PHYSICAL REVIEW D 84, 104043 (2011) Photon polarization and geometric phase in general relativity
Aharon Brodutch,1 Tommaso F. Demarie,1 and Daniel R. Terno1,2,3 1Department of Physics & Astronomy, Macquarie University, Sydney NSW 2109, Australia 2Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo ON N2L 2Y5, Canada 3Centre for Quantum Technologies, National University of Singapore, Singapore 117543 (Received 17 August 2011; published 29 November 2011) Rotation of polarization in an external gravitational field is one of the effects of general relativity that can serve as a basis for its precision tests. A careful analysis of reference frames is crucial for a proper evaluation of this effect. We introduce an operationally-motivated local reference frame that allows for a particularly simple description. We present a solution of null geodesics in Kerr space-time that is organized around a new expansion parameter, allowing a better control of the series, and use it to calculate the resulting polarization rotation. While this rotation depends on the reference-frame con- vention, we demonstrate a gauge-independent geometric phase for closed paths in general space-times.
DOI: 10.1103/PhysRevD.84.104043 PACS numbers: 04.20.Cv, 04.25.Nx, 03.65.Vf
I. INTRODUCTION technology is used in precision tests of relativity [16]or quantum information processing on the orbit [17]. Electromagnetic waves—visible light and other bands of Gravity causes polarization to rotate. This effect is the spectrum—are our prime source of information about known as a gravimagnetic/Faraday/Rytov-Skrotski rota- the Universe [1]. Since measurements of the light deflec- tion [18–21]. Helicity is invariant under rotations, but tion near the Sun [2] were made in 1919 wave propagation states of a definite helicity acquire phases e i . is used to test general relativity (GR). Two of the ‘‘classical Depending on the context we refer to either as a tests’’ of GR, light deflection and time delay, can be under- polarization rotation or as a phase. Once evaluated this stood in terms of geometric optics [3]. The first post- phase can be encapsulated as a quantum gate [12] and eikonal approximation [4] allows to track the evolution incorporated into (quantum) communication protocols or of electric and magnetic fields along the light rays and thus metrology tasks. discuss polarization. In the Schwarzschild space-time, as well as at the lead- In this approximation we speak about photons with a ing order of the post-Newtonian approximation, this phase null four-momentum k and a transversal four-vector polar- is known to be zero [18,19,22,23]. It is higher-order gravi- ization f. Both vectors are parallel-transported along the tational moments that are held responsible for the rotation trajectory, which is a null geodesic [5,6] of the polarization plane. In particular, the GR effects were k k ¼ 0; rkk ¼ 0; (1) shown to dramatically alter polarization of the X-ray ra- diation that is coming from the accretion disc of the (then presumed) black hole in Cyg X 1 [24]. Numerous ana- k f ¼ 0; rkf ¼ 0; (2) lytical and numerical studies of trajectories and polariza- tions in different models and astrophysical regimes were where rk is a covariant derivative along k. performed ([19–21,25–29], and references therein). The In the last decades polarization has been yielding im- scenarios included fast-moving gravitating bodies, influ- portant astrophysical and cosmological data. Cosmic mi- ence of gravitational lenses and propagation through gravi- crowave background [7], blazar flares [8,9], astrophysical tational waves. jets [9,10], and searches for dark matter [11] are just The results are often contradictory. Some of the contra- several examples where polarization conveys crucial dictions result from genuine differences in superfluously information. similar physical situations [20,28,29]. On the other hand, Photons are commonly used as physical carriers of polarization rotation is operationally meaningful if the quantum information. The abstract unit of quantum infor- evolving polarization vector is compared with some stan- mation is a qubit (a quantum bit) [12], and two linearly dard polarization basis (two linear polarizations, right- and independent polarizations encode the two basis states of a left-circular polarizations, etc.) at each point along the ray qubit. One of the branches of quantum information is [30,31]. quantum metrology, which aims to improve precision mea- Setting up and aligning detectors requires alignment of surements by using explicitly quantum effects, such as local reference frames. Much work has been done recently entanglement [13]. Relativistic properties of the informa- on the role of reference frames in communications, espe- tion carriers [14,15] become important when quantum cially in the context of quantum information [32]. Partial
1550-7998=2011=84(10)=104043(13) 104043-1 Ó 2011 American Physical Society BRODUTCH, DEMARIE, AND TERNO PHYSICAL REVIEW D 84, 104043 (2011) ^ ^ knowledge of reference frames can lead to loss of commu- such as k , and the four-vector itself as k ¼ k eð Þ, where nication capacity, and mistakes in identifying the informa- eð Þ are vectors of a local orthonormal tetrad. tion content of a physical system. The lack of definition for polarization standards and an ad hoc introduction of the angle adjustments is one reason for the variety of quoted II. NULL GEODESICS values for the phase . Even the consensual result of a A. Null geodesics in the Kerr space-time zero phase in the Schwarzschild space-time should be The Kerr metric in the Boyer-Lyndquist coordinates is qualified. Without specifying the appropriate reference given by [33] frame it is either meaningless or wrong. 2 In this article we investigate the role of local reference 2 2Mr 2 2 2 2 ds ¼ 1 dt þ dr þ d frames in defining . The result is obviously gauge- 2 dependent, and additional considerations should be used 2 2 to fix the gauge. Communicating reference frames is some- þ 2 þ 2 þ Mra sin2 sin2 2 r a 2 d times a difficult procedure which may involve a high communication cost. On a curved background of GR it 4Mar sin2 dtd ; (3) may require some knowledge of the metric at each point 2 along the trajectory. Using the method presented by us in [31] we build a local reference frame in stationary space- where M is the mass and a ¼ J=M the angular momentum times, and then fix the standard polarizations in a single per unit mass of the gravitating body. We use the standard construction using what we call the Newton gauge. Our notations construction does not require communication between the 2 ¼ r2 þ a2cos2 ; ¼ r2 2Mr þ a2: (4) parties, gives a precise meaning to the idea that there is no polarization rotation in the Schwarzschild space-time, and Thanks to the three conserved quantities—the energy reproduces the absence of phase in the Minkowski space- E ¼ k0, the z-component of the angular momentum L time. From an operational point of view it allows us to set and the constant [6,34]—the geodesic equations in Kerr up detectors at any point in space based only on the local space-time are integrable in quadratures. When dealing properties at that point. We illustrate the use of this gauge with photons it is convenient to set the energy to unity by studying polarization rotation in the Kerr space-time, and to rescale other quantities [6]. Explicit form and the and derive an explicit expression for in the scattering asymptotic expansions of D ¼ L=E and are given in scenario that is described below. The calculations are based Appendix A. Fixing the energy and using the null vector on the results of [6,21,25] with the added value of the condition leaves us with only two components of the four- Newton gauge. Careful bookkeeping is required with re- momentum k. In specifying the initial data we usually take spect to the various coordinate systems used in different k1 and k1 as independent. parts of the calculation, as well as different orders of the We discuss polarization from the point of view of static series expansion. To simplify the latter, we introduce a new observers that are at rest in the ‘‘absolute’’ space t ¼ const: expansion parameter. At every point (outside the ergosphere, if the model rep- Closed paths result in a gauge-invariant gravity-induced resents a black hole) we introduce a chronometric ortho- phase. We discuss them in Sec. IV. The rest of the paper is normal frame [5,35] (Appendix A), which will be used to organized as follows: First we discuss null trajectories in express the initial conditions and observed quantities. the Kerr space-time. In Sec. III we review the Wigner’s The Hamilton-Jacobi equation for a null geodesic sepa- construction of setting standard polarizations in a local rates and the trajectories can be deduced from it [6]. We frame, set up the Newton gauge and explicitly calculate label their initial and final points as ðr1; 1; 1Þ and the rotation in the scattering scenario, where the light is ðr2; 2; 2Þ, respectively. The gauge convention that we emitted and observed far from the gravitating body (r1, adapt (Sec. III B) makes the knowledge of redundant r2 M, a), but can pass close to it. Finally, we discuss for discussions of polarization rotation in the scattering gauge-dependent and gauge-invariant aspects of our results scenario (for the calculation of see Appendix A). In from mathematical and operational points of view particular we are interested in the integrals (Sec. IV). Summary of the important facts about the Kerr Z Z r dr d space-time, as well as detailed calculations and special R :¼ pffiffiffiffi ¼ pffiffiffiffiffi ; (5) cases are presented in the appendices. R We use þþþ signature, set G ¼ c ¼ 1 and use Einstein’s summation convention in all dimensions. where Three-dimensional vectors are written in boldface and R ¼ r4 þða2 D2 Þr2 the unit vectors are distinguished by carets, such as b^ . þ 2 ð þð Þ2Þ 2 Local tetrad components are written with carets on indices, M D a r a ; (6)
104043-2 PHOTON POLARIZATION AND GEOMETRIC PHASE IN ... PHYSICAL REVIEW D 84, 104043 (2011) ¼ þ a2cos2 D2cot2 : (7) Equating the radial and angular expressions for R we obtain Null geodesics in Kerr space-time are classified accord- pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ing to the sign of the constant . The scattering scenario 4M sin cos in 2 0 2 out ¼ in þ Oð Þ; corresponds to > , and typically r1. Following sin in [25] we also assume that the angle reaches its extremal (13) value (either maximum or minimum) only once, 1 ! max= min ! 2. Series expansions of the above integrals with the higher-order terms and special cases described in are most conveniently written with the help of a new Appendix B. The plus sign corresponds to the trajectory in constant which first decreases with r and reaches min before increasing to out. 2 :¼ D2 þ : (8) þ We use the parametrization D ¼ cos , and ¼ B. Null geodesics in 1 3 formalism 2sin2 , with 0 . In this notation the minimal In stationary space-times a tetrad of a static observer is coordinate distance from the center is naturally related to Landau-Lifshitz 1 þ 3 formalism [33]. 3 2 2 cos 2cos2 Static observers follow the congruence of timelike Killing ¼ M þ aM a vectors that defines a projection from the space-time mani- rmin M 2 2 fold M onto a three-dimensional space 3, : M ! 3. þ Oð 2Þ: (9) In practice it is performed by dropping the timelike coordinate of an event, and vectors are projected by a We perform the integration by the methods of [6,25] push-forward map k ¼ k in the same way. (Appendix B). In the scattering scenario r1, r2 !1, while Contravariant vector components satisfy the constants of motion are kept finite. Expansion in terms m m m m of gives ðkÞ ð kÞ ¼ðkÞ ¼ dx =d ; m ¼ 1; 2; 3; (14) 1 2sin2 R ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c þ c a ð3ðc þ c Þ 2 2 1 2 2 1 2 where is the affine parameter. a 4 The metric g on M can be written in terms of a three- 4 þ sinc 1 cosc 1 þ sinc 2 cosc 2ÞÞ þ Oð Þ ; dimensional scalar h, a vector g, and a metric on 3 as
2 0 m 2 2 (10) ds ¼ hðdx gmdx Þ þ dl ; (15) cosc ¼ cos 2 where = þ and þ is defined in the where the three-dimensional distance is given by dl ¼ appendix. m n mndx dx . The metric components are To separate conceptual issues from the computational details we make two simplifying assumptions. First, the g0mg0n mn ¼ gmn ; (16) initial and final points are taken to lie in the asymptotically g00 flat regions, ri !1, while and D are kept finite. Second, in these regions the polar angle (nearly) reaches its and asymptotic values in and out, respectively. The asymptotic form of momentum can be deduced h ¼ g00;gm ¼ g0m=g00: (17) from the equations of motion [6]. In the chronometric tetrad basis the outgoing momentum tends to The inner product of three-vectors will always refer to this m n metric, k f ¼ mnk f . Vector products and differential s out D k ^ ! 1; 1; ; ; (11) operators are defined as appropriate dual vectors [33]. r r sin out Finally, the spatial projection of a null geodesic has a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi length l that is related to the affine parameter as where out :¼þ ð outÞ and s ¼ 1. This expression is true in any scenario where an outgoing photon reaches the dl 2 k2 asymptotically flat region. ¼ 0 ¼ k2 ¼: k2: (18) The asymptotic expression for the incoming momentum d h is similar, For a static observer the three spatial basis vectors of the ^ s in D local orthonormal tetrad are projected into an orthonormal k ! 1; 1; ; ; (12) e ¼ ^ ^ ^ ¼ r r sin in triad, ðmÞ eðmÞ, eðmÞ eðnÞ mn. We adapt a gauge pffiffiffiffiffiffiffiffiffiffiffiffiffiffi in which polarization is orthogonal to the observer’s four- where in :¼þ ð inÞ, and s ¼ 1 correspond to in ! velocity, u f ¼ 0. In a three-dimensional form this condi- max= min ! out, respectively. tion reads as k f ¼ 0.
104043-3 BRODUTCH, DEMARIE, AND TERNO PHYSICAL REVIEW D 84, 104043 (2011) In stationary space-times the evolution equations Eq. (1) tion gauge relates 3- and 4- polarization vectors, ^ and (2) can be reduced into a convenient three-dimensional b ¼ð0; b1;2Þ1; 2. form [20]. Using the relations between four- and three- The standard Lorentz transformation can be taken as1 dimensional covariant derivatives, r and D , respec- m ðkÞ¼ ð ^ Þ ð Þ tively, the propagation Eqs. (1) and (2) are brought to a L R k Bz k ; (23) three-dimensional form [19,20], where BzðkÞ is a pure boost along the z-axis that takes kS to ^ ðk; 0; 0;kÞ, and the standard rotation Rðk^ Þ brings the z-axis Dk Eg k Df ^ ¼ k þ k; ¼ f: (19) ^ ð Þ d k2 d to the desired direction k ; first by rotating by around the y-axis and then by around the z-axis, The angular velocity of rotation is ^ RðkÞ¼Rzð ÞRyð Þ: (24) ¼ 2! ð! k^ Þk^ E k; (20) g The standard polarization vectors for an arbitrary momen- where g and h play the roles of a vector potential of a tum are defined as gravimagnetic field Bg, ^ ^ ^ S b iðkÞ¼RðkÞbi ; (25) 1 1 and a general real four-vector of polarization can be written ! ¼ k0curl g k0Bg; (21) 2 2 as and a scalar potential of a gravielectric field, f ¼ cos b1ðkÞþsin b2ðkÞ: (26) rh Helicity is invariant under Lorentz transformations. The E ¼ ; (22) g 2h corresponding polarization vectors are respectively [36]. ^ 1 ^ ^ b ðkÞ¼pffiffiffi ðb1ðkÞ ib2ðkÞÞ: (27) 2 III. LOCAL REFERENCE FRAMES AND Under a Lorentz transformation a state of a definite POLARIZATION ROTATION helicity acquires a phase e i ð ;kÞ, which can be read off To discuss how a photon’s polarization rotates it is from Wigner’s little group element 1 necessary to define the standard polarization directions Wð ; kÞ¼L ð kÞ LðkÞ¼Rzð ÞTð ; Þ: (28) along its trajectory. Only after physically defining two standard polarizations it is meaningful to talk about polar- The little group element W leaves the standard momentum k ð Þ ization rotation, and only in a particular gauge the S-invariant. Here T ; form a subgroup which is iso- ð Þ Schwarzschild space-time induces a zero phase on an morphic to the translations of a Euclidean plane and Rz open trajectory. is a rotation around the origin of that plane, which in this case is also a rotation around the z-axis. If the transformation in question is a pure rotation R, A. Wigner phase—polarization convention then the little group element is a rotation Rzð Þ, and the Transversality of electromagnetic waves makes the polarization three-vector is rotated by R itself [30]. choice of two standard polarization directions momentum- The phase has a simple geometric interpretation: it is ^ ^ dependent. On a curved background it also depends on the the angle between, say, b1ðRkÞ and Rb1ðkÞ. It is zero if location. Wigner’s construction of the massless represen- the standard polarizations of the new momentum are the tation of the Poincare group [37,38] is the basis for classi- same as the rotated standard polarizations of the old fication of states in quantum field theory. We use it at every momentum. space-time point to produce standard polarization vectors. This is what happens if a rotation R2ð!Þ is performed The construction—part of the induced representation of ^ around the current b2ðkÞ: the resulting phase ðR2ð!ÞÞ is ´ 0 the Poincare group [38]—consists of a choice of a standard zero [31]. Indeed, if k^ ¼ k^ ð ; Þ, then by setting k^ ¼ reference momentum kS and two polarizations, a standard ^ R2ð!Þk and using the decomposition Lorentz transformation LðkÞ that takes kS to an arbitrary k ^ 1 ^ momentum , and a decomposition of an arbitrary Lorentz R2ð!Þ¼RðkÞRyð!ÞR ðkÞ; (29) transformation in terms of the standard transformations and Wigner’s little group element W. The standard refer- 1 ence three-momentum is directed along the z-axis of an When it does not lead to confusion we use the same letter to arbitrarily chosen reference frame, with its x and y axes label a four-dimensional object and its three-dimensional part. In ^ S particular, R stands both for a Lorentz transformation which is a defining the two linear polarization vectors b1;2, respec- pure rotation, and for the corresponding three-dimensional rota- tively. Hence kS ¼ð1; 0; 0; 1Þ, and imposing the polariza- tion matrix itself.
104043-4 PHOTON POLARIZATION AND GEOMETRIC PHASE IN ... PHYSICAL REVIEW D 84, 104043 (2011) we find that k^ 0 ¼ k^ 0ð þ !; Þ, and the little group ele- C. Kerr space-time examples ment is We impose the temporal gauge in local frames, hence 1 ^ 0 ^ 1 W ¼ R ðk ÞR2ð!ÞRðkÞ¼R ð þ !ÞR ð ÞR ð!Þ¼1; 2 y y y ^ 2Marf sin f0 0 , ft (30) 2 2Mr ^ indicating the absence of rotation with respect to standard 2Maf3 sin polarization basis. ¼ þ Oðr 3Þ; (34) r2 In a curved space-time one has to provide the standard ðxyzÞ directions at every point. The next section deals with 0^ 0^ where we used f ¼ f eð Þ. This transversality reduces to this problem. a familiar three-dimensional expression k f k f ¼ 1^ 1^ þ 2^ 2^ þ 3^ 3^ ¼ 0 B. Newton gauge f k f k f k . This also implies The three-dimensional propagation Eqs. (19) in station- ^ 0^ ft ¼ eð Þtf ¼ eð0Þtf ¼ 0: (35) ary space-times result in a joint rotation of polarization and unit tangent vectors In the Kerr space-time the Walker-Penrose quantity [39] K2 þ iK1, ^ Df^ Dk ¼ k^ ; ¼ f^: (31) d d K1 ¼ rB aA cos ; K2 ¼ rA þ aB cos ; (36) In the Schwarzschild space-time test particles move in the where plane passing through the origin [5,6,33], ¼ Eg k 2 and the covariant time-time component of the metric is A ¼ðktfr krftÞþaðkrf k frÞsin ; (37) related to the Newtonian gravitational potential ’ as h ¼ ¼ 1 þ 2 ð Þ¼1 2 g00 ’ r M=r. Requiring the resulting B ¼ððr2 þ a2Þðk k k f Þ aðktf k ftÞÞ sin ; phase to be zero, as in [18,19,26], constrains a choice of local reference frames. We use the zero phase condition to (38) make a physically motivated choice of standard polariza- is conserved along null geodesics [6]. Transversality and tions that does not require references to a parallel transport the gauge (34) make the Walker-Penrose constants func- or communication between the observers. ^ tions of only two polarization components, for example f2 Orient the local z-axis along the direction of the free-fall 3^ acceleration, z^ k r’ and the standard polarizations as and f . For a generic outgoing null geodesic in the asymptotic ^ ^ ^ ^ ^ ^ y^ b2 :¼ w k=jw kj; b1 :¼ k b2; (32) regime the constants become where w is a free-fall acceleration in the frame of a static ¼ 2^ 3^ ¼ ^ K1 outfout s outfout; observer. Then b2, and the propagation induces no (39) 2^ 3^ phase. K2 ¼ s outfout þ outfout; In a general static space-time we take the z-axis along the local free-fall direction as seen by a static observer. In where the Kerr space-time its components are [35] pffiffiffiffi :¼ csc sin 2 2 2 out D out a out; (40) Mð 2r Þ M M 4 w1^ ¼ ¼ þ Oðr Þ; 3ð 2 2MrÞ r2 r3 and the analogous expression gives the constant in terms of 2 sin2 sin2 (33) the initial data. Hence, ¼ Mra ¼ 2 þ Oð 5Þ w2^ 3 2 a M 4 r ; ð 2MrÞ r 1 1^ ¼ 0 3^ ¼ ð þ Þ ¼ 0 fout ;fout 2 2 s outK1 outK2 ; w3^ : out þ out 1 This convention, that we will call the Newton gauge, is 2^ ¼ ð Þ fout 2 2 s outK2 outK1 : (41) consistent: if we set z^ ¼ r^ in the flat space-time, then no out þ out phase is accrued as a result of the propagation. In addition to being defined by local operations, the Newton gauge has To determine the polarization rotation we take the ^ in two further advantages. First, it does not rely on a weak initial polarization to be, say, fin ¼ b1 , which often con- field approximation to define the reference direction. siderably simplifies the expression for K1 and K2. Since K1 Second, if the trajectory is closed or self-intersecting, the and K2 are linear functions of polarization, expressing the ^ out ^ out reference direction z^ is the same at the points of the above result in the basis ðb1 ; b2 Þ the desired rotation intersection. angle is observed from
104043-5 BRODUTCH, DEMARIE, AND TERNO PHYSICAL REVIEW D 84, 104043 (2011) 1 1 cos Motion in the equatorial plane corresponds to ¼ 0 ! T ¼ ; (42) 0 0 sin ( sin ¼ 0) and is qualitatively similar to the motion in Schwarzschild space-time. If the trajectory starts there but where T is an orthogonal matrix that is described below. eventually moves outside, then the polarization is rotated We assume fixed D and . Using Eq. (32) we find that at by the limit rin !1 the initial standard linear polarization 8M2a directions are sin ¼ s sin þ Oð 4Þ: 3 (52) 1 ^ in ¼ ð0 sin Þ b1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ;s in in;D ; In the case of initial propagation along the z-axis agrees 2 þ 2 sin2 D in in (43) with both [18,20], if we take into account the respective 1 definitions of reference frames. On the other hand, in a ^ in ¼ ð0 sin Þ b2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ;D; s in in : generic setting our Eq. (49) differs by a power of from 2 þ 2 sin2 3 D in in [21]or[19] (both works predict but disagree on the prefactors). The origin of this disagreement is in the Similarly, the final standard polarizations are following. In addition to using a more transparent series 1 ^ out expansion, we use a different polarization basis from the b1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð0;s out sin out; DÞ; 2 2 2 one in [21]. In particular, we do not require the knowledge D þ outsin in (44) of out in. Moreover, [19] considered only a Machian 1 ^ out b2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð0;D;s out sin outÞ: effect (Sec. IV), which is dominated by the reference- 2 2 2 D þ outsin in frame term [31].
The rotation matrices Nin and Nout from the polarization bases to ð ;^ ^ Þ basis are given in the Appendix C. V. GEOMETRIC PHASE Following [21] using the asymptotic relationship be- A more geometric perspective on polarization rotation is tween Walker-Penrose constants and the components of possible both in a static space-time, where we use the 1 þ 3 polarization one can introduce a transformation matrix R, formalism, as well as in arbitrary space-times. While the ! ! 2^ 2^ phase is gauge-dependent for an open trajectory, we fout ¼ fin 3^ R 3^ : (45) will show that it is gauge-invariant on a closed path. fout fin Consider the differential equations for polarization ro- f ¼ b1 The transformation matrix has the form tation [31]. By setting at the starting point and using the parallel transport equations Eqs. (1) and (2) we arrive at 1 1 x the equation R ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; (46) 1 þ 2 x 1 x d 1 1 ¼ rkðf b2Þ¼ f rkb2: (53) where the parameter x is given by d cos f b1
in out out in In a static space-time projection of the polarization on 3 x ¼ s ; (47) in out þ in out results in 1 ð^ ^ Þ with in :¼ D csc in a sin in. Finally, d D f b2 ¼ ; (54) 1 d cos d T ¼ NoutRNin : (48) hence the desired equation is Expansion in the inverse powers of results in ^ ^ d 1 Df ^ ^ Db2 4Ma 3 ¼ b2 þ f T21 ¼ sin ¼ cos in þ Oð Þ: ^ ^ 2 (49) d f b1 d d 1 ^ This is our main new result. Few special cases are of ¼ ! ^ þ ^ Db2 k ^ ^ f : (55) interest. If the initial propagation is parallel to the z-axis f b2 d b and the impact parameter equals , then The first term corresponds to the original Machian effect 2 ¼ b2 a2;D¼ 0; (50) that was postulated in [23], but the observable quantity involves both the Machian and the reference-frame terms and the polarization is rotated by [31]. 2 Now we discuss the geometric meaning of these equa- 4Ma 15M a sin ¼ þ þ Oð 4Þ; (51) tions. First consider a basis of one-forms ð 1; 2; 3Þ that 2 4 3 ^ ^ ^ is dual to the orthonormal polarization basis ðb1; b2; kÞ and the antiparallel initial direction gives the opposite sign. at every point of the trajectory. A matrix of connection
104043-6 PHOTON POLARIZATION AND GEOMETRIC PHASE IN ... PHYSICAL REVIEW D 84, 104043 (2011) 1-forms ! is written with the help of Ricci rotation coef- space-time. As a result of the gauge invariance of ficients !{^ as !{^ ¼ !{^ j [5,40], and a linear polariza- Eq. (61), no gravitationally-induced phase is accrued along |^ l^ l^ |^ l^ tion is written as a closed trajectory in the Schwarzschild space-time, re- gardless of the gauge convention. ^ 1^ ^ 2^ ^ ^ ^ f ¼ f b1 þ f b2 ¼ cos b1 þ sin b2: (56) In a general space-time we introduce an orthonormal tetrad such that k ¼ ke0 þ ke3 at every point of the tra- Its covariant derivative equals to jectory. We again impose a temporal gauge and set e1;2 ¼ ^ ^ b1 2, where the local polarization basis is chosen according Df ¼ a^ c ^ þ 3 c þ ð c^Þ ^ ¼ 1 2 ; k!3^ ^f ba !^ ^f k k f bc;a;c; : d c 3 c to some procedure. Then from Eq. (26) it follows that (57) df ^ r f ¼ ^ ^ ^ þ k k eð Þ! ^ ^ f eð Þ Taking into account the antisymmetry of the connections d {^ ! ^ ¼ !^ ^ ¼ !^^, we find 2^ 1^ d 1^ 1^ l { l l { ¼ð f eð1Þ þ f eð2ÞÞ ð! þ ! Þk d 0^ 2^ 3^ 2^ ^ ^ ^ Df ^ 2 ^ 1 d 1 ^ ^ ^ ^ ¼ð b1f þ b2f Þ ! k 0 0 c^ 3 3 c^ 3^ 2^ þðkð! þ ! Þf Þeð0Þ þðkð! þ ! Þf Þeð3Þ d d 0^^ c 3^^ c 0^^ c 3^^ c þ ^ ð 3^ 1^ þ 3^ 2^ Þ c ¼ 1; 2; (63) kk !3^ 1^ f !3^ 2^ f : (58) A comparison with Eq. (31) leads to the identification where we used again the antisymmetry of the connection. 1 ^ 3^ 2 3^ From the parallel transport condition rkf ¼ 0, we can see ¼ b1 ¼ k!^ ^ , ¼ k!^ ^ and to an alternative 3 2 3 1 that equation for the polarization rotation, ^ ^ ^ ^ ð!0 þ !0 Þfc^ ¼ð!3 þ !3 Þfc^ ¼ 0 (64) d ¼ ! ^ þ 1^ 0^ c^ 3^ c^ 0^ c^ 3^ c^ k !3^ 2^ k: (59) d and
Given a trajectory with a tangent vector k one can define a 1^ 1^ ^ ¼ð þ Þ :¼ ¼ 1 d !0^ 2^ !3^ 2^ kd : (65) SO(2) line bundle with the connection ! !3^ 2^ kd , simi- larly to the usual treatment of geometric phase [40,41]. We cannot have a closed trajectory in chronologically ^ ^ Freedom of choosing the polarization frame ðb1; b2Þ at protected space-time, so to obtain a gauge-invariant result every point of the trajectory is represented by a SO(2) we consider two future-directed trajectories that begin and ðc ð ÞÞ rotation Rk^ . Under its action the connection trans- end in the same space-time points. This layout is similar to forms as ! ! R!R 1 þ R 1dR [40], so the two arms of a Mach-Zender interferometer [4]. We also c align the initial and final propagation directions of the d ! ! ^ þ 1^ þ d k !3^ 2^ k : (60) beams and use the same rules to define the standard polar- d d izations. Similarly to the previous case the curvature is just In a static space-time we can consider a closed trajectory ¼ d , and the Stokes theorem gives the phase as in space. Then the resulting phase is gauge-invariant, since Z Z I ZZ the last term above is a total differential and drops out upon ¼ ¼ ¼ : (66) the integration on a closed contour, 1 2 I I ¼ ! k^ d þ !: (61) V. CONCLUSIONS AND OUTLOOK We can formalize it with the help of a curvature 2-form is Rotation one ascribes to polarization as light propagates introduced as :¼ d! þ ! ^ !: For the SO(2) bundle it on a curved background depends on the gauge conventions reduces to ¼ d! , so by using Stokes’ theorem the that are used along the way. However, the closed-loop reference-frame term can be rewritten as a surface integral phase is gauge-independent (assuming we use the same of the bundle curvature as reference frame for transmission and detection). Similarly I ZZ to other instances of a geometric phase, the phase is ! ¼ : (62) given by the integral of the (bundle) curvature over the surface that is bounded by the trajectory. The Newton A more practical expression follows from our previous gauge provides a convenient local definition of the stan- ^ ^ discussion: ¼ arcsinfout b2. Conservation of K1 and dard polarizations. It is motivated by its relative simplicity K2 in the Kerr space-time ensures that if a trajectory is and path-independence of the reference frames. It is also ^ ^ closed as a result of the initial conditions, then fout ¼ fin the gauge in which the statement of a zero accrued phase and ¼ 0. The Newton gauge is designed to give a along an arbitrary path in the Schwarzschild space-time is zero phase along any trajectory in the Schwarzschild correct.
104043-7 BRODUTCH, DEMARIE, AND TERNO PHYSICAL REVIEW D 84, 104043 (2011) pffiffiffiffi ^ Our results characterize the behavior of polarization k1 ¼ kr = ; (A5) qubits on curved backgrounds. Specific regimes and real- istic scenarios, both on the Kerr background and beyond, 2^ ¼ are to be investigated farther. Of special interest are those k k ; (A6) scenarios that will be experimentally feasible in the near sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi future, such as sending photons between satellites [17]. 3^ ¼ sin Once the expression for is obtained, it can be represent k k 2 2 ; (A7) a sin it as the action of a quantum gate [12], similarly to the the 2 2 m^ special-relativistic scenarios [14,42]. This will allow us to while k ¼ k ¼ k km^ satisfies use the full toolbox of quantum optics to design experi- 2Mr 2M ments. Understanding of the polarization phase in GR 2 ¼ 1 þ ¼ 1 þ þ Oð 2Þ k 2 2 r : (A8) opens new possibilities for optics-based precision mea- Mr r surements [31], both classical and quantum.
ACKNOWLEDGMENTS 2. Constants of motion The rescaled (E ¼ 1) z-component of the angular mo- We thank P. Alsing, B.-L. Hu, A. Kempf, and N. mentum is Menicucci for discussions and comments. Research at Perimeter Institute is supported by the Government of 1 D ¼ ðk ða2 þ r2Þð 2 2MrÞ Canada through Industry Canada and by the Province of 2 2Mr Ontario through the Ministry of Research & Innovation. 2 2Marð1 k asin ÞÞ: (A9)
APPENDIX A: SOME ASPECTS OF THE KERR The rescaled Carter’s constant SPACE-TIME :¼ K ðD aÞ2; (A10) 1. Chronometric tetrad where the constant K is most conveniently expressed as We fix an orthonormal tetrad at every space-time point 2 2 2 (outside the static limit) by demanding that e0 is the four- K :¼ða sin D= sin Þ þð k Þ : (A11) velocity of a static observer, and the vectors e1 and e2 are The asymptotic expressions when r !1and the momen- proportional to the tangent vectors @r and @ . Their cova- tum components are fixed are riant components eð Þ are 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ¼ sin þ Oð 1Þ 2 D k3^ r r ; (A12) @ 2Mr 2Marsin A e ð0Þ ¼ 1 ; 0; 0; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (A1) 2 2 2 Mr ¼ð 2cos2 þ 2Þ 2 þ cos2 ð ð 2 þ 2 1Þ k3^ k2^ r a a k2^ k3^ and 1 4Mk^ sin ÞþOðr Þ; (A13) pffiffiffiffi 3 eð1Þ ¼ð0; = ; 0; 0Þ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 1 eð2Þ ¼ð0; 0; ;0Þ; (A2) ¼ r k^ þ k^ þ Oðr Þ; (A14) 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 3 @ A eð3Þ ¼ 0; 0; 0; sin 2sin2 k3^ sin a cos ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ Oðr 2Þ: (A15) 2 þ 2 k2^ k3^ Setting E ¼ k0 ¼ 1 leads to 2 2 Scattering with the impact parameter b that measures the 2Mrak sin t ¼ coordinate distance r cos from the z-axis is most conven- k 2 2 ; (A3) Mr iently described with the help of a fiducial Cartesian sys- and kr is expressed from the null condition k2 ¼ 0, tem. Its axes are ’’parallel’’ to the fictitious Cartesian axes of the Byer-Lindquist coordinates, and in the asymptotic 2 2 2 2 2 2 ½1 ðk Þ ð a sin Þ ðk Þ sin region to the global Cartesian grid. We introduce the ðkrÞ ¼ : ^ 2sin2 momentum components pi that are related to the spherical a ^ (69) components ki the the usual relations. In this case for the initial momentum parallel to the z-axis we have The spatial components of the momentum in the coor- pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m^ ðmÞ ¼ Oð 3Þ ¼ 2 2 þ Oð 2Þ dinate and tetrad bases are related to k ¼ e k as D r1 ; b a r1 : (A16)
104043-8 PHOTON POLARIZATION AND GEOMETRIC PHASE IN ... PHYSICAL REVIEW D 84, 104043 (2011) Some care is needed in treating RðrÞ as a function of the Defining initial conditions when r1 !1. Introducing the constants ð1 þ x2Þ c0, c1, c2, we write it as ð Þ :¼ f x ð 2 2Þ ; 4 2 2 2 2 2 a RðrÞ¼: r c r r þ c1r r þ c0r ; (A17) (B7) 2 1 1 1 ðD aÞ2 þ 1 þ x þ x2 gðxÞ :¼ 2 ; where 2 a2 1 þ x 2 : 2 2 2 2^ 2 3^ 2 2 c2 ¼ð a Þ=r1 ¼ðk1Þ þðk1Þ þ Oðr1 Þ (A18) we finally get Z 1 dr 2 2 R ¼ 2 pffiffiffiffi c1 :¼ 2Mð þ a 2 a cos Þ=r1 1 rmin R ^ 2 2 sin 2 Z 1 ¼ 2 2 ak1 1 þ Oð 2Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 2Þ M c2 r1 ; (A19) 2 2 dx x r1 a 0 a2 M 1=2 2 2 2 1 þ ð Þþ ð Þ c0 :¼ a sin 2 f x g x : (B8) rmin rmin 2 2^ 2 2 3^ 2 2 ¼ a ððk Þ cos 1 þðk Þ ÞþOðr Þ: (A20) 1 1 1 Expanding the integrand up to the third order in gives Eq. (15). 1 4 15 3 APPENDIX B: TRAJECTORIES M 2 2 2 R1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ þ M a sin 2 2 2 4 4 1. The r integral a 8aM M 128 r cos þ 2 þ 2ð10cos2 The integral over in Eq. (5) is split into three parts, 2 3 3 M a Z : r dr R ¼ pffiffiffiffi ¼ R1 R1 R2; (B1) 6sin2 Þþ15 cos þ Oð 4Þ R Ma : (B9) where The second and third integrals can be performed as an Z Z 1 dr 1 dr approximation in powers of r1, using Eq. (A17). Since R 1 :¼ 2 pffiffiffiffi ; R :¼ pffiffiffiffi : (B2) i 2 rmin R r R ð þ Þ i ð Þ¼ð 4 2 2 2Þ 1 þ c0 c1r r1 R r r c2r1r 2ð 2 2 2Þ ; (B10) We present a corrected expression for Ri and evaluate r r c2r1 the term R1 to the fourth order in . Starting from R1 in using the method of [25]. the leading terms in the expansion are Factoring out (r rmin) and substituting r ¼ rmin=x Z 1 1 pdrffiffiffiffi ¼ arcsinc2r1 allows to write RðrÞ as r R c2r1 r i i 2 3 4 2 2 2 2 rmin Bx Cx rmin c1 2 2r c r R ¼ r2 þ Ax2 þ þ ¼: R~ðxÞ; (B3) þ iqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 1 þ Oð 5Þ 4 min 2 4 2 ð 2 Þ2 r1 ; x rmin rmin x c2r1 4 2 2 2 2 c2rir1 ri c2r1 where the constants A, B, C are determined by the poly- (B11) nomial division. Noting that R~ð1Þ¼0 we have with r being either r1 or r2, where in the latter case we 2 ¼ B C i rmin A 2 (B4) assume that r2 * r1. rmin rmin which gives 2. The integral R pffiffiffiffiffi 4 1 ~ð Þ¼ B C þ 2 þ B þ Cx We calculate the integral = d using the method of R x A Ax 2 : (B5) rmin rmin rmin rmin [6]. Noting that 2 2 Rewriting the equation and inserting the values for A, B ¼ þða DÞ ða sin D csc Þ ; (B12) and C gives us :¼ cos we perform a change of variables and obtain r2 a2 ð Þ¼ min ð 2 2Þð1 2Þ 1þ ð1þ 2Þ Z d Z d R r 4 a x 2 2 2 x ¼ pffiffiffiffiffi ¼ x rminð a Þ I qffiffiffiffiffiffiffiffi ; (B13) 2 ð Þ2 þ 1þ þ 2 M D a x x 2 2 : (B6) rmin a 1þx where for >0
104043-9 BRODUTCH, DEMARIE, AND TERNO PHYSICAL REVIEW D 84, 104043 (2011) 2 2 2 2 2 2 2 2 ¼ a ð þ Þð þ Þ; 0 þ; 2 ¼ 15M =4 8Ma cos ; (B25) (B14) 2 2 and 3 ¼ Mð 6a 128M þ 45Ma cos 0 1 2 2 2 24a cos2 þ a ð3 þ cos2c 1Þsin Þ: (B26) 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 :¼ @ ð 2 2Þ2 þ 4 2 ð 2 2ÞA 2 a a a : 2a Similarly, setting 2 ¼ 1 þ , (B15) X # ¼ k ; (B27) k The calculation is performed using the approximations k to the auxiliary integral Z and expanding the both sides of þ d 1 qffiffiffiffiffiffiffiffi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðc 2Þ F ;k ; (B16) cos 1 2 2 cosð Þ¼ cos arccos a þ þ þ c ; (B28) þ ðc 2Þ where F ;k is the elliptic integral of the first kind, and we find 2 :¼ 2 ð 2 þ 2 Þ cosc :¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k þ= þ ; = þ; (B17) 2 2 sin cos 1 qffiffiffiffiffiffiffiffi #1 ¼ 4M : (B29) 2 sin with 0 c and þ ¼þ þ. 1 The asymptotic expansion in the powers of gives In the special case of scattering with the initial momen- a2 tum parallel (antiparallel) to the z-axis the angular mo- 2 ¼ sin2 þ sin22 þ Oð 4Þ ¼ 1 ¼ c ¼ 0 þ 4 2 (B18) mentum is zero, so þ and in in , , and it is easy to see that out ¼ c , c , respectively.
2 2 2 2 a 2 4 ¼ cos þ sin 2 þ Oð Þ (B19) 3. The motion a2 2 2 The equation for is given by [6] so Z Z Z r dr r rdr 2 r dr a2sin2 ¼ D pffiffiffiffi þ 2Ma pffiffiffiffi a D pffiffiffiffi k2 ¼ þ Oð 4Þ; (B20) R R R 2 Z Z d d D pffiffiffiffiffi pffiffiffiffiffi : (B30) sin2 1 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 a 2 4 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 sin þ Oð Þ : 2 2 2 2 2 If const: the first and the last terms cancel thanks to a þ þ a Eq. (5). For brevity we write the remaining terms as (B21) Z r Z : Trajectories of the type 1 ! min= max ! 2 corre- 2 1 ¼ R ðrÞdr Tð Þd : (B31) spond to cos min ¼ þ, and cos max ¼ þ < 0, respec- tively. In both cases the integration leads to We again decompose the radial integral as R ¼ 1 2 2 R1 R1 R2 , where R ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½Fðc 1;k ÞþFðc 2;k Þ ; (B22) 2 2 Z Z a þ þ 1 1 R ¼ 2 R ð Þ R ¼ R ð Þ 1 r dr; i r dr: and the expansion in powers of leads to Eq. (10). rmin ri (B32) Using that at the zeroth order (the flat space-time) c 1 þ c ¼ c ¼ c þ 2 , we expand 2 1 c as Following the same procedure as in Sec. B2we find X k 8 cos c 2 ¼ c 1 þ : (B23) a M a k R 1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 2 a2 4 Equating the two expressions for R1 in the scattering Mð3M 8a cos Þ þ þ Oð 3Þ : (B33) scenario leads to 6 2
¼ 4 1 M; (B24) Expanding R ðrÞ for r r1 we obtain
104043-10 PHOTON POLARIZATION AND GEOMETRIC PHASE IN ... PHYSICAL REVIEW D 84, 104043 (2011) 2aM R ðrÞ¼qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 2 c1r1 c0r1 2 2 ðr c2r1Þð 1 þ 2 2 2 þ 2 2 2 2 Þðr þ a 2MrÞ rðr c2r1Þ r ðr c2r1Þ a2D þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ð 2 2 2Þð 1 þ c1r1 þ c0r1 Þð 2 þ 2 2 Þ r r c2r1 rðr2 c2r2Þ r2ðr2 c2r2Þ r a Mr 2 1 2 1 a 2M aD ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2M 1 þ þ þ Oðr 5Þ; (B34) 2 2 2 2 r r r r c2r1
2 where the coefficients ci are defined in Appendix A2. where ðq; c ;k Þ is the elliptic integral of the third kind. Hence Expanding the prefactor in the powers of gives Z 1 2aM R ðrÞdr ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D 1 sec 2 2 2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 ri þ þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ Oð Þ; (B41) ri ri c2r1 ri 2 2 1 2 2 2 0 a þ þ þ a 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a D @ 2 2 2 c2r1 c r þ r 2 3 3 3 2 1 i while c2r1ri 0 11 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 arctan@ c2r1 AA þ Oð 5Þ ð p; c ; 0Þ¼pffiffiffiffiffiffiffiffiffiffiffiffiffi arccotð 1 þ p tanc Þ ri qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r1 : 1 þ 2 2 2 þ 2 p c2r1 ri :¼ 0ðp; c Þ (B42) (B36) R Integral Td leads to the elliptic integral of the third and cos ¼ ¼ þ cost kind. A standard change of variables 2 and the identity 2 k ð p; c ;k Þ¼ 0ðp; c Þþ ðc 0ðp; c ÞÞ 2 2 2 2p sin ¼ð1 þÞð1 þ psin tÞ; (B37) þ Oðk4Þ: (B43) where 2 2 Taking into account that :¼ þ ¼ tan2 1 þ a þ Oð 4Þ p 2 2 (B38) 1 þ 1 a2 pffiffiffiffiffiffiffiffiffiffiffiffiffi ¼jcos j 1 sin2 þ Oð 4Þ; (B44) lead to 1 þ p 2 Z 1 ð Þ ¼ D T d qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 min 2 2 1 þ we get for the scattering scenario a þ þ Z c X dt D i pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; out in ¼ þ ; (B45) 2 2 2 j j i 0 ð1 þ psin tÞ 1 k sin t D i (B39) where the first two terms of the series are with cosc ¼ cos = þ as in Appendix B2. Hence Z cos 1 2 1 ð Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð c Þ 1 ¼ ; (B46) T d 2 p; ;k ; cos2 c cos þ sin2 c sec min 2 2 1 þ a þ þ (B40) and
3 2 2 2ð3 þ cos2 Þsec þ 4 sec ð 1 sin2c 1 2 cos2c 1Þtan 1 2 2 ¼ þ a cos þ 2Ma; (B47) ð1 þ sec2 cos2c tan2 Þ2 4 where i are given in Appendix B2.
104043-11 BRODUTCH, DEMARIE, AND TERNO PHYSICAL REVIEW D 84, 104043 (2011) APPENDIX C: NEWTON GAUGE RELATIONSHIPS Projecting f^ on these directions gives
In this Appendix we present a relationship between the 1 components of polarization in the chronometric tetrad and y ¼ ð 2^ ð Þ f N f w1^ k1^ k3^ w2^ k3^ k2^ the Newton gauge basis. This relationship allows to calcu- k1^ late the Walker-Penrose conserved quantity from the op- 3^ 2 2 þ f ðw1^ k1^ k2^ þ a1^ ðk^ þ k^ ÞÞÞ; (C5) erationally meaningful polarization information and in the 3 1 asymptotic regime gives the matrices N of Eq. (48). The transversally of polarization k^ f^ ¼ 0 ensures the 1 ^ ^ 2^ 3^ x y x 2 3 linearity of the relationship between ðf ;f Þ and ðf ;f Þ, f ¼ ðf ðw2^ k1^ w1^ k2^ Þþf ð w1^ k3^ ÞÞ: (C6) N 1^ where k
^ x ^ y ^ f ¼ f bx þ f by: (C1) We define the transformation matrix N as ^ ! Since b ¼ w^ k=jw^ kj and b ¼ b k, by using ^ y x y f2 fy Eq. (33) we find ^ ¼ N : (C7) f3 fx 1 b ¼ ðw^ k^ ; w^ k^ ; w^ k^ þ w^ k^ Þ; (C2) y N 2 3 2 3 2 1 1 2 ^ In the asymptotic regime where f1 ! 0 it becomes or- thogonal. Taking km^ ¼ð1; s =r; D=r sin Þ we obtain in 1 2 2 b ¼ ðw^ k^ k^ w^ ðk þk Þ; the leading order 1=r x N k 2 1 2 1 3^ 2^ ð 2 þ 2Þ ð þ ÞÞ w2^ k2^ k1^ w1^ k1^ k2^ ; k3^ w1^ k1^ w2^ k3^ ; (C3) 1 D= sin s N ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : (C8) 2 þ 2 sin2 s D= sin 2 2 2 2 D = N ¼ðw2^ k3^ Þ þðw2^ k3^ Þ þð w2^ k1^ þ a1^ k2^ Þ : (C4)
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