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], 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 ! 3, : 3. (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 k ¼ k in the same way. For a static rate values of the rotation angle 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- em ¼ e^ m, e^ m e^ n ¼ mn, m, n ¼ 1,2,3. lishing , 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 3 [17]as expected Machian term [6,15], this equation contains a 2 ¼ ð 0 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 ;g¼ sin2 ; (3) d 1 Df^ Db^ 2 0i i 2 ¼ b^ þ f^ 2 ^ ^ 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 ! k^ þ f^ 2 : (8) ^ ^ 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- ! 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 k^ ; ¼ f^; (4) d In general, a rigid rotation of momentum and polariza- d tion leads to a nonzero polarization rotation 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 e i , respectively. ¼ 2! ð! k^ Þk^ E k; (5) g The statement of zero accrued phase ¼ 0 in the where Schwarzschild space-time [5,6,11], where a ¼ 0,so ! ¼ ¼ r 0 and Eg k, is correct only in a particular ! ¼ 1 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 1ðR ^ 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 1ð ^ Þ We adapt a gauge in which polarization is orthogonal Rb^ R k Ry R k ; (10) u 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 f^ ¼ 0. matrix is The net polarization rotation that results from a Lorentz 1 ^ 0 ^ 1 Rzð Þ¼R ðk ÞR ^ ð ÞRðkÞ¼Ry ð þ ÞRyð ÞRyð Þ transformation can be read off either by using the definition b2 of the corresponding little group element or by referring ¼ 1: (11)
121501-2 RAPID COMMUNICATIONS
POLARIZATION ROTATION, REFERENCE FRAMES, AND ... PHYSICAL REVIEW D 84, 121501(R) (2011)
B. Newton gauge of the azimuthal angle out, and the conservation laws In the Schwarzschild space-time, we see that if the determine the outgoing momentum as z-axis is oriented along the direction of the free-fall accel- k ^ ¼ð1; 1; =r; D=r sin Þ; (14) ^ / ¼ out out eration Eg, and b2 Eg k , no polarization phase is accrued. As a photon propagates along its planar trajec- where D ¼ Lz=E is a scaled angular momentum along the ^ ^ 2 þ 2 2 2 2 tory, its momentum k and linear polarization b1 are rotated z-axis and out a cos out D cot out. Initial ^ momentum in the scattering scenario satisfies a similar around the direction b2, which is perpendicular to the plane of motion. expression. In a general stationary space-time, we construct the Using the transversality of polarization and the gauge polarization basis by defining a local z-axis along the condition f0^ ¼ 0, we select f ^ and f ^ as two independent free-fall acceleration, z^ w=jwj, and define the standard components. Walker-Penrose constants are bilinear in the ^ polarization direction b2 to be perpendicular to the mo- components of polarization and momentum. Hence, one mentum and the local z-axis, obtains a transformation matrix R, 0 1 0 1 ^ w k^ =jw k^ j: (12) ^ ^ b 2 fout f @ A ¼ R@ in A: (15) This convention, that we will call the ‘‘Newton gauge,’’ is ^ ^ fout fin consistent: if we set z^ ¼ r^ in the flat space-time, then no ^ ^ phase is accrued as a result of the propagation. In addition to Polarization is expressed in the basis b1, b2. Matrices N being defined by local operations, the Newton gauge has two connect ðf ^;f ^ Þ with ðf1;f2Þ. As a result, the evolution is further advantages. First, it does not rely on a weak field represented by the matrix approximation for definition of the reference direction. ¼ 1 Second, if the trajectory is closed or self-intersecting, the T NoutRNin : (16) reference direction z^ is the same at the points of the ^ ¼ ^ in intersection. If initial polarization is f b1 , then the rotation angle is given by sin ¼ T21 [16]. We present a special case of the scattering scenario that IV. EXAMPLES allows an easy comparison with the literature [16] and We discuss polarization rotation in a Kerr space-time in highlights the importance of a proper treatment of refer- two different settings, giving the calculational details in ence frames. If the initial propagation direction is parallel Ref. [16]. To simplify the expressions, we define the affine to the z-axis with the impact parameter s, then the polar- ¼ ¼ ization is rotated by parameter [3,4] by fixing the energy E k0 1. Trajectories are calculated using Carter’s conserved quan- 4Ma 15M2a tity [4,17]. sin ¼ þ þ Oð 4Þ; (17) ð Þ 2 4 3 The light is emitted from the point rin; in; 0 and ob- ð Þ served at rout; out; out . To simplify the exposition, we where !1 assume a distant observation point (rout ). In one scenario, we consider a trajectory in the outgoing principal 2 D2 þ ¼ s2 a2: (18) null geodesic congruence [4,7]. In a ‘‘scattering’’ scenario, the light is emitted and observed far from the gravitating The antiparallel initial direction gives the opposite sign. At the leading order in s 1, we can assume f^ b^ and body (rin;rout M; a). 1 Outgoing geodesics in principal null congruence directly integrate Eq. (8). The result shows that the leading have km ¼ð1; 0;a= Þ, r2 2Mr þ a2, along the contribution to the polarization rotation comes from the entire trajectory, and can be easily integrated [4]. Taking reference-frame term, while the Machian term ! k^ at this into account that the Machian term ! k^ Oðr 3Þ in order does not contribute to the integral. The latter result Eq. (8) is dominated by the reference-frame term, the was used to justify the view that the polarization rotation is 3 leading order of polarization rotation is found by a direct 1=rmin effect, where rmin s is the minimal distance from integration: the gravitating body. a ¼ þ Oð 2Þ sin cos in rin ; (13) rin V. GAUGE-INVARIANT PHASE in agreement with Ref. [7]. Consider a basis of one-forms ð 1; 2; 3Þ that is dual to ð ^ ^ ^ Þ Evolution of polarization along a general trajectory is b1; b2; k . A matrix of connection one-forms ! is written a a ¼ most easily obtained by using Walker-Penrose conserved with the help of Ricci rotation coefficients !cb as ! b þ a c quantity K1 iK2 [4]. We adapt the calculation scheme of !cb [3,20]. Taking into account Eq. (7) and the anti- a ¼ ¼ Ref. [8]. Trajectories reach the asymptotic outgoing value symmetry of the connections ! b !ab !ba, we find
121501-3 RAPID COMMUNICATIONS
AHARON BRODUTCH AND DANIEL R. TERNO PHYSICAL REVIEW D 84, 121501(R) (2011) ¼ Df^ d initial angle 2 0. After the second scattering and ap- ¼ð b^ f2 þ b^ f1Þ !1 k d 1 2 d 32 propriate reflections, it is returned to the initial position with the initial value of the momentum. Then þ ð 3 1 þ 3 2Þ k !31f !32f : (19) 1 ¼ ð 2 2ÞþOð 3Þ A comparison with Eq. (4) leads to the identification ¼ 4aM 1 2 : (22) ^ ¼ 3 2 ¼ 3 ¼j j b1 k!32, k!31, where k k , and an al- ternative equation for the polarization rotation,
d ¼ ! ^ þ 1 VI. OUTLOOK k !32k: (20) d Observation of the frame-dragging effects is the last of ð ^ ^ Þ Freedom of choosing the polarization frame b1; b2 at the ‘‘classical’’ tests of GR [22] that has not yet been every point of the trajectory is represented by a SO(2) performed with a sufficient accuracy. In addition to ðc ð ÞÞ Gravity Probe-B [23] and LAGEOS [24] experiments, rotation Rk^ . Under its action, the connection trans- forms as ! ! R!R 1 þ R 1dR [20], so d =d ° there are proposals to use Sagnac interferometry with d =d þ dc =d . ring-laser gyroscopes [25], or matter or quantum optical kin ¼ kout m ¼ m interferometry [26]. The main difficulty in these tests is the A closed photon trajectory (i.e. , xin xout) may occur ‘‘naturally’’ (as a result of the initial conditions) necessity to isolate a much larger geodetic effect that is or with judiciously positioned mirrors. The resulting caused by the Earth’s mass. gauge-invariant phase is Polarization phase (along a closed trajectory or in the I I Newton gauge for an open path) is insensitive to it. We take ¼ ! ^ þ 1 Eq. (13) to estimate the polarization rotation in the near- kd !32kd : (21) Earth environment. The Earth angular momentum is J ¼ 40 2 1 Given a trajectory with a tangent vector k, one can define a 5:86 10 cm g sec [27]. We send the photon on a SO(2) line bundle with the connection ! ¼ !1 k, similarly geodetic from the principal null congruence starting at 32 ð Þ to the usual treatment of geometric phase [20,21]. Using some rin; in and detect it at infinity (otherwise, we ¼ Stokes’ theorem, the reference-frame term can be rewritten have to correct by the term a sin in=rout). Assuming rin as a surface integral of the bundle curvature ¼ d! ,as 12270 km (the semimajor axis of the LAGEOS satellite R RR ! ¼ . A more practical expression follows from our orbit [24]), we obtain 55arc msec in a single run. For previous discussion: ¼ arcsinf^ b^ . comparison, the relevant precession rates for the Gravity out 2 Probe-B and LAGEOS experiments are 39 arc msec/yr and Conservation of K1 and K2 in the Kerr space-time ensures that if a trajectory is closed as a result of the initial 31 arc msec/yr, respectively. We showed that polarization rotation in a gravitational conditions, then f^ ¼ f^ and ¼ 0. The Newton gauge out in field is obtained from both the Machian Coriolis accelera- is designed to give a zero phase along any trajectory in the tion and a reference-frame term, and the latter may be Schwarzschild space-time. As a result of the gauge invari- dominant. It is responsible for a gauge-independent geo- ance of Eq. (21), no phase is accrued along a closed metric phase that is accrued on a closed trajectory in sta- trajectory in the Schwarzschild space-time, regardless of tionary space-times. The effect is proportional to the the gauge convention. angular momentum of a gravitating body and possibly As an example of a nonzero phase along a closed tra- may be used in future gravity probe experiments. jectory, consider the following combination of the scatter- ing scenarios. The trajectory starts parallel to the axis of ACKNOWLEDGMENTS rotation with an impact parameter s1 (and the initial angle ¼ 1 ). Far from the gravitating body (so its influence We thank B.-L. Hu, A. Kempf, G. Milburn, N. can be ignored), the outcoming photon is twice reflected Menicucci, K. Singh, C. Soo, and V. Vedral for comments and sent in again with the impact parameter s2 and the and discussions.
[1] M. Born and E. Wolf, Principles of Optics (Cambridge [4] S. Chandrasekhar, The Mathematical Theory of Black University Press, Cambridge, England, 1999). Holes (Oxford University Press, Oxford, 1992). [2] T. Padmanabhan, Theoretical Astrophysics (Cambridge [5] G. V. Skrotskii, Sov. Phys. Dokl. 2, 226 (1957). University Press, Cambridge, England, 2000), vol. I. [6] M. Nouri-Zonoz, Phys. Rev. D 60, 024013 [3] C. W. Misner, K. S. Thorn, and J. A. Wheeler, Gravitation (1999); M. Sereno, Phys. Rev. D 69, 087501 (Freeman, San Francisco, 1973). (2004).
121501-4 RAPID COMMUNICATIONS
POLARIZATION ROTATION, REFERENCE FRAMES, AND ... PHYSICAL REVIEW D 84, 121501(R) (2011) [7] F. Fayos and J. Llosa, Gen. Relativ. Gravit. 14, 865 (1982). [19] N. H. Lindner, A. Peres, and D. R. Terno, J. Phys. A 36, [8] H. Ishihara, M. Takahashi, and A. Tomimatsu, Phys. Rev. L449 (2003). D 38, 472 (1988). [20] T. Frankel, The Geometry of Physics (Cambridge [9] P.P. Kronberg, C. C. Dyer, E. M. Burbidge, and V.T. University Press, Cambridge, England, 1999). Junkkarinen, Astrophys. J. 367, L1 (1991); C. C. Dyer [21] M. V. Berry, Nature (London) 326, 277 (1987);R. and E. G. Shaver, Astrophys. J. 390, L5 (1992). Ołkiewicz, Rep. Math. Phys. 33, 325 (1993). [10] C. R. Burns, C. C. Dyer, P.P. Kronberg, and H.-J. Ro¨ser, [22] C. M. Will, Theory and Experiment in Gravitational Astrophys. J. 613, 672 (2004). Physics (Cambridge University Press, Cambridge, [11] S. M. Kopeikin and B. Mashhoon, Phys. Rev. D 65, England, 1993); Am. J. Phys. 78, 1240 (2010). 064025 (2002). [23] Gravity Probe-B Science Results—NASA Final Report [12] V. Faraoni, New Astronomy 13, 178 (2008). (2008); C. W. F. Everitt et al., Classical Quantum [13] F. Tamburini, B. Thide´, G. Molina-Terriza, and G. Gravity 25, 114002 (2008). Anzolin, Nature Phys. 7, 195 (2011). [24] I. Ciufolini and E. C. Pavlis, Nature (London) 431, 958 [14] J. Barbour and F. Pfister, Mach’s Principle: From (2004). Newton’s Bucket to Quantum Gravity (Birkha¨user, [25] M. O. Scully, M. S. Zubairy, and M. P. Haugan, Phys. Boston, 1994). Rev. A 24, 2009 (1981); G. E. Stedman, K. U. Schreiber, [15] B. B. Godfrey, Phys. Rev. D 1, 2721 (1970). and H. R. Bilger, Classical Quantum Gravity 20, 2527 [16] A. Brodutch, T. Demarie, and D. R. Terno, (2003). arXiv:1108.0973 [Phys. Rev. D (to be published). [26] A. Camacho and A. Camacho-Galva´n, Rep. Prog. Phys. [17] L. D. Landau and E. M. Lifshitz, The Classical Theory of 70, 1937 (2007); C. Feiler et al., Space Sci. Rev. 148, 123 Fields (Butterworth-Heinemann, Amsterdam, 1980). (2009). [18] E. Wigner, Ann. Math. 40, 149 (1939); S. Weinberg, The [27] I. Ciufolini and J. A. Wheeler, Gravitation and Quantum Theory of Fields (Cambridge University Press, Inertia (Princeton University, Princeton, New Jersey, Cambridge, England, 1996), vol. I. 1995).
121501-5 Information for Referees http://publish.aps.org/refinfo.html
American Physical Society
Log in | Create Account (what's this?) RSS Feeds | Email Alerts
APS Journals APS » Journals » Information for Referees Article Lookup Journal Search Site Search About the Journals Information for Referees Browse the Journals Search the Journals General Information APS Home Review Process Join APS Guidelines for Professional Conduct PACS Scheme Annual Index Instructions Sent to Referees (Referee Package) BAPS Physical Review (PDF) APS arXiv Mirror Physical Review Letters (PDF)
Authors Notice to Referees General Information Physical Review A (PDF) Submit a Manuscript Physical Review B (PDF) Publication Rights Physical Review B Rapids (PDF) Open Access Physical Review D (PDF) Policies & Practices Physical Review E (PDF) Tips for Authors Professional Conduct Advice/Suggestions to Referees
Physical Review Referees Physical Review Letters General Information Physical Review Special Topics - Accelerators and Beams Submit a Report Physical Review Special Topics - Physics Education Research Update Your Information Policies & Practices Referee Response Forms Referee FAQ Physical Review (PDF) Advice to Referees Physical Review Letters (PDF) Outstanding Referees Physical Review Special Topics - Accelerators and Beams (PDF)
Librarians Referee Query Form (PDF) General Information Short Papers Subscriptions Short Papers (general information) (PDF) Online License Agreement Comments/Replies (general information): Usage Statistics PRA (more) Your Account PRB PRC Students PRD Physical Review Focus PRE PhysicsCentral PRL (more) Student Membership ST-AB (more) ST-PER APS Members Subscriptions Comments/Replies (specific instructions):
Article Packs PRA and E (PDF) Membership PRB (PDF) APS News PRC (PDF) Meetings and Events PRD (PDF) PRL (PDF)
APS | Journals | Privacy | Policies | Contact Information | Join APS | Feedback
Use of the American Physical Society websites and journals implies that the user has read and agrees to our Terms and Conditions and any applicable Subscription Agreement. Physical Review ®, Physical Review Letters ®, Reviews of Modern Physics ®, and Physical Review Special Topics ® are trademarks of the American Physical Society.
1 of 1 2/05/2011 3:25 PM