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 eÆiÁ. is used to test general relativity (GR). Two of the ‘‘classical Depending on the context we refer to Á 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 Á 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 À 1 [24]. Numerous ana- k Á 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 eðÞ, where nication capacity, and mistakes in identifying the informa- eðÞ 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 Á. 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 Á ds ¼ 1 À dt þ dr þ d frames in defining . The result is obviously gauge- 2 Á 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 À sin2dtd; (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; Á ¼ r2 À 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 ¼k0, 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 Á 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 Á 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.