θ

θ = 0

e

+

−

Figure 4: The parametrized curve r ( θ , t ) is, for each fixed
time t , a closed loop in space representing a fiber loop. θ
is a comoving coordinate, designating fixed material points on the
fiber.
Shown are the tangent e
at a point θ of the loop. The two counter-propagating beams
are marked with the red arrows and the ± signs. The black dot
represents the co-moving beam splitter located at θ = 0 and
the two dashed red arrows the co-moving source and detector.
We choose a parametrization where the beam splitter is located at
θ = 0 . It is
useful to imagine that the black triangle in the figures
represents a pair of (putative) source and detector co-located
at θ = 0 with world-line ( t , r ( 0 , t ) )
and rest frame S 0 . The
source emits monochromatic waves with frequency ω 0 (as
measured in S 0 ).

Two light beams, emanating from the beam splitter, propagate in opposite
directions and make one round each along the loop. The phase difference
Δ Φ = Φ + − Φ − between the beams as they return to
the beam splitter depends, among other things, on the fiber’s shape
and motion. Being measurable, Δ Φ must be a Lorentz-invariant
quantity.

Consider an infinitesimal fiber’s segment d θ , and denote its
momentary rest frame by S ′ . The full differential of r ( θ , t )
takes the form

d r = ( ∂ θ r ) d θ + ( ∂ t r ) d t ≡ e d θ + v d t
(8)

where e is the tangent vector and ^ e
the corresponding unit vector. We denote the infinitesimal spatial displacement vector d ℓ = e d θ .
The segment’s velocity with respect
to S (and hence the velocity of S ′ with respect to S )
is v (| v | < c ).
Note that e
depends on parametrization, but v and ^ e
are parametrization invariant.

We denote the length of the infinitesimal segment in the lab frame
by

The segment’s proper length (namely its length in its own rest
frame S ′ ) is denoted d ℓ ′ . Owing to Lorentz contraction,
these two quantities are related by

d ℓ ′ = ( d ℓ 2 ⊥ + γ 2 d ℓ 2 ∥ ) 1 / 2 = γ γ ⊥ d ℓ ,
(10)

where d ℓ ⊥ , d ℓ ∥
denote the projections of d ℓ
perpendicular and parallel to v . We use here
γ = ( 1 − v 2 / c 2 ) − 1 / 2 and γ ⊥ = ( 1 − v ⊥ 2 / c 2 ) − 1 / 2 , where v ⊥ is the
component of v perpendicular to e .
By its very definition d ℓ (like d ℓ ′ ) is parametrization invariant.

If the fiber is non-stretching, d ℓ ′ of any segment
is independent of time. In the case of stretching the local
stretching rate — or simply the stretch — is naturally
defined in the Newtonian approximation by

We can re-write this as

s = ∂ t log d ℓ = ∂ t log | e | = ^ e ⋅ ∂ ℓ v .
(12)

The last equality follows from ∂ t e = ∂ t ∂ θ r = ∂ θ v .
The partial derivative ∂ ℓ is taken at fixed t (namely
∂ ℓ = | e | − 1 ∂ θ ).

In the relativistic framework, the stretch is naturally defined in
the same manner as above, as a scalar, but in the local rest frame, namely

s ≡ ( d ℓ ′ ) − 1 ∂ t ′ ( d ℓ ′ ) = γ ∂ t log ( d ℓ ′ ) = γ ∂ t log ( γ γ ⊥ | e | )

which differs from the Newtonian expressions (11 ,12 )
in second order of v / c .

The phase velocity of the wave with respect to the segment’s rest
frame S ′ , shall be denoted by u ′ . It is determined by
the fiber’s refractive index n , through

We allow n to vary along the fiber, but we assume that it is time-independent
(at any given segment), polarization independent and dispersion free.
Under these assumptions, the wave’s velocity at a given segment
is the same for the two directions of light propagation, namely u ′ + = u ′ − ≡ u ′ .

Our main concern is the phase difference Δ Φ = Φ + − Φ −
between waves in the two directions of propagation. This phase difference
results from the difference in travel times in the
two directions along the loop. We denote these travel times by T ± ,
and their difference by Δ T :

With ω the lab-frame frequency of the source,

In a relativistic treatment, the lab frequency
ω and the rest-frame frequency ω 0
differ by time dilation: ω = ω 0 / γ 0 , where γ 0 ≡ γ ( θ = 0 ) .
The relativistic expression for the phase difference is therefore

All that is left is to calculate the travel-time difference Δ T .

We point out that Eq. (14 ) is
accurate to order ( v / c ) 3 . This is because γ 0 is generally
time dependent, and may undergo tiny changes during the short time
interval Δ T . In experiments this ambiguity
in the value of γ 0 is totally unimportant — even its
very deviation from 1 is too small to be detected.
For the sake of completeness we present the precise
expression for Δ Φ , taking into account the time variation
of γ 0 , in the Appendix. One can write this precise expression
in the form (14 ) but with γ 0 replaced by
an appropriately averaged quantity ¯ γ 0 . The precise
expression for Δ Φ is given in Eq. (44 ).

3 Calculation of travel times and phase difference
In this section we first analyze the travel times of the two
counter-propagating signals along an infinitesimal fiber’s segment.
Then we use this result to calculate the integrated time difference
Δ T and phase difference Δ Φ along the closed loop.

3.1 Contribution of infinitesimal segment
Let us consider an infinitesimal fiber’s segment d θ . The segment
has a momentary velocity v with respect to S , its
lab-frame length is d ℓ , Eq. (9 ), and its proper
length is d ℓ ′ , Eq. (10 ). The two counter-propagating
waves visit the segment d θ at two different moments t ± ,
and have two different (lab-frame) traversal
times which we denote d t ± . We
shall now explicitly calculate d t + , the calculation of d t −
will then follow analogously. For notational simplicity, we shall
now omit the “+ ” index in the various quantities, and recover
it later at the end of this subsection, adding the “± ”
suffix to the relevant quantities.

It is easiest to calculate the crossing time in the rest frame S ′ ,
because it is in that frame that the phase velocity takes its canonical
form u ′ = c / n . We obtain right away

However, to compute T ± we need to calculate the corresponding
lab-frame travel time d t . This quantity is easily obtained
from d t ′ via Lorentz transformation, as we now describe.

For concreteness let us consider a nodal point of the propagating wave, and follow its travel
across the segment d θ . Let A denote the event that the
node enters the segment and B the event that it leaves it, as shown
in Fig. 5 . In the lab frame, the time difference between
these two events is d t (≡ d t A B = t B − t A ),
and their spatial difference will be denoted d r A B ≡ r B − r A .
The standard Lorentz-transformation formula for d t ′ , applied to
the spacetime interval A-B, is

x

t

θ

θ + d θ

A

B

C

d ℓ

d r A B

v d t

Figure 5: Space-time diagram describing the travel of a nodal point across an
infinitesimal fiber’s segment d θ . The diagram
shows the projections of all 3D vectors in one direction.
The two parallel black lines are the world-lines of the two segment’s
edges. The red line is the world-line of the node. The lines intersect
at the events A and B when the node enters and leaves the segment.
Also shown are the vectors e d θ = d ℓ ,
d r A B and v d t which participate in Eq. (17 ).
We denote by d ℓ the spatial displacement
vector between the two edges of the segment (at an S-frame moment
of simultaneity); namely d ℓ ≡ d r A C = e d θ .
From Fig. 5 it is clear that d r C B = v d t ,
and hence

d r A B = d r A C + d r C B = d ℓ + v d t .
(17)

Extracting d t from Eq. (16 ) and substituting Eq. (17 )
we obtain

d t = d t ′ γ + v ⋅ d r A B c 2 = d t ′ γ + v ⋅ d ℓ c 2 + v 2 d t c 2 .
(18)

We can now extract d t once again:

d t = γ d t ′ + γ 2 v ⋅ d ℓ c 2 = γ d ℓ ′ u ′ + γ 2 v ⋅ d ℓ c 2 .

So far we considered the travel time for the “+ ” node. When
considering a “− ” node the only change is that now d ℓ
is to be replaced by − d ℓ . We therefore
obtain the exact, fully relativistic, expression for the contribution
of an infinitesimal segment:

d t ± = γ ± d ℓ ′ ± u ′ ± γ 2 ± v ± ⋅ d ℓ ± c 2 .
(19)

Recall that the ± nodes visit the segment at two different times
t ± ( θ ) . The “± ” suffix in the various time-dependent
quantities in Eq. (19 ) reflects this difference between
values at t + and t − . The phase velocity u ′
in a given segment is independent of time.

We denote by δ t the contribution of the infinitesimal segment
to the overall time difference Δ T :

The overall travel-time difference is thus

3.2 The schedules t ± ( θ )
The expression (19 ) for d t ±
can be interpreted as a first-order ODE which determines the exact
phase schedule, namely the unknowns t ± ( θ ) .
In the Appendix we describe in more details
this ODE and its boundary condition
and construct an exact formal expression for Δ Φ .
However, for the rest
of this paper we shall not really need this formulation.
Instead, we shall now restrict the analysis to first order in the
small parameter v / c (which provides an excellent approximation
for all lab experiments so far).

3.3 δ t to first order in v / c
To first order in v / c we may set γ ± ≅ γ 0 ≅ 1 and d ℓ ′ ≅ d ℓ in Eq. (19 ) which then reduces to

The contribution of the infinitesimal segment to the overall time
difference Δ T is

δ t = d t + − d t − = d ℓ + − d ℓ − u ′ + v + ⋅ d ℓ + + v − ⋅ d ℓ − c 2 .
(23)

In general, the quantities d ℓ ± , d ℓ ± , v ± on the R.H.S. may be time dependent, and are different for the ± beams hence their “± ’’ suffix.
However, typically the light cycle time is so short that the velocity and length of a given segment hardly change.

δ t ≅ d ℓ + − d ℓ − u ′ + 2 v ⋅ d ℓ c 2 .
(24)

We shall refer to the two terms in the R.H.S. as the stretch and Wang’s terms, which we shall respectively denote δ t s t and δ t w . In the next two sections we address these two contributions.

4 Non-stretching fibers: Wang formula
In the non-stretching case d ℓ of a given segment is fixed and
hence d ℓ + = d ℓ − . The first term
in the R.H.S. of Eq. (24 ) then cancels out, yielding

Accumulating the δ t from the infinitesimal segments of the loop we obtain

therefore we proved Wang formula, Eq. (5 ):

Δ Φ = 2 ω c 2 ∮ v ⋅ d ℓ ≡ Δ Φ w a n g .
(27)

The integration is carried out with lab time set to the detection time t d e t .
Note that to leading order in v / c we do not need to distinguish between ω and ω 0 .

5 Stretching media
In the case of stretching media d ℓ changes
with time, and d ℓ + − d ℓ − although small, no longer vanishes.
However, the smallness of v compared to c (or more precisely,
compared to u ′ = c / n ) implies that these quantities hardly change
during the short time interval [ t + ( θ ) , t − ( θ ) ] . We can then approximate

d ℓ + − d ℓ − ≅ ( t + − t − ) ∂ t ( d ℓ ) .
(28)

From Eqs. (11 ) and (12 )

∂ t ( d ℓ ) = s d ℓ = d ℓ ^ e ⋅ ∂ ℓ v .
(29)

This shows that s d ℓ is first order in v and hence to first order in v / c

δ t s t = d ℓ + − d ℓ − u ′ ≅ ( t + − t − ) s u ′ d ℓ .
(30)

We still need to evaluate t + − t − . Since
s is proportional to v , we only need to evaluate t ±
at zeroth order in v / c , that is, we may pretend that the fiber
is static. We thus simply use d t ± = ± d ℓ / u ′ = ± ( | e | / u ′ ) d θ ,
yielding

t ± ( θ ) ≅ ± ∫ θ | e ( θ ′ ) | u ′ d θ ′ ,
(31)

with boundary conditions

This integral may be viewed as the zeroth order of the scheduling
equation (42 ). We therefore obtain

t + ( θ ) − t − ( θ )
= ∫ θ 0 | e ( θ ′ ) | u ′ ( θ ′ ) d θ ′ − ∫ θ m a x θ | e ( θ ′ ) | u ′ ( θ ′ ) d θ ′
= ∫ θ m a x 0 sgn ( θ − θ ′ ) | e ( θ ′ ) | u ′ ( θ ′ ) d θ ′ .

which we may conveniently re-express as

t + ( ℓ ) − t − ( ℓ ) = ∫ L 0 d ~ ℓ sgn ( ℓ − ~ ℓ ) u ′ ( ~ ℓ )
(32)

using the length parameter ℓ along the fiber, along with the
corresponding integration variable ~ ℓ .

The quantity δ t s t in Eq. (30 ) is
the contribution from a given infinitesimal segment d ℓ . Integrating along
the fiber and multiplying by ω we obtain the overall stretch
contribution

Δ Φ s t r e t c h ≅ ω ∫ L 0 [ t + ( ℓ ) − t − ( ℓ ) ] s ( ℓ ) u ′ ( ℓ ) d ℓ .
(33)

Recall that to leading order in v / c we need not
distinguish between ω and ω 0 .
Substituting Eq. (32 ) in the integrand, we bring this expression to the more explicit form

Δ Φ s t r e t c h
≅ ω ∫ L 0 d ℓ ∫ L 0 d ~ ℓ s ( ℓ ) sgn ( ℓ − ~ ℓ ) u ′ ( ~ ℓ ) u ′ ( ℓ )
= ω 2 ∫ L 0 d ℓ ∫ L 0 d ~ ℓ s ( ℓ ) − s ( ~ ℓ ) u ′ ( ~ ℓ ) u ′ ( ℓ ) sgn ( ℓ − ~ ℓ )
(34)

which is equivalent to Eq. (7 ).

We make the following observations:

Δ Φ s t r e t c h = 0 if s and u ′ are symmetric, i.e. s ( ℓ ) = s ( L − ℓ ) , and u ′ ( ℓ ) = u ′ ( L − ℓ ) .

In the special case where n is constant along the fiber the R.H.S.
of Eq. (32 ) becomes ( 2 ℓ − L ) / u ′ and Eq. (33 )
reduces to

Δ Φ s t r e t c h ≅ ω n 2 c 2 ∫ L 0 s ( ℓ ) ( 2 ℓ − L ) d ℓ .
(35)

It is proportional to the first moment of s ( l ) relative to the
middle of the fiber.
One can further verify that for the validity of the last equation it is sufficient that n is constant throughout the support of s .

If n is constant and the stretch is localized to a single point, s ( ℓ ) = v 0 δ ( ℓ − ℓ 1 ) , then

Δ Φ s t r e t c h = ω n 2 v 0 c 2 ( 2 ℓ 1 − L ) .
(36)

It thus gives information on the distance of the stretching point from the mid-point L / 2 .

6 The Fizeau experiment
The framework we have described is sufficiently general
to accommodate Fizeau’s experiment.
We consider
a straight pipe section in which the water flows.
The laser beam enters
the pipe through a glass window located at ℓ = ℓ 1 , and leaves
it at another glass window at ℓ = ℓ 2 with ℓ the length
coordinate along the light beam. The optical path then has a section with
flowing water and a complementary static section where the beam-splitter and
mirrors are. The entire system is stationary.

We shall only care here about the tangential velocity component v ∥ = ^ e ⋅ v
of the water (the transversal velocity has no contribution at linear order).
Note that v ∥ ( ℓ ) vanishes at the two windows at ℓ = ℓ 1 , 2 .

From Eq. (27 ) we obtain for Wang’s term:

Δ Φ w a n g = 2 ω c 2 ∫ ℓ 2 ℓ 1 v ∥ ( ℓ ) d ℓ .
(37)

Turning next to analyze Δ Φ s t r e t c h , we first observe that
the stretch occurs entirely inside the water so we can use Eq. (35 ). Since the laser beam
is straight line inside the water, ^ e is constant there
and Eq. (12 ) yields
s = ∂ ℓ v ∥ . Equation (35 ) therefore reads

Δ Φ s t r e t c h ≅ ω n 2 c 2 ∫ ℓ 2 ℓ 1 ( 2 ℓ − L ) ∂ ℓ v ∥ d ℓ .
(38)

In a typical Fizeau experiment the pipe has a uniform cross section (as in Fig. 2 ), hence v ∥ has an approximately constant value v 0 along the pipe—except near the windows where it abruptly drops to zero. In such a case the entire stretch contribution comes from the near-window region, where ∂ ℓ v ∥ has a δ -function shape (with amplitude v 0 at ℓ ≈ ℓ 1 and − v 0 at ℓ ≈ ℓ 2 ), which allows a simple evaluation of the last integral. It is simpler, however, to integrate Eq. (38 ) by parts:

Δ Φ s t r e t c h = − 2 ω n 2 c 2 ∫ ℓ 2 ℓ 1 v ∥ ( ℓ ) d ℓ .
(39)

(recalling that v ∥ always vanishes at the two windows).

Summing the two contributions (37 ,39 ) we obtain the overall Fizeau phase difference

Δ Φ = Δ Φ w a n g + Δ Φ s t r e t c h = 2 ω c 2 ( 1 − n 2 ) ∫ ℓ 2 ℓ 1 v ∥ ( ℓ ) d ℓ .
(40)

This holds for an arbitrary v ∥ ( ℓ ) .
Hence the last expression reduces to

where L w ≡ ℓ 2 − ℓ 1 is the length of the light’s
orbit in water,
and v 0 is v ∥ ( ℓ ) averaged over length (between ℓ 1 and ℓ 2 ).
This reproduces Laue’s [9 ] classic result, Eq. (2 ).

7 Conclusion
We have described a unified framework for the phase shift in interferometers
where counter-propagating beams share a common optical path,
and gave a rigorous proof of Wang formula. We have shown that, neglecting dispersion, the phase shift to leading order in v / c has, in general, two contributions:

The Wang term Δ Φ w a n g , reflecting the geometry of Minkowski
space and Lorentz transformations. It is accurate to first order in v / c , is independent of n and holds if the fiber does not stretch.

The stretch term Δ Φ s t r e t c h which, to order v / c is given by Eq. (7 ). For constant n the stretch term is proportional to n 2 .

Our framework thus encompasses
the experimental setting of Sagnac, Fizeau and Wang and sheds light
on all three.

Appendix: Exact expression for Δ Φ
We shall present here the exact relativistic expression for the phase
difference Δ Φ , which accounts for the time variation
of the various quantities during the wave propagation along the fiber.

Suppose that we want to predict the value of Δ Φ at a given
detection moment t = t d e t . Then we need to follow the two constant-phase
curves t ± ( θ ) backward in time, along the entire closed
loop, from t = t d e t to the moments of emission t e ±
at which the two counter-propagating constant-phase curves have left
the source. ( Both the “source” and the “detector”
are realized by the beam-splitter, located at θ = 0
or equivalently θ m a x .)

The schedule equation is a first-order ODE for the unknowns t ± ( θ )
in the range 0 ≤ θ ≤ θ m a x , which follows from Eqs. (19 ,10 ):

d t ± d θ = ± γ 2 ± ( | e | ± ( γ ⊥ ) ± u ′ ± v ± ⋅ e ± c 2 ) .
(42)

The differential equation is non-linear since the quantities on the R.H.S may be rather arbitrary functions of t ± and θ (u ′ only depends on θ .)

Recalling that the “+ ” and “− ” directions respectively correspond to
increasing and decreasing θ , the boundary conditions are

t + ( θ m a x ) = t − ( 0 ) = t d e t .
(43)

The curves t ± ( θ )
are determined, throughout the interval 0 ≤ θ ≤ θ m a x ,
by the first-order ODE (42 ) along with the boundary
condition (43 ). The emission moments t e ±
are then given by

Note that the two travel times are T ± = t d e t − t e ± ,
and hence Δ T = t e − − t e + .

Knowledge of t e ± will allow the precise calculation of
Δ Φ . We assume that the source emits a wave with fixed rest-frame
frequency ω 0 . The desired quantity Δ Φ is the
difference in the phase of the emitted wave between the moments t = t e +
and t = t e − . This phase difference is ω 0 multiplied
by the proper time Δ τ of the source between these two moments.
Since the source speed v 0 is in general time-dependent, so is
it Lorentz factor γ 0 , and Δ τ is thus given by
an integral of 1 / γ 0 ( t ) between the two relevant moments.
Therefore,

Δ Φ = ω 0 ∫ t e − t e + d t γ 0 ( t ) .
(44)

This result may also be re-expressed, similar to Eq. (14 ),
as

where 1 / ¯ γ 0 is defined to be the time average of 1 / γ 0 ( t )
in the range between t e + and t e − (that is, the integral
in Eq. (44 ) divided by Δ T ). Note that a fully-precise
determination of Δ Φ requires knowledge of the two emission times
t e ± , not just their difference Δ T .

Acknowledgment
The research is supported by ISF. We thank R. Wang, S. Lipson and O. Kenneth for useful discussions.

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