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Polarization Rotation, Reference Frames, and Mach's Principle

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Polarization Rotation, Reference Frames, and Mach's Principle RAPID COMMUNICATIONS PHYSICAL REVIEW D 84, 121501(R) (2011) Polarization rotation, reference frames, and Mach’s principle Aharon Brodutch1 and Daniel R. Terno1,2 1Department of Physics & Astronomy, Macquarie University, Sydney NSW 2109, Australia 2Centre for Quantum Technologies, National University of Singapore, Singapore 117543 (Received 5 August 2011; published 2 December 2011) Polarization of light rotates in a gravitational field. The accrued phase is operationally meaningful only with respect to a local polarization basis. In stationary space-times, we construct local reference frames that allow us to isolate the Machian gravimagnetic effect from the geodetic (mass) contribution to the rotation. The Machian effect is supplemented by the geometric term that arises from the choice of standard polarizations. The phase accrued along a close trajectory is gauge-independent and is zero in the Schwarzschild space-time. The geometric term may give a dominant contribution to the phase. We calculate polarization rotation for several trajectories and find it to be more significant than is usually believed, pointing to its possible role as a future gravity probe. DOI: 10.1103/PhysRevD.84.121501 PACS numbers: 04.20.Cv, 03.65.Vf, 04.25.Nx, 95.30.Sf I. INTRODUCTION to the net rotation in the examples we consider. Only a particular choice of the standard polarizations allows stating Description of electromagnetic waves in terms of rays that the mass of the gravitating body does not lead to a phase and polarization vectors is a theoretical basis for much of along an open orbit, and we demonstrate how this gauge can optics [1] and observational astrophysics [2]. Light rays in be constructed by local observers. On the other hand, on general relativity (GR) are null geodesics. Their tangent closed trajectories, the reference-frame term results in a four-vectors k, k2 ¼ 0, are orthogonal to the spacelike gauge-independent contribution. polarization vectors f, f2 ¼ 1. Polarization is parallel- Taking Kerr space-times as an example, we calculate transported along the rays [3,4], Á for several trajectories and find the phase to be greater r kk ¼ 0; rkf ¼ 0; k (1) than it was commonly believed. As a result, polarization rotation may become a basis for alternative precision tests r k where k is a covariant derivative along . These equations of GR. were solved for a variety of backgrounds ([4–13]), ranging We use þþþ signature, set G ¼ c ¼ 1, and use from the weak field limit of a single massive body to Einstein summation convention in all dimensions. Three- gravitational lenses and gravitational waves. dimensional vectors are written in boldface, and the unit Solutions of Eq. (1) predict polarization rotation in a vectors are distinguished by carets, such as e^. gravitational field. Mass of the gravitating system is thought to have only a trivial effect, while its spin and higher moments cause the gravimagnetic/Faraday/Rytov- II. TRAJECTORIES AND ROTATIONS Skrotski rotation (phase) [5–8]. A version of Mach’s To every local observer with a four-velocity u, we attach principle [14], interpreted as presence of the Coriolis ac- an orthonormal tetrad, with the time axis defined by celeration due to frame-dragging around a rotating massive e0 u [3]. In stationary space-times, a tetrad of a static body [3,4], was first proposed to calculate polarization observer is naturally related to the Landau-Lifshitz 1 þ 3 rotation in Ref. [15]. It provides a convenient conceptual formalism [17]. Static observers follow the congruence of framework for the gravimagnetic effect [6]. timelike Killing vectors that defines a projection from the Polarization is operationally meaningful only if its di- space-time manifold M onto a three-dimensional space rection is compared with some standard polarization basis M ! Æ3, : Æ3. (two linear polarizations, right- and left-circular polariza- Projecting is performed in practice by dropping the tions, etc.). Insufficient attention to the definition of the timelike coordinate of an event. Vectors are projected by standard polarizations is one of the reasons for the dispa- a push-forward map Ãk ¼ k in the same way. For a static rate values of the rotation angle Á that are found in the observer, the three spatial basis vectors of the local ortho- literature (see Ref. [16] for the discussion). In this paper, normal tetrad are projected into an orthonormal triad, we investigate the role of local reference frames in estab- Ãem ¼ e^ m, e^ m Á e^ n ¼ mn, m, n ¼ 1,2,3. lishing Á, focusing primarily on stationary space-times. The metric g on M can be written in terms of a three- We derive the equation for polarization rotation for an dimensional scalar h, a vector g, and a three-dimensional arbitrary choice of the polarization basis. In addition to the metric on Æ3 [17]as expected Machian term [6,15], this equation contains a 2 ¼ ð 0 À mÞ2 þ 2 reference-frame term, which makes a dominant contribution ds h dx gmdx dl ; (2) 1550-7998=2011=84(12)=121501(5) 121501-1 Ó 2011 American Physical Society RAPID COMMUNICATIONS AHARON BRODUTCH AND DANIEL R. TERNO PHYSICAL REVIEW D 84, 121501(R) (2011) ¼ ¼ f0 where h g00, gm g0m=g00 and the three- the resulting polarization vector to the new standard 2 m n 0 dimensional distance is dl ¼ mndx dx . The inner polarization vectors biðk Þ [19]. On a curved background, ð ^ ^ ^ Þ product of three-vectors will always refer to this metric. the standard polarization triad b1; b2; k should be defined Vector products and differential operators are defined as at every point. appropriate dual vectors [17]. Assume that two standard linear polarizations are ^ ¼ ^ For example, the Kerr metric in the Boyer-Lyndquist selected. By setting f b1 at the starting point of the coordinates [3,4,17] leads to ð Þ¼^ð Þ ^ ð Þ trajectory, we have sin f b2 ,so 2Mr 2Mar h ¼ 1 À ;g¼ sin2; (3) d 1 Df^ Db^ 2 0i i 2 ¼ Á b^ þ f^ Á 2 ^ Á ^ 2 d f b1 d d where M is the mass of a gravitating body, a ¼ J=M is the 2 2 2 2 1 Db^ angular momentum per unit mass, ¼ r þ a cos , and ¼ ! Á k^ þ f^ Á 2 : (8) ^ Á ^ ij is Kronecker’s delta. f b2 d Using the relationship between four- and three- This is the desired phase evolution equation. The term r dimensional covariant derivatives, and Dm, respec- ! Á k^ is the Machian effect as it was postulated in tively, the propagation equations (1) in a stationary Ref. [15]. The second term—the reference-frame contribu- space-time [6,7] result in a joint rotation of unit polariza- tion—was missing from the previous analysis. However, in tion and tangent vectors [16], the examples below, the reference-frame term dominates the ^ Df^ Machian effect by one power of the relevant large parameter. Dk ¼ Â k^ ; ¼ Â f^; (4) d In general, a rigid rotation of momentum and polariza- d tion leads to a nonzero polarization rotation Á with where is the affine parameter and k^ ¼ k=k. The angular respect to the standard directions [18,19]. Right- and left- velocity of rotation is given by circular polarizations remain invariant, but acquire Wigner phase factors eÆiÁ, respectively. ¼ 2! ð! Á k^ Þk^ À E Â k; (5) g The statement of zero accrued phase Á ¼ 0 in the where Schwarzschild space-time [5,6,11], where a ¼ 0,so ! ¼ ¼ Â r 0 and Eg k, is correct only in a particular ! ¼1 ¼ h k0curl g; Eg : (6) gauge. Defining this phase to be zero allows us to make a 2 2h choice of standard polarizations that does not require refer- ences to a parallel transport or communication between the III. REFERENCE FRAMES observers. This gauge construction is based on the follow- ing property. For each momentum direction k^ polarization is specified with respect to two standard polarization vectors b^ .Ina 1;2 A. Property 1 flat space-time, these are uniquely fixed by Wigner’s little A rotation R ^ ð Þ around the b^ -axis, where the polar- group construction [18,19]. The standard reference mo- b2 2 ð ^ ^ ^ Þ ð Þ mentum is directed along an arbitrarily defined z-axis, ization triad b1; b2; k is obtained from x^; y^; z^ by the with x- and y-axes defining the two linear polarization standard rotation Rðk^ Þ, does not introduce a phase. vectors. A direction k^ is determined by the spherical angles ð; Þ. The standard rotation that brings the z-axis to k^ is Proof of property 1 ð Þ defined [18] as a rotation around the y-axis by Ry that is If the triad ðb^ ; b^ ; k^ Þ is rigidly rotated by R, it typically ð Þ 1 2 followed by the rotation Rz around the z-axis, so results in an angle between, say, Rb^ and b^ ðRk^ Þ. This is ð ^ Þ ð Þ ð Þ 2 2 R k Rz Ry . Standard polarizations for the direc- the Wigner’s phase of photons. It can be read off from the ^ ^ ð ^ Þ ^ ð ^ Þ tion k are defined as b1 R k x^ and b2 R k y^.An definition [18,19], additional gauge fixing promotes b^ to four-vectors, so a i ð Þ À1ðR ^ ÞR ð ^ Þ general linear polarization vector is written as Rz R k R k : (9) f ¼ b ðkÞþ b ðkÞ However, if R ¼ R ^ ð Þ, then using the decomposition cos 1 sin 2 : (7) b2 ð Þ¼ ð ^ Þ ð Þ À1ð ^ Þ We adapt a gauge in which polarization is orthogonal Rb^ R k Ry R k ; (10) u Á f ¼ 2 to the observer’s four-velocity, 0, so in a three- 0 0 so k^ ¼ R ^ ð Þk^ ¼ k^ ð þ ; ’Þ, the Wigner rotation dimensional form, the transversality condition reads as b2 k Á f^ ¼ 0.
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