The Sagnac effect is a phase shift observed between two beams of light traversing in opposite directions the same closed path around a rotating object. A description of this experiment is obtained within the context of general relativity. In this context the effect provides an operational definition of rotation. An expression for the magnitude of the phase shift is derived under fairly general conditions. The general definition of rotation provided by this experiment is shown to reduce, in certain particular cases, to the usual definitions available. It is observed that the Sagnac effect represents a gravitational analog of the Aharanov−Bohm effect in electrodynamics.

## REFERENCES

1.

M. G.

Sagnac

, C. R. Acad. Sci. (Paris),

157

, 708

, 1410

(1913

).This experiment is also described in most text books on relativity theory. See, e.g., Landau & Lifshitz,

*Classical Theory of Fields*(Addison‐Wesley, Reading, Mass., 1969).2.

We mean the framework of Newtonian mechanics together with the hypothesis of constancy of speed of light.

3.

Throughout this paper we use units where $c\u2009=\u2009G\u2009=\u20091.$

4.

Since the glass tube is rigid, it is natural to demand that the 4‐velocity vector field, $\xi a$ of the tube be Born‐rigid on μ [i.e., $L\xi (gab+\xi a\xi b)\u2009=\u20090$]. This requirement is automatically satisfied if the tube moves along the trajectories of a Killing field.

5.

Note that we only require that $ta$ be a timelike Killing vector field on $(\mu ,hab).$ The space‐time $(M,gab)$ may not have any Killing field [e.g., the locally nonrotating vector field which we use in Case 2 is a Killing field on $(\mu ,hab)$ but not on $(M,gab)$].

6.

Note that since $\nu \u2009=\u2009\u2212\lambda \u22121/2$ and since the scalar field λ remains constant along the integral curves of $ta,$ the frequency of the light rays represented by $C+$ and $C\u2212$ remains constant along the worldline of the mirror.

7.

It is convenient, on an

*n*‐manifold, only to integrate*n*‐forms, so that the volume element is independent of the metric.8.

The curve

*C*appearing in the definitions of*I*and $I\u2032$ is an element of the first homology group on μ. The curve*S*in Eq. (1) is homologous to*C*.9.

An interesting geometrical property of $(\mu ,hab)$ is the following. Suppose we start at the event $m\u2032$ (see Fig. 1.) and move orthogonal to the Killing field $ta$ everywhere. Let $m\u2033$ be the event (in future of $m\u2032$) where we would first meet the world‐line

*M*of the mirror. Then $m\u2033$ is the midpoint of*m*and $m\u2032$ on*M*. To see this, choose for*S*in Eq. (1) the closed curve $m\u2032m\u2033m\u2032$ obtained by moving orthogonal to $ta$ from $m\u2032$ to $m\u2033$ and along*M*from $m\u2033$ to $m\u2032.$ Then, since the tangent vector to the curve_ from $m\u2032$ to $m\u2033$ is orthogonal to $ta,$ $\Delta \tau \u2009=\u20092\lambda M1/2\u222bm\u2032m\u2032\lambda \u22121tadSa\u2009=\u20092$ (distance between $m\u2033$ and $m\u2032$ as measured along*M*). The result is immediate if one recalls that δτ is the distance between*m*and $m\u2032$ measured along*M*.10.

In particular, this could be done by fixing the tube to the stationary object. Note also that by requiring the tube to follow trajectories of $ta$ we have restricted ourselves to the experiments which can be performed only outside the ergosphere.

11.

However, in the post‐Newtonian approximations there does exist such a relationship. For example, to the first post‐Newtonian approximation Δτ is essentially the flux of the angular momentum of the source through the tube. For details see, e.g., S. Chandrasekhar in

*Relativity*, edited by Carmeli, Fickler, Witten (Plenum, New York, 1970).12.

$Ta$ is that Killing vector field which is timelike at spatial infinity. We have assumed throughout that the two Killing vectors commute, i.e., $LRTa\u2009=\u20090.$ If the re are no other independent Killing vectors and if there is an open region in which $Ta$ is timelike and $Ra$ spacelike with closed integral curves, this condition is always satisfied. Proof: let $LRTa\u2009=\u2009aTa+bRa,$ with

*a, b*, constants. Consider the scalar field $\alpha \u2009=\u2009(aTa\u22122bRa)Ta$ then $LR\alpha \u2009=\u20092a2TaTa\u22122b2RaRa\u2a7d0.$ But the integral curves of $Ra$ are closed. Since α is a well‐defined (single‐valued) function, $LR\alpha \u2009=\u20090,$ i.e., $a\u2009=\u2009b\u2009=\u20090.$ Thus the two Killing fields commute in an open region and hence everywhere. This result is due to P. S. Jang (private communication).13.

Note that we have let each point on the tube evolve along a trajectory of $Ta$ only because we wish to compare the Sagnac criterion for presence of rotation of the object itself with the usual one. Note also that we have restricted ourselves to experiments which can be performed only outside the ergosphere.

14.

This vector field was first introduced by Bardeen. See, e.g.,

J.

Bardeen

, Ap. J.

162

, 71

, (1970

). Since the locally nonrotating vector field remains timelike down to the horizon, the “zero rotation” Sagnac experiment can be performed in the ergosphere unlike the previous ones.15.

More precisely, consider a body which can be approximated by a worldline together with a spatial triad attached to each point of this worldline. Given a body, its description by a worldline with spatial triads becomes more and more accurate as the scalar curvature becomes smaller and smaller.

16.

Thus the “Vector fields” $Ua$ and $Va$ are defined only along the worldline of the center of the tube.

17.

For example, a test applicable to bodies of finite size.

18.

19.

This prediction has been experimentally verified. See, e.g.,

R. G.

Chambers

, Phys. Rev. Lett.

5

, 3

(1960

).
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© 1975 American Institute of Physics.

1975

American Institute of Physics

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