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P7. General Sagnac Effect and Invalid Relativistic Explanation
C P Viazminsky and P K Vizminiska
Abstract
We show that the experimentally verified formula of Wang et el [1] underlying
Sagnac interference is an immediate result of the theory of universal space and time
(TUST). The general Sagnac effect which includes rotational and translational types is
discussed. We demonstrate that a certain form of Sagnacs effect occurs under
rotation even if the optical circuit is not closed, and under translation provided the
circuit is not closed and its enclosing segment is not perpendicular to its velocity [2].
We finally show that the Sagnac effect cannot be explained within the framework of
the special relativity theory.
1.. Introduction
We start by a review of basic results pertaining to the optical length of a rod
discussed in P5 [3], which is needed to study a more general type of Sagnac
interference. Next, we show that the Wang formula which quantifies the general
Sagnac effect is an immediate consequence of the scaling transformation of second
type (STII) in the theory of universal space and time.
Suppose that a rod ob of geometric length L is translating in the universal
frame S at a constant velocity v that makes an angle with the vector ob. We found
in P5 [3] that the optical length of the light trip
(Figure 1a) is given by either
side of the scaling transformation of the second type STII:
(1.1)
In both figures the rode ob is translating in S at a constant velocity v. The red arrow
points in the direction of a light pulse that starts from one end (an emitter) and ends
at the other (receiver). The angle between the radius vector of the emitter relative to
the receiver and the velocity v is in Fig. 1a and
in Fig.1b.
where
,
(1.2)
is the Euclidean factor, and is the duration of the trip. Note that
is
the angle between the velocity v and the radius vector of the lights source b
relative to the receiver o.
The duration of the latter lights trip can also be written in the form
(1.3)
where
(1.4)
is the scaling factor.
1
2
We have subscripted the duration t of the trip by (bo) to emphasize the lights
trips beginning b and end o. It is easily seen that the duration of the trip
is
(1.5)
where
is the angle between the radius vector of the emitter o
relative to the receiver b.
2. Deriving Wang Formula from the STII
We show here that the experimentally verified formula of Wang [1] is an
immediate result of the scaling transformation of the second type.
Suppose that at the same instant
a lights pulse emanates from each end
of the rod ob and propagates towards the other end (Figure 2). By equations (1.5) and
(1.3) the difference in the durations of the two trips
is
(2.1)
which is positive for
, negative for
and vanishes for
The difference
reduces to
(2.2)
for
and to
for
In both latter cases the pulse travelling
opposite to the rods velocity arrives at the other end before the pulse travelling along
the velocity direction.
The difference (2.1) can be expressed in a form of inner product
(2.3)
The latter relation is valid even if the rod ob is of infinitesimal geometric length .
Consider thus the situation in which light is guided to follow an arbitrary path (optical
circuit) which itself is moving in S, and think of two light beams traveling in opposite
directions along the path, and imagine that the path is divided into infinitesimal line
elements of geometric lengths dl. The difference between the durations of two
opposite lights beams traversing one such line element is
(2.4)
The total difference between the durations of two lights trips each starting from an end and following the given path till arriving at the other is obtained through integrating the latter expression over the whole path:
(2.5)
The Wang formula for Sagnac interference
(2.6)
approximates equation (2.5) for low velocities. Indeed, for
we have
(2.7)
2
3
which reduces (2.5) to Wang formula. However for high velocities Wang formula must be replaced by equation (2.5) which was derived from STII in the theory of universal space and time.
It is important to note however that equation (2.5) is valid as long as the line element is invariant under the paths motion, which implies that the optical circuit moves like a rigid body keeping the distances between all the circuits points constant. Noting that the line element, or the metric, is invariant only under translation or rotation, or any transformation composed of them, we deduce that (2.5) is valid only when the circuit executes such motions, and thus is moving like a rigid body.
In a uniform rectilinear motion all points of the path moves at a constant vector velocity, and equation (2.5) takes the form
(2.8)
Example: Electromagnetic Transmission Round the Equator
In [5] the TUST was employed to calculate the time delay (advance) for an
electromagnetic wave heading eastward (westward) and making a full revolution
round the Earths equator. The TUST agrees fully with the common expression of
Sagnac interference equation where both predict 207 ns advance (delay)[6]. By the
TUST the time difference (2.8), in which we set
is
.
3. Translational Sagnacs Effect. It is demonstrated here that “a certain form” of Sagnacs effect is intimate to
non-closed optical paths translating at a constant velocity relative to a universal frame S (i.e. in a uniform rectilinear motion in S). Indeed, if the optical circuit, shown in red (Figure 3), is translating at a constant velocity v in S, then equation (2.8) takes the form
Figure 3. An open optical circuit translating at a constant velocity v. At the same instant of time a pulse of light originates from each end and travels to the other.
(3.1)
where
denote the position vector of the beginning b of line integral (the
circuit) and its end e. Note that
-The path need not be planar.
- The time difference in (3.1) refers to (time recorded at e when receiving the
light)- (time recorded at b when receiving the light). Thus if and denote the
instants at which light emitted at
from both ends is received at b and e
respectively then
(3.2)
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The quantity in (3.1), which is positive for figure 3, will of course reverse sign if
the beginning and end of the line integral were interchanged.
-The phrase “a certain form of Sgnac effect” we have used in the abstract and at
beginning of this section refers to the difference in time arrival of the lights beams
that started simultaneously from the opposite ends. Only when the circuits two ends
coincide one may observe fringes shift.
Equation (3.1) shows that
(a)The Sagnac effect is intimate to any constant translational motion of a non-closed
optical circuit; it vanishes however when its closing vector is perpendicular to its
velocity v.
(b) When it is closed, the beginning and end of the circuit are coincident, and
on the account of
in (3.1). Therefore, there is no Sagnac effect associated with
a closed circuit executing a uniform translational motion.
(c) Two optical circuits with respective ends
and
yields the same time
difference if
. It follows that, under uniform translational motion, every
circuit is equivalent to its closure, and all circuits that have equal vector closure are
Sagnac equivalent,
4. Rotational Sagnac Effect
We mentioned in section 2 that equation (2.5) holds for any motion in which
the circuit moves as a rigid body. We retrieve here the familiar expression that
quantifies the rotational Sagnac effect [4],
(4.1)
which pertains to a polygonal circuit of area rotating about an axis with a constant
angular velocity . The relation which we shall derive using (2.5) applies however to
any shape of a planar circuit including that which is not even closed.
Consider a rigid planar optical circuit of any shape that is rotating at a
constant angular velocity about an axis . The axis of rotation, which is not
necessarily perpendicular the circuits plane, intersects the latter at o (Figure 4). The
velocity of a point of the circuit is
and hence
(4.2)
The quantity
represents twice the area of the triangle with base and
vertex at the point o. Substituting in (2.5) yields
(4.3)
which is the expression of rotational Sagnac interference as implied by TUST.
Although the velocity magnitude is a function in the position (and ),
in some cases the factor
can be pulled out of the integral;
these include the following:
(a). All the circuits points are at equal distances from the center of rotation, and
hence their velocities have the same magnitude.
(b) The dimensions of the circuit are much smaller that its distance from the axis of
rotation. In this case one can neglect the variation in
and deal with it as a
constant function. The interferometer of Michelson and Gale experiment, consisting
of 4 mirrors situated on Earth surface, and rotating with the earth about the (South-
North pole) axis is an example.
(c) The velocities of the circuits components are small .
In all cases above we may make the approximation
(4.4)
or even set
4
5
Figure 4.
where A is the area subtended by the
circuit and the lines connecting its ends e and b with the center of rotation.
It is noted that the circuit considered need not be closed and A is the area
subtended by the circuit and the lines
, where b and e are the beginning and
end points of the circuit. When the circuit is closed A reduces to the circuits area and
the familiar relation (4.1) approximates (4.4) on neglecting terms of order
and
higher.
5. Point Circuits
Examples of optical point circuits are the Sagnac interferometer which consists
of a number of small mirrors
, or a set of satellites revolving in one
orbit, or a set of transmission stations on a great circle on Earths surface.
By the STII the time duration of a lights trip along the side
of an
n-polygonal circuit in S and in any other frame is
(5.1)
where
is the geometric length of the side
and
If the polygon is regular, and the rotation is taking place about its center the
difference between the durations of the trips
and
is given
by
(5.2)
where
is the speed of any vertex (in the unit c). The latter expression is the
familiar translational Sagnac effect as given by (2.3), and which corresponds to the
rod making an angle
with its velocity.
From (5.2) we may proceed to reach either the usual expression of the
rotational Sagnac effect (4.4), or the Wangs formula. To reach the expression (4.4) of
a circuit rotating at an angular velocity , we proceed from (5.2) as follows
(5.3)
where we identify
by
. The latter relation yields the time difference
between the durations of two light beams propagating in opposite directions along the
ring interferometer:
(5.4)
Another interesting expression for this time difference is the following:
(5.5)
where n is the number of the sides of the regular polygon and u is the speed of any vertex of the interferometer when rotating about its center. Indeed,
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6 (5.6)
Figure 5. Geometric representation of (5.6)
where
is a vector whose length is equal to the circumference nL of the
polygon and is a vector velocity that makes an angle
with
and
whose magnitude
is equal to of the speed of any vertex when the
interferometer rotates about its center (Figure 5).
6. Special Relativistic Explanation of Sagnac Effect is Invalid We show here that the special relativistic explanation of Sagnac effect as
given in Wikipedia (prior to July 12, 2020) is incorrect, and in the next section we demonstrate that the special relativity theory cannot explain this effect.
We quote from Wikipedia: “Relativistic derivation of Sagnac formula[edit]
Fig. 4: A closed optical fiber moving arbitrarily in space without stretching.
Consider a ring interferometer where two counter-propagating light beams share a common optical path determined by a loop of an optical fiber, see Figure 4. The loop may have an arbitrary shape, and can move arbitrarily in space. The only restriction is that it is not allowed to stretch. (The case of a circular ring interferometer rotating about its center in free space is recovered by taking the index of refraction of the fiber to be 1.)
Consider a small segment of the fiber, whose length in its rest frame is . The time intervals, , it takes the left and right moving light rays to traverse the segment in the rest frame coincide and are given by
Let
be the length of this small segment in the lab frame. By the relativistic length
contraction formula,
correct to first order in the velocity v of the segment.
The time intervals for traversing the segment in the lab frame are given by Lorentz
transformation as:
correct to first order in the velocity v. In general, the two beams will visit a given segment at slightly different times, but, in the absence of stretching, the length dl is the same for both beams. It follows that the time difference for completing a cycle for the two beams is
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” - quotation finished.
The above argument which is sound in a relativistic sense contradicts
experimental findings. According to special relativity the time intervals taken by
the left and right moving light in the rest frame to traverse the given segment
coincide:
. Therefore the durations of a full trip round the ring in opposite
directions are equal:
and the two beams arrive back at the point from which they started at the same instant. This means that the Sagnac effect is not observed in the rest frame, which is not true. Indeed, the fringes shift is a physical reality and the Sagnac effect is observed in the lab frame as well as in the rest frame. The Michelson and Gale experiment is a cogent proof that the Sagnac effect is observed in the rest frame. The location of the latter experiment on the surface of the rotating Earths is the rest frame.
7. Sagnac Effect is not Explicable by Special Relativity
We abide throughout this section with special relativity theory and show that it
is incapable of explaining Sagnac effect. Consider again the circuit drawn in the
previous section and let s be the rest frame of the given segment (line element) drawn
there. We name the segments left and right ends by 1 and 2 respectively. The rest
frame s is moving in the lab frame S at velocity v and can be considered inertial
during the extremely short duration of the experiment.
Consider the following events in the lab frame S:
- The right moving light,
passes by the left end (1) of the segment at
:
, and
arrives at the right end (2) of the segment at :
- The left moving light
passes by end (2) of the segment at :
, and
arrives at end (1) at
:
Note that the left moving light may pass from the end 2 at an instant that differs
slightly (by ) from the instant at which the right moving light passes through end 1.
However, this difference is so small and can be neglected (i.e.
with no bearing
on subsequent results.
The 4-interval between the first two events is represented by the 4-vector
in Minkowski space . Similarly, the 4-interval between the last two
events is the 4-vector
The difference between the latter vectors is the
4-vector
(7.1)
where we set
(7.2)
Lets take the inner product of the latter vector by the velocity 4-vector of the
given circuit element,
, and denote the result by I:
(7.3)
We remind the reader that the inner product of two 4-vectors
(7.4)
in is defined by
(7.5)
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An important property of the inner product is that: It is invariant under orthogonal
transformations in and thus in particular under Lorentz transformation. In the rest
frame s of the given segment, the 4-velocity of the segment is
, and the vector
corresponding to (7.1) is
(7.6)
However, in the rest frame s,
because light takes the same duration to
traverse the segment from either end to the other, and
Thus the vector
(7.1) is transformed in s to
. Equation (7.3) and the invariance of the inner
product yield
(7.7)
In the frame S, and are the vectors from one end of the segment to the other
but at different instants of time. Each vector joins the location of the end when light
enters the segment and the location of the other end when it leaves. Neglecting
quantities of high order in smallness yields
, and hence
(7.8)
The total difference in time arrival of the two beams to their common starting point in
the circuit is obtained through integrating over the circuit:
(7.9)
However, in the rest frame of each segment of the circuit there is no difference in time arrival of the two beams at the opposite ends. As a result, the two beams return to their starting point at the same time, and no displacement in interference fringes will be observed in a frame attached rigidly to the circuit. The latter result is at contradiction with experiment.
Conclusion The Wang formula for Sagnac interference follows directly from the scaling transformation of the second type in the theory of universal space and time. In contrast, the special theory of relativity is utterly incapable of explaining Sagnac effect. It was also demonstrated that that a certain form of Sagnacs effect occurs under rotation even if the optical circuit is not closed, and under translation provided
the circuit is not closed and its enclosing segment is not perpendicular to its velocity.
References [1] Wang, R.; Zheng, Y.; Yao, A.; Langley, D (2006). "Modified Sagnac experiment for measuring travel-time difference between counter-propagating light beams in a uniformly moving fiber". Physics Letters A. 312 (12): 7 [2] Viazminsky C P., November 1, 2009: The Scaling Theory - X: The General Sagnac Effect https://www.gsjournal.net/Science-Journals/Research%20Papers/View/1959 [3] Viazminsky C P and Vizminiska P, May 15, 2020: Optical Length of a Moving Rod https://www.gsjournal.net/Science-Journals/Research%20Papers/View/8249 [4] Viazminsky C P and Vizminiska P, May 16, 2020: The Sagnac's Interference, and Michelson and Gale Experiment. https://www.gsjournal.net/Science-Journals/Research%20Papers/View/8252 [5] Viazminsky C P, April 5, 2007: On the Sagnac Effect. https://www.gsjournal.net/Science-Journals/Research%20PapersMechanics%20/%20Electrodynamics/Download/1942 [6] Russo Daniele, (2007), Stellar aberration: the contradiction between Einstein and Bradley", Apeiron, 14, no 2, 95-112.
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