75-watt transmitting tube. This discharge

produced a uniform steady glow over nearly the whole diameter of the inner

tube, and the interference rings were completely free from the fluctuations

in brightness and diameter which were visible with the ordinary arc. During

a run, and for some time in advance of it, the tube was kept in continuous

operation in order that all conditions should be steady. It was found that

the frequency of the light depended on the temperature of the cooling liquid

and the voltage applied to oscillator, so these factors had to be closely con-

trolled. These green line; its

e"fffreecqtusenpcryo,ba"blays

arise from the complicated structure of the inferred from the interference pattern, is of

course a sort of mean of the frequencies of its components, weighted accord-

ing to their intensities. It is to be mentioned that each of several attempts to

410

R. J. KENNEDY' AND E. M. THORNDIXE

use sealed-off tubes failed; after a few minutes of operation with such tubes the rings would disappear, presumably because the oscillatory discharge readily excited a green band in traces of oxygen which probably remain in tube.
In view of the theorem of Lorentz previously discussed, the usual theory of interference for stationary systems can be applied directly to the present
8 situation. In Fig. 5, A represents the surface of one end-mirror and the
1
R

B

Fig. 5.

Fig. 6.

image of the other at distance / from A. Since A and B are parallel, the ray
R impinging on both at angle 0 produces on reHection the two parallel rays EI and R2. If these are brought to a focus, the difference between the lengths of their paths will evidently be ah+be. Now
+ + ab = l/cos 8, be = ab cos 28. ab be = (l/cos 8) (1 cos 28) = 2l cos 8.

For constructive interference, this path-difference must contain an integral

number of waves; hence the cones of rays for which 2l cos 8;(i=1, 2, 3,

)

equals a series of consecutive integers' can be brought to a focus as a series

of concentric rings of radii r;=(sin 8,)/k~, where k~ is a constant depending

on magnification of lens system producing the interference pattern. Now

ah+be is n =2v'l(1

—thek&'qr,u')a'n"t/icty(1

ds in Eq.
—P')'~'. It

(4); hence As;
is convenient

= 21 cos 8; =
to consider

2l(1 —k&'r
only the

)'~' and central

ray, and to express its phase in terms of the radii of the rings. For this ray
r=0, so

I n. = 2vll/c(1 —I3')"' = +. p

(6)

where no is an integer
ence n =no —i. Then

and

p a fraction.

In general,

for constructive

interfer-

no —i = [2v'l/c(1 —P')'"](1 —kg'rP)'" = (No+ p)(1 —kg'rP)'".

i+ p = (No —i)/(I —ki'r')'" —No = no — a(eo —i)kPr —e,

= (k/2)r 2

(7)

approximately; here k is a new constant. The approximations since n is of order 10'&(i and kIr; has a maximum value of about

are 10

'jufsotirfitehde

rings measured. From Eq. (7) we find on differentiating

3 When the distances are expressed in wave-lengths.

RELATIVITY OF TIME

bp = kr;Sr; = kr;8r; =

(8)

If measurements br; of the values of the variations in r; are made for each of a number of rings of orders m to p, the mean value of Sp computed from
them is

(9)

It will be convenient to have 6p in another form. In the final summary of data
there are many values of Sp to be averaged for each value of the hypothetical
velocity. It is clear, then, that the final average will be unaffected if we re-
1/r„it. place the variations br; in (8) by their individual measured values hr, , so that
r;5r; =r;br;. Multiplying and dividing the right side of Eq. (9) by g~
becomes

when the expressions (r;(r;)8r; that appear in the product are replaced by pr;. Since we are dealing with extremely small variations in the radii, the radii can be measured and Z1/r; computed once for all for a given adjustment of apparatus; then the variations 8p are simply proportional to the sums of the variations in the several radii. This possibility greatly expedites the labor of measurement of plates; the way in which it was employed is discussed in connection with the description of the comparator designed for the purpose.
It should be remarked that in this procedure insufficient weight is given to
the somewhat greater precision of measurements on the larger, sharper, rings; however the final weighting of data is based on mean deviations of the computed values of 6p, and the conclusions as to precision are not vitiated by this approximation.
The principle of the comparator is as follows:
A diametral section of each photograph to be measured is made to appear juxtaposed with a similar section of a nearly identical photograph which is used as a standard of reference for the whole series. In this way very small differences between reference and measured plates reveal themselves. The juxtaposition is along a diameter of each of the systems of concentric rings, and the comparison is made by moving the standard until one side of a ring on one plate appears to be continuous with the corresponding ring on the other plate, and then noting the distance along the line of demarcation which the standard must be moved in order to make the other sides of same rings coalesce similarly. This distance is evidently the difference between the diameters of the two rings. On the shaft of a fine micrometer screw which moves the reference plate, and concentric with it, is mounted a graduated slip-ring arranged so as to be held stationary when the portions of interference rings on one side of center are matched, and to rotate with the screw when the matching is done on the other side of center; the angle through which slip-ring is rotated during settings on a number of rings is thus evidently

412

R. J. KENNEDY AND E. 2'. TIIORNDIKE

proportional to the sum of the differences of their diameters from those on corresponding rings on reference plate. Hence in view of Eq. (10) a single reading (of this angle) summarizes the measurement of the whole exposure.
Nine or ten rings, alternately dark and light, and near the center, were
usually measured.
The device is diagrammed in Fig. 6; there I' is a pair of similar right-
angled prisms cemented together on their diagonal faces (one of which is halfsilvered) and mounted on a carriage which can be moved by the micrometer screw on ways perpendicular to the section represented. The heavy lines s&
and s2 represent thin metal strips covering half the right and bottom faces
of the prism combination; the lower edge of s~ and left edge of s2 are ground accurately straight and the strips are cemented to the prisms in such a way that the image of the former edge in the diagonal mirror exactly coincides with the latter. After traversing a water-cell a beam of light from right of
figure passes through the reference plate R, which is mounted on the carriage and has its em'ulsion side in contact with screen s~ along a diameter of the
ring system; another beam, by way of mirror 3EI, illuminates a similar part
8 of the photograph on the plate which is to be measured, and both parts
are viewed from above through a lens system magnifying about four times. The latter plate is held by springs against stops which fix position of the emulsion side regardless of thickness of plate and'of course at such a distance as to eliminate parallax. The exposures can be compared in turn by sliding the plate to right or left of diagram (toward or away from operator). Since the ring system may not be exactly circular and also in order to expedite placing the plates in position for comparison, a sharp notch was cut
in each end of the slit behind which the plate is held during exposure; this
leaves a sharp point at each end of the photograph which serves for setting
accurately a1ong the same diameter. It is to be noted that the comparator is
automatically compensated for temperature (both reference and measured photographs being on same material); this compensation was not particu-
larly important for the present purpose because the scheme of interleaving photographs taken twelve hours apart secured the same result.
So accurately and quickly can the settings be made that the measurement of a photograph can be made after some practice with a probable error
of a thousandth of a fringe (i.e., a thousandth of the shift that would be pro-
duced by changing path-difference by one wave-length) in about five minutes. The labor of comparing the 48 exposures comprising a day's run is thus not
great. It was particularly desirable to be able to make rapid measurements
during the numerous preliminary adjustments of apparatus, tests of effects of varying the several experimental conditions, etc.
Two precautions were taken in order to keep the operator from being influenced in making settings on the comparator. The slip-ring, on which could be read the average differences of the diameters at any stage of comparison of a particular exposure, was kept covered until the final setting was made, thus preventing unconscious corrections during the later settings. Also, the plates were marked in such a way that the operator was in com-

RELATIVITY OF TI3M

413

piete ignorance of times of day at which they were exposed; not until a full
day's readings were finished were they arranged in chronological order for
computation.

DATA AND RESULTS FOR DAILY EFFECT

It was intended when the experiment was proposed to look chiefiy for an

effect of a change of velocity due to the orbital rather than the rotational

motion of the earth. However with the first apparatus constructed, in which

the mirrors were mounted in invar frames, it was found impossible to elimi-

nate a slow, rather irregular variation in the interference pattern which

would have masked the effect sought; hence it was decided to concentrate

on the possible rotational effect. Three series of data were taken with this

apparatus (in April and October, 1929 and January, 1930); after an interrup-

tion of over a year, during which the apparatus was rebuilt in its final form,

three more series were taken in May, July and August 1931.The same form

of light source was used in all six series. A large amount of data previously ob-

tained with the water-cooled arc and under less carefully controlled conditions

are ignored in the summary because of necessity of applying doubtful cor-

rections to it. No corrections have been applied to the data here presented.

Where results of the several series are combined, they are weighted in ac-

cordance with the usual theory of errors in terms of probable errors corn-

puted from the mean deviations.

a

Each of the time we may

series regard

— extended over
sin (8~ co&) in

a period
Eq. (5)

of only a few days; during such as virtually constant. From this

equation and Eq. (6), 8m =8p+a constant. Since 8q is proportional to 8& we

have from Eq. (5)

— + + = 8p a sin (8& u&z)

N2

k'

where e, b and k' are constants, the last two including any slow uniform

variation such as might result from stresses in the apparatus. Letting com-

puted values 6p; correspond to angles 0;, we have according to the principle

of least squares the condition that the most probable values of a and co2 are

those for which

+ — Q — — = Z(8p 8p~)'

[a sin (8; (ag)

M; —8p;]'

is a minimum. When account is taken of the fact that the data are distributed uniformly over the day, we infer from this condition that

— + m
a = (2/m) +8p; sin (8„. o&~)

2b cos A&2

1
— + = tan &o2 Z8p; cos 8f/(Z8p; sin 8, mb).

The constant b can be computed by comparing mean values of 8p on successive days; m is a number of exposures per day, usually 48.
Incidentally, the last two equations show the importance of the pro-
cedure of interleaving the exposures so that adjacent ones on any plate are made twelve hours apart. For it is known that the photographic emulsion

414

J. A. E'ENNEDY AND Z. 3II. THORNDII('A

is subject to shrinkage which varies from plate to plate, the plates may be slightly curved, and there are probably variable stresses in the apparatus due to different weights of plates; there is also a slight effect, on measured diameters due to varying densities of photograph such as mould result from different treatment and sensitiveness of plates. All of these errors are evidently automatically eliminated in the process of computing, however, since each is multiplied into a sine or cosine term in 0; and then added to a similar product into a term of opposite sign. Compensation is made also for the greater shrinkage of emulsion near ends of plates, because the plates were started alternately one and two slit-widths from the end.

TABLE I.

2.0

2.4

2.0

(4)

0.7 0.2

2. 1

3.2

(8)

0.9 3.4

2.0

1.0

(12) 1.4

1.3

2. 0

1.7

(16)

1.0 0.8

0.3

(20) —01..22

1.6

1.5 1.7

(24)

0.8

(25)

2. 2 1.6

2. 2
1.1 1.1

2. 1 1.4
(32) 0.6
2. 2
0.0 0.9 (36) 1.0
2. 1 0.2

2' 1

(40)

0.4 1.1

1.3

2.5
(44} 0.7 0.3 0.6

2.4

(48)

2. 7

B

D

—2. 1 0. 1 0.7
——011'..938 0.8 0.0 1.8 1.6
—01..49

—0.26 0.03 0.27
— —001...491520 0.63 0.00
1 ~ 56
1.48 —01..4803

1.7 — —111...571
0.0 1.3
2. 8
0.6 0.6 —21..47 1.2

1.70 — —111...406976
O. OQ
1.03 1.98 0.37 0.30 0.92
Q
0. 16

ZN; sin 0;=3.57 Zu; cos 0;=4.78 tan co2 = —4.78/3. 57, co2 = 207' sin co2= —80, cos ~2=+0.60

a = (2/48) (0 60 X3.57+0.80 X~8) =0.26

A sample of data for a period of three days and the computations for the resultant amplitude and phase of the sine curve to which it most closely
conforms is given in Table I. The numbers in the two columns at the left are
means of the three values of 8p at the same hour of each day, arranged in
chronological order. Column A contains sums and differences of the four
terms in the previous columns for which the sines of the corresponding phase
angles are equal or opposite. 4 Column 8 contains products of the terms of

4 The summation in the formula above for the amplitude can evidently be expanded as follows.

— = — m Sp5 sin (05 555)

48

48

cos 555+ 8p5 sin e; sin 555+ bp5 cos e;

1

1

1

12
= cos 5555+ (Sp;+Sp55
1

'5p5555 8p45;) sin e;

RELATIVITY OF 1'IMZ

column A into the sines of the corresponding phase angles. Columns C and
D contain the corresponding quantities to A'and B, using the cosine instead
of the sine.
The results for the daily effect are summarized in Table II. The column
headed ~ contains the phase angles corresponding to the sidereal time of the maximum value of 5n. The amplitudes are expressed in thousandths of a
fringe.
TABLE II.

Time of year

Weighted amplitude

January

0. 16

890

April

0, 27

273

May
July August October

0. 18 0. 14 0.30 0.22

18 43
128 183

Since the total velocity of the earth could vary during the year by no

more than twice the orbital velocity it is probably as well to average these

results without reference to the first term in Eq. (5) that is, by simply adding

them vectorially. When that is done, the amplitude of the resulting sine

curve is velocity

0.06+0.05. Substituting Vp = 24+ 19 kilometers per

in (5) second.

t'his

is found

to correspond

to a

SEARCH FOR LONG PERIOD EFFECT
Because the apparatus in its final form appeared to be permanently in adjustment and the average values of the ring diameters proved to be nearly constant, it became feasible to test whether an effect exists due to orbital
motion, i.e., to determine the coefficient V in first term bracketed in Eq. (5).
The direct way of doing this would evidently be like that for daily effects
which has just been discussed, i.e., to determine 8e for a large part of a year
and fit the data to a curve of the required form. Instead, a modification of this procedure was adopted in order to make it unnecessary to keep all the
experimental conditions the same for long times. It is based on the assump-

-) — — 12 sin au2+ (sp; &s26, hpM+4—+&n4s cos 8. 1

The terms in parentheses in the last two summations are the quantities in the columns A an

C,

respectively.
' There is a

superficial

appearance

that this result conflicts with the assumption

introduced

in Eq. (5), i.e., that v& and v2 are riegligible in comparison with e„and vp, since the value just

determined for vp is even less than the orbital velocity v&. The contradiction is merely apparent

however; it arises from the adoption of a corresponding velocity as a means of expressing the

accuracy of the result, as has been customary in discussions of the Michelson-Morley experi-

ment. From that experiment it is not inferred that the velocity of the earth is but a few kilome-

ters per second, but rather that the dimensions of the apparatus vary very nearly as required

by relativity. From the present experiment we similarly infer that the frequency of light varies

conformably to the theory.

416

R. J. KENNEDY AND E. 3II. TIIORNDIKE

tion that the most probable rate of variation of 5n (computed from measured values of 8p over short times) is equal to the derivative of the most probable first term in Eq. (5). Each of three series of data, taken for periods varying from eight days to a month, and at intervals of three months, was used to compute the daily rate of change of p8 at those times of year. This rate was found by averaging arithmetically the readings of each day of a given series and determining by the method of least squares the slope of the most probable straight line represented by them. Similarly the most probable sine curve corresponding to these three derivatives is computed. Some 300 exposures comprised the three series.
The three computed rates of change were 0.050+ 0.020, 0.007+ 0.013 and
—0.015+0.021, all expressed in thousandths of a fringe per day. The com-
puted sine curve has an amplitude of 2.96 thousandths and this corresponds
to a velocity V = 15+4 km per sec. Since the relatively small probable error
is based only on the internal consistency of the data and is therefore not
to be taken very seriously, this result can scarcely be regarded as indicating a real velocity. Furthermore the direction of the computed velocity is 123'
away from that computed above. As we have used only 300 exposures in the application of this method, it
is evident that the accuracy could be increased by a large factor if data were taken steadily for a few months. The proverbial brevity of life, however,
argues against laboring the point. If the last result and that for the rotational effect are given the same
weight and combined vectorially (ignoring difference of direction of V and
Vp' their resultant is 10+10 km per sec. In view of relative velocities
amounting to thousands of kilometers per second known to exist among the nebulae, this can scarcely be regarded as other than a clear null result; it is of the same order of precision as that of the Michelson-Morley experiment.
It is perhaps best expressed as at present in terms of a velocity, although of
course the conclusion to be drawn is that the frequency of a spectral line varies in the way required by relativity. This appears to be the only investigation in which a quantum phenomenon is shown to conform to Einstein's theory.
Insofar as the radiating atom may be regarded as a typical clock, the result of this experiment can be combined with assumption (b) to derive the Lorentz-Einstein transformations. Throughout the foregoing discussion we have dealt with time regarded as measured only at a fixed place in the moving
system S; in order to specify unambiguously the time at another point of S it is necessary to specify the operations which define it. Perhaps the most
natural meaning to attach to the concept is that the time at any point is the indication of a clock which has been moved with infinitesimal velocity to the
' The two results can be combined only by making some approximation.
'f lt is of course altogether possible that there is a real (inherently observable} velocity
which is so nearly perpendicular to the orbital and equatorial planes as to have components in them small enough to have escaped observation, but the probability seems small in view of the nebular velocities mentioned above.

RELATIVITY OF TIME

417

point, and from the same location as an identical clock with which it was originally in agreement; it turns out that this definition is equivalent to that
of synchronizing by means of light signals. We have shown that the frequency v' of an atom moving with velocity
v bears the relation v' = (1 —v'/c')""v to that of a fixed atom. Let us assume
that the indications of clocks under similar conditions bear the same ratio.
Suppose that at time t=t'=0, the origins of parallel coordinates in S and S'
(previously defined) coincide, and that the S-clock passes through the origin
with a small velocity with respect to S. Because of this motion, the velocity of the clock with respect to S' will have components, say, v+u„u„, u„hence the times t' and t indicated by a clock in S' and the clock in S will thereafter
stand in the relation

1—(v+u )2+ u 9+ u 2 //2
C

p2

t"I' — 2t&2u v

—1/2

C

C

Now hence

Nu, t,'t'i=s (e1qu—alv'/tco')'S"'x-m. eaIfsutrheis

of is

distance x substituted

traversed in the

in S by the
second term

clock; of the

right side of the above equation and u is made to approach zero,

1

+ + 'vx

~2 x2 1/2

t1

(1 v2/c2) 1/2 c2

g4 t2

Here x/t is the velocity of clock with respect to 5, and it approaches zero
with u; hence the coefficient of t in the last expression is unity, and
t' = [1/(1 —v'/c')'"] [t + (v/c')x]

The statement that the systems are in uniform relative velocity, together with fact that t' is independent of y and s implies

+ x' = x'(x vt); hence Bx'/Bx = (1/v) (Bx'//tt) .

(12)

The measurement in S' of the lenth of an interval 8s in S is obtained by observing the distance 5s' between points in S' with which the ends of the
interval coincide at same S'-time. For measurement along x' axes we have,

because of Lorentz-Fitzgerald contraction

— — — + = — 5x'

Bx 6x

Bx 5t = 1

~2 1/2
8x

Bx

Bt

C2

when

— — — "' St'

— =
(1.

—v'/c')

+ 8t

'V
c' 8x

= 0, i.e. ,

J R KENNEDY AND E. 3II THORNDIKE

9= when

—(v/c')5x. Hence

— Bx 5x —8 8$ 5x =

Bx

c2 Bt

| — — c'

1/2
bx.

From this and (12)

1

8$

and

(1 s2/g2) t/2

(1 vQ/c2) I/9

Hence
x' = (1 —s'/c') '"(—x + vt) .
This equation and (11) together with y'=y, s'=s, are the Lorentz-Einstein
transformations; because they are known to possess the group property, the
system S' which has been used as a tentative standard of reference evidently
loses all trace of uniqueness. The research set forth in this paper has been carried on over a period of
several years, during which many obligations have been incurred. Preliminary
work on it served as basis for the senior author's doctoral thesis at Johns Hopkins University. The main work was done at the California Institute of
Technology with the aid of fellowships granted by the National Research
Council, the Guggenheim Memorial Foundation and the Institute; it was
completed during leave of absence granted by the University of Washington.
Particularly acknowledgment is made to Professors E. T. Bell, R. C. Tolman and R. A. Millikan, whose interest and encouragement have made the
work possible, and to Mr. Julius Pearson to whom several essential re6nements of the apparatus are due.