The gravitational spin Hall effect of light

The propagation of electromagnetic waves in vacuum is often described within the geometrical optics approximation, which predicts that wave rays follow null geodesics. However, this model is valid only in the limit of infinitely high frequencies. At large but finite frequencies, diffraction can still be negligible, but the ray dynamics becomes affected by the evolution of the wave polarization. Hence, rays can deviate from null geodesics, which is known as the gravitational spin Hall effect of light. In the literature, this effect has been calculated ad hoc for a number for special cases, but no general description has been proposed. Here, we present a covariant WKB analysis from first principles for the propagation of light in arbitrary curved spacetimes. We obtain polarization-dependent ray equations describing the gravitational spin Hall effect of light. We also present numerical examples of polarization-dependent ray dynamics in the Schwarzschild spacetime, and the magnitude of the effect is briefly discussed. The analysis presented here is analogous to the spin Hall effect of light in inhomogeneous media, which has been experimentally verified.


Introduction
The propagation of electromagnetic waves in curved spacetime is often described within the geometrical optics approximation, which applies in the limit of infinitely high frequencies [48,58]. In geometrical optics, Maxwell's equations are reduced to a set of ray equations, and a set of transport equations along these rays. The ray equations are the null geodesics of the underlying spacetime, and the transport equations govern the evolution of the intensity and the polarization vector. In particular, the geometrical optics approximation predicts that the ray equations determine the evolution of the polarization vector, but there is no backreaction from the polarization vector onto the ray equations. However, this model is valid only in the limit of infinitely high frequencies, and there has been interest in calculating the light propagation more accurately. At large but finite frequencies, diffraction can still be negligible but rays can deviate from geodesics. This is known as the gravitational spin Hall effect of light [50].
The mechanism behind the spin Hall effect is the spin-orbit interaction [13], i.e., the coupling of the wave polarization (spin) with the translational (orbital) motion of the ray as a particle, resulting in polarization(spin)-dependent rays. Related phenomena are found in many areas of physics. In condensed matter physics, electrons travelling in certain materials experience a spin Hall effect, resulting in spin-dependent trajectories, and spin accumulation on the lateral sides of the material [26,62]. The effect was theoretically predicted by Dyakonov and Perel in 1971 [28,27], followed by experimental observation in 1984 [6] and 2004 [40]. In optics, the polarization-dependent deflection of light travelling in an inhomogeneous medium is known as the spin Hall effect of light [13,43]. The effect was predicted by several authors [20,42,52,10,11,22,23], and has recently been verified experimentally by Hosten and Kwiat [38], and by Bliokh et al. [12]. The spin Hall effect of light provides corrections to the geometrical optics limit, which scale roughly with the inverse of frequency. This, and several other effects, can be explained in terms of the Berry curvature [66,7,12,13].
There are several approaches aiming to describe the dynamics of spinning particles or wave packets in general relativity. Using a multipole expansion of the energy-momentum tensor, the dynamics of massive spinning test particles has been extensively studied in the form of the Mathisson-Papapetrou-Dixon equations [47,53,68,15,16]. A massless limit of these equations was derived by Souriau and Saturnini [63,57], and particular examples adapted to certain spacetimes have been discussed in [25,24,46]. Another commonly used method is the Wentzel-Kramers-Brillouin (WKB) approximation for various field equations on curved spacetimes. For massive fields, this has been done in [5,54] by considering a WKB approximation for the Dirac equation. For massless fields, using a WKB approximation for Maxwell's equations on a stationary spacetime, Frolov and Shoom derived polarizationdependent ray equations [30,31] (see also [71,19,18,36]). With methods less familiar in general relativity, using the Foldy-Wouthuysen transformation for the Bargmann-Wigner equations in a perturbative way, Gosselin et al. derived ray equations for photons [35] and electrons [34] travelling in static spacetimes (see also [61,60,51]). The gravitational spin Hall effect of gravitational waves was also considered in [71,70]. However, as discussed in [50], there are inconsistencies between the predictions of these different models, and some of them only work in particular spacetimes.
In this work, we are concerned with describing the propagation of electromagnetic waves in curved spacetime, beyond the traditional geometrical optics approximation. We carry out a covariant WKB analysis of the vacuum Maxwell's equations, closely following the derivation of the spin Hall effect in optics [12,55,56], as well as the work of Littlejohn and Flynn [44]. As a result, we derive ray equations that contain polarization-dependent corrections to those of traditional geometrical optics, and capture the gravitational spin Hall effect of light. As in optics, these corrections can be interpreted in terms of the Berry curvature. To illustrate the effect, we give some numerical examples of the effective ray trajectories in the Schwarzschild spacetime.
Our paper is organized as follows. In Section 2 we start by introducing the variational formulation of the vacuum Maxwell's equations. Then, we present the specific form of the WKB ansatz to be used, we discuss the role of the Lorenz gauge condition, and we state the assumptions that we are considering on the initial conditions. In Section 3, we present the WKB approximation of the field action, and the corresponding Euler-Lagrange equations. Analyzing these equations at each order in the geometrical optics parameter , we obtain the well-known results of geometrical optics. The dynamics of the polarization vector is expressed in terms of the Berry phase. Finally, we derive an effective Hamilton-Jacobi system that contains O( ) corrections over the standard geometrical optics results. In Section 4, we use the corrected Hamilton-Jacobi equation to derive the ray equations that account for the gravitational spin Hall effect of light. The gauge-invariance of these equations is discussed, and noncanonical coordinates are introduced. In Section 5, we present some basic numerical examples. We consider the effective ray equations on a Schwarzschild background, and compare with the results of Gosselin et al. [35]. The magnitude of the effect is estimated numerically. A summary of the result, including the effective Hamiltonian and the effective ray equations, can be found in Section 6.
Notations and conventions. We consider an arbitrary smooth Lorentzian manifold (M, g µν ), where the metric tensor g µν has signature − + ++. The absolute value of the metric determinant is denoted as g = | det g µν |. The phase space is defined as the cotangent bundle T * M , and phase space points are denoted as (x, p). The Einstein summation convention is assumed. Greek indices represent spacetime indices, and run from 0 to 3. Latin indices from the beginning of the alphabet, (a, b, c, ...), represent tetrad indices and run from 0 to 3. Latin indices from the middle of the alphabet, (i, j, k, ...), label the components of 3-vectors and run from 1 to 3.

2.
Maxwell's equations and the WKB approximation 2.1. Lagrangian formulation of Maxwell's equations. Electromagnetic waves in vacuum can be described by the electromagnetic tensor F αβ . This is a skew-symmetric, real 2-form, which satisfies the vacuum Maxwell's equations [48] Solutions to Maxwell's equations can also be represented by introducing the electromagnetic four-potential A α , which is a real 1-form. Then, the electromagnetic tensor can be expressed as 2) and equations (2.1) become [48] This equation can be obtained as the Euler-Lagrange equation of the following action: where the last equality is obtained using integration by parts.
2.2. WKB Ansatz. We assume that the vector potential admits a WKB expansion of the form where S is a real scalar function, A α is a complex amplitude, and is a small expansion parameter. The gradient of S is denoted as Note that we are allowing the amplitude A α to depend on k µ (x). In physical terms, the limit 1 indicates that the phase of the vector potential rapidly oscillates, and its variations are much faster than those corresponding to the amplitude A α (x, k, ).
The role of the expansion parameter becomes clear if we consider a timelike observer, traveling along the worldline τ → x α (τ ), with proper time τ . This observer measures the frequency where t α = dx α /dτ is the velocity vector field of the observer. The phase function S and are dimensionless quantities. Working with geometrized units, such that c = G = 1 [69, Appendix F], the velocity t α is dimensionless, and k α has dimension of inverse length. Hence, ω has the dimension of the inverse length, as expected for frequency. Then, an observer sees the frequency going to infinity as goes to 0.
We illustrate the validity condition of the geometrical optics approximation on a Schwarzschild black hole, with Schwarzschild radius r s . For a source of light that is falling into the black hole, the gravitational redshift formula implies that the frequency ω ∞ measured by an observer at infinity in the rest frame of the central object is smaller than the frequency measured by an observer at finite distance from the black hole. Then, a criterion for the high-frequency limit to hold is = (ω ∞ r s ) −1 1.
(2.8) Note that we could have taken any observer at finite distance of the black hole as a criterion. The choice of the observer at infinity provides the simplest expression.
2.3. Lorenz gauge. As commonly known, Maxwell's equations in the form (2.3) are not hyperbolic. In particular, they admit pure gauge solutions. We eliminate this problem by introducing a gauge, specifically, the Lorenz gauge (2.9) Using the identityD (2.14) Assume furthermore that the initial data for the wave equation (2.14) satisfy The initial data (2.15) for the wave equation (2.14) imply that, initially, Observe that the condition is automatically satisfied, where T is a unit future-oriented normal vector to the hypersurface on which initial data are prescribed. Hence, the equation satisfied by the Lorenz gauge source function (2.16) admits initial data as in (2.17) and (2.18) vanishing at the appropriate order in (at O( 1 ) and O( 0 ), respectively). This implies that Maxwell's equationŝ 19) are satisfied in the Lorenz gauge 2.4. Assumption on the initial conditions. In this paper, we consider solutions of the vacuum Maxwell's equations assuming a WKB ansatz (2.5), with initial conditions satisfying the following properties: (1) The Lorenz gauge (2.20) is satisfied initially.
(2) The initial phase gradient k α is a future-oriented null vector. This assumption is in fact a compatibility condition resulting from the dispersion relation (3.8) below, and the Lorenz gauge condition (2.20). (3) Initially, the beam has circular polarization (see Section 3.4).

Higher-order geometrical optics
3.1. WKB approximation of the field action. We compute the WKB approximation for our field theory by inserting the WKB ansatz (2.5) in the field action (2.4): Since e iS/ is a rapidly oscillating function, the Riemann-Lebesgue lemma implies, for sufficiently regular f , Upon expanding the derivative terms in equation (3.1), and keeping only terms of the lowest two orders in , we obtain the following WKB approximation of the field action (for convenience, we are shifting the powers of , such that the lowest-order term is of O( 0 )): Here, D β α represents the symbol [32] of the operatorD β α , evaluated at the phase space point (x, p) = (x, k), and we are using the notation v ∇ µ D β α for the vertical derivative (Appendix A) of D β α , evaluated at the phase space point (x, p) = (x, k). The action depends on the following fields: Following the calculations in Appendix B, the Euler-Lagrange equations are In the above equations, the symbol D β α and its vertical derivatives are all evaluated at the phase space point (x, k).

3.2.
0th-order geometrical optics. Starting with equations (3.5)-(3.7), and keeping only terms of O( 0 ), we obtain Equation (3.8) can also be written as: This equation admits nontrivial solutions if and only if A 0β is an eigenvector of D β α with zero eigenvalue. Two cases should be discussed: k is a null vector, or k is not a null vector.
Assume first that k is not a null vector, k µ k µ = 0. Then, equation (3.8) leads to In other words, when k is not a null vector, the corresponding solution is, at the lowest order in , a pure gauge solution. Since the corresponding electromagnetic field vanishes, we do not consider this case further.
This is consistent with the Lorenz gauge condition (2.20) at the lowest order in . A similar argument can be applied for the complex-conjugate equation (3.9), from which we obtain k α A 0 * α = 0.
Using equations (3.8)-(3.10), we obtain the well-known system of equations governing the geometrical optics approximation at the lowest order in : where I 0 = A 0 * α A 0α is the lowest-order intensity (more precisely, I 0 is proportional to the wave action density [67]). Equation (3.17) is obtained from equation (3.10) by using the orthogonality condition (3.16). Using equation (2.6), we have and we can use equation (3.15) to derive the geodesic equation for k µ : 3.3. 1st-order geometrical optics. Here, we examine equations (3.5) and (3.6) at order 1 only: Using equation (3.4), we can also rewrite equation (3.20) as follows: Using equation (3.18), we can rewrite the last two terms as With the results obtained at O( 0 ), we also have Then, equation (3.22) becomes The last term can be eliminated by using the Lorenz gauge (2.20). The same steps can be applied to the complex-conjugate equation (3.21): Furthermore, using the lowest-order intensity I 0 , we can write the amplitude vectors in the following way: where a 0α is a unit complex vector (i.e. a 0 * α a 0α = 1) describing the polarization. Then, from equation (3.26), together with equation (3.17), we obtain The parallel propagation of the complex vector a 0α along the integral curve of k µ is another well-known result of the geometrical optics approximation.
3.4. The polarization vector in a null tetrad. We observed that the polarization vector satisfies the orthogonality condition Consider the Newman-Penrose tetrad {k α , n α , m α ,m α } satisfying Since the Newman-Penrose tetrad is adapted to the vector k α , the orthogonality conditions imply that m α andm α are functions of k α . Since the polarization vector a 0α is orthogonal to k α , we can decompose it as where z 1 , z 2 , and z 3 are complex scalar functions. Since a 0α is a unit complex vector, the scalar functions z 1 and z 2 are constrained by It is important to note that the decomposition (3.31), and more specifically, the choice of m α , requires choosing a null vector n α . Fixing n α is equivalent to choosing a timelike vector field t α , that represents an observer. Once n α is fixed, the remaining SO(2) gauge freedom in the choice of m α is described by the spin rotation Altogether, the gauge freedom in the decomposition (3.31) is described by the little group, that is to say the subgroup of the transformations leaving k invariant. Using equations (3.31) and (3.19), the parallel-transport equation for the polarization vector becomes (3.34) Contracting the above equation withm α , m α , and n α , we obtain Recall that in the above equations, the vectors m α andm α are functions of x and k(x). The covariant derivatives are applied as follows: where h ∇ µ is the horizontal derivative (Appendix A). It is convenient to introduce the following 2-dimensional unit complex vector, which is analogous to the Jones vector in optics [29,9,55,56]: and we shall also use the hermitian transpose z † , defined as follows: Then, the equations for z 1 and z 2 can be written in a more compact form: where σ 3 is the third Pauli matrix, and B µ is the real 1-form extending to general relativity the Berry connection used in optics [9,55] Furthermore, if we restrict z to a worldline x µ (τ ), withẋ µ = k µ , we can writė Integrating along the worldline, we obtain where γ represents the Berry phase [9,55] γ(τ 1 ) = Using either equation (3.35) or equation (3.42), we see that the evolution of z 1 and z 2 is decoupled in the circular polarization basis, and the following quantities are conserved along k µ : Thus, we have s = ±1, depending on the choice of the initial polarization state. The results described in this section are similar to the description of the polarization of electromagnetic waves travelling in a medium with an inhomogeneous index of refraction [9].
3.5. Extended geometrical optics. Now, we take equations (3.5)-(3.7), but without splitting them order by order in . Our aim is to derive an effective Hamilton-Jacobi system that would give us O( ) corrections to the ray equations.
3.5.1. Effective dispersion relation. By contracting equation (3.5) with A * α and equation (3.6) with A β , and also adding them together, we obtain the following equation: Using equations (3.4), (3.15) and (3.16), we can rewrite this as follows: (3.48) Using equation (3.16), we obtain so we can write: where the latter equality is due to (3.18). Then, equation (3.47) becomes (3.51) Let us introduce the O( 1 ) intensity: Then, we can rewrite the amplitude as where a α is a unit complex vector. Then, equation (3.51) can be expressed as follows: This can be viewed as an effective dispersion relation, containing O( ) corrections to the geometrical optics equation (3.15). Finally, let us assume the notation and rewrite the effective dispersion relation takes the following form: This equation can also be obtained directly from the effective field action (3.3), specifically by varying the latter with respect to I. (3.57) We can perform the following replacements in the above equation: After rearranging terms, the effective transport equation becomes (3.59) The following term vanishes due to the symmetry of the Ricci tensor: (3.60) Furthermore, using the Lorenz gauge condition (2.20), we are left with the following form of the effective transport equation: (3.61) Introducing the intensity I, and the vector K µ , we obtain This is an effective transport equation for the intensity I, which includes O( ) corrections to the geometrical optics equation (3.17).

Effective ray equations
4.1. Hamilton-Jacobi system at leading order. The lowest-order geometrical optics equations (3.15) and (3.17) can be viewed as a system of coupled partial differential equations: where k µ = ∇ µ S. The first equation is a Hamilton-Jacobi equation for the phase function S, and the second equation is a transport equation for the intensity I 0 [49]. The Hamilton-Jacobi equation can be solved using the method of characteristics. This is done by defining a Hamiltonian function on T * M , such that It is obvious that in this case, the Hamiltonian is and Hamilton's equations take the following form: Once the Hamilton-Jacobi equation is solved, the transport equation (4.2) can also be solved, at least in principle [49]. However, our main interest is in the ray equations governed by the Hamiltonian (4.4). The corresponding Hamilton's equations (4.5) and (4.6) describe null geodesics. These equations can easily be rewritten as x µ + Γ µ αβẋ αẋβ = 0, (4.8) or in the explicitly covariant form: 4.2. Effective Hamilton-Jacobi system. The effective dispersion relation (3.56), together with the effective transport equation (3.62) introduce O( 1 ) corrections over the system discussed above: Using equation (3.31), the effective dispersion relation becomes (4.12) where B µ = B µ (x, k) is the Berry connection introduced in equation (3.41), and s = ±1, depending on the initial polarization. Using equation (3.43), together with the assumption on the initial polarization, we can write: Since the value of s is fixed by initial conditions, the only unknowns are the phase function S and the Berry phase γ. We can write an effective Hamilton-Jacobi equation for the total phaseS = S + sγ: (4.14) The corresponding Hamiltonian function on T * M is and we have the following Hamilton's equations: We can also write these ray equations in a more compact form where the constant matrix on the right-hand side is the inverse of the symplectic 2-form, or the Poisson tensor [45].
4.2.1. Noncanonical coordinates. The Hamiltonian (4.15) contains the Berry connection B µ , which is gauge-dependent. The latter means that B µ depends on the choice of m α andm α ; for example, the transformation m α → m α e iφ causes the following transformation of the Berry connection: This kind of gauge dependence was considered by Littlejohn and Flynn in [44], where they also proposed how to make the Hamiltonian and the equations of motion gauge-invariant. The main idea is to introduce noncanonical coordinates such that the Berry connection is removed from the Hamiltonian and the symplectic form acquires the corresponding Berry curvature, which is gauge-invariant. This is similar to the description of a charged particle in an electromagnetic field in terms of either the canonical or the kinetic momentum of the particle. The Berry connection and Berry curvature play a similar role as the electromagnetic vector potential and the electromagnetic tensor [14].
We start by rewriting the Hamiltonian (4.15) as (4.20) where H 0 = 1 2 g µν p µ p ν . Following [44], the Berry connection can be written in the following way, by using the definition of the horizontal derivative (4.21) The Berry connection can be eliminated from the Hamiltonian (4.15) by considering the following change of coordinates on T * M Since (X, P ) are noncanonical coordinates, the symplectic form transforms nontrivially under this change of coordinates. The Hamiltonian (4.15) is as a scalar, so we obtain (4.24) In the new coordinate system (X, P ), we obtained the following Hamiltonian: The corresponding Hamilton's equations can be written in a matrix form as: where we have the following Berry curvature terms (4.28) Simplified expressions for these terms can be found in appendix C. Now we can write Hamilton's equations in the new variables: The last term on the right-hand side of equation (4.29) is the covariant analogue of the spin Hall effect correction obtained in optics, (ṗ × p) /|p| 3 , due to the Berry curvature in momentum space [12,55]. This term is also the source of the gravitational spin Hall effect in the work of Gosselin et al. [35]. In equation (4.30), the second term on the right-hand side contains the Riemann tensor, and resembles the curvature term obtained in the Mathisson-Papapetrou-Dixon equations [16].

Numerical examples
To illustrate how the polarization-dependent correction terms modify the ray trajectories, we provide here some numerical examples. For convenience, we use canonical coordinates and treat x 0 as a parameter along the rays. Hence, equations (4.16) and (4.17) becomė x 0 = 1, (5.1) and p 0 is calculated from This equation can be solved explicitly, using the fact that the velocityẋ α is future oriented: while the green trajectory represents a null geodesic. We take r s = 1, and we start with the initial position x i (0) = (−50r s , 15r s , 0), and initial normalized momentum p i = (1, 0, 0). The wavelength λ is set to a sufficiently large value to make the effect visible on this plot.
Note that in general B µ depends on p 0 . However, since this is an O( 1 ) term, we can replace the O( 0 ) expression for p 0 in B µ . In order to compare with the results of Gosselin et al. [35], we consider a Schwarzschild spacetime in Cartesian isotropic coordinates (t, x, y, z): where r s = 2GM c 2 is the Schwarzschild radius, and R = x 2 + y 2 + z 2 . We also define the following orthonormal tetrad: where t µ = (e 0 ) µ is our choice of observer. The Berry connection B µ can be explicitly computed by introducing a particular choice of polarization vectors. Using the orthonormal tetrad, we can easily adapt the polarization vectors used in optics [55]. We can write p µ = P a (e a ) µ , v µ = V a (e a ) µ , and w µ = W a (e a ) µ , where the components of these vectors are given by where The vectors v µ and w µ are real unit spacelike vectors that represent a linear polarization basis, satisfying (C.2). We now have all the elements required for the numerical integration of equations (5.1)-(5.3). For this purpose, we are using Mathematica [39], with the NDSolve function. The default settings for integration method, precision and accuracy are being used. As a first step, we numerically compare our ray equations (5.1)-(5.3) with those predicted by Gosselin et al. [35]. Up to numerical errors, we obtain the same ray trajectories with both sets of equations. However, while the equations obtained by Gosselin et al. only apply to static spacetimes, equations (5.1)-(5.3) do not have this limitation.
The results of our numerical simulations are shown in Figure 1, which illustrates the general behavior of the gravitational spin Hall effect of light around a Schwarzschild black hole (the actual effect is small, so the figure is obtained by numerical integration of equations (5.1)-(5.3) for unrealistic parameters). Here, we consider rays of opposite circular polarization (s = ±1) passing close to a Schwarzschild black hole, together with a reference null geodesic (s = 0). Except for the value of s, we are considering the same initial conditions, x i (0), p i (0) , for these rays. Unlike the null geodesic, for which the motion is planar, the circularly polarized rays are not confined to a plane.
As another example, we used initial conditions x i (0), p i (0) , such that the rays are initialized as radially ingoing or outgoing. In this case (not represent by any figure, since it is trivial), the gravitational spin Hall effect vanishes, and the circularly polarized rays coincide with the radial null geodesic.
Using these numerical methods, we can also estimate the magnitude of the gravitational spin Hall effect. As a particular example, we consider a similar situation to the one presented in Figure 1, where the black hole is replaced with the Sun. More precisely, we model this situation by considering a Schwarzschild black hole with r s ≈ 3 km. We consider the deflection of circularly polarized rays coming from a light source far away, passing close to the surface of the Sun, and then observed on the Earth. This situation is illustrated in Figure 2. The numerical results are based on the initial data presented in the caption of Figure 2. When reaching the Earth, the separation distance between the rays of opposite circular polarization depends on the wavelength. For example, taking wavelengths of the order λ ≈ 10 −9 m results in a separation distance of the order d ≈ 10 −15 m, while for wavelengths of the order of λ ≈ 1 m we obtain a separation distance of the order d ≈ 10 −6 m. Although the ray separation is small (about six orders of magnitude smaller than the wavelength), what really matters is that the rays are scattered by a finite angle, so the ray separation grows linearly with distance after the re-intersection point. This means that the effect should be robustly observable if one measures it sufficiently far from the Sun. Furthermore, massive compact astronomical objects, such as black holes or neutron stars, are expected to produce a larger gravitational spin Hall effect.

Conclusions
In summary, we have presented a first comprehensive theory of the gravitational spin Hall effect that occurs due to the coupling of the polarization with the translational dynamics of the light rays. The ray dynamics is governed by the corrected Hamiltonian Here, the first term represents the geometrical optics Hamiltonian, and the second terms represents a correction of O( 1 ) that is due to the Berry connection, which is given by Assuming the noncanonical coordinates (4.22), the corresponding ray equations arė 3) where the terms F px andF xx and the timelike vector t are given in appendix C. The term containing m [νmµ] is the covariant analogue of spin Hall correction term usually encountered in optics [12,35], while the Riemann curvature term is reminiscent of the Mathisson-Papapetrou-Dixon equations [16]. The resulting deviation of the ray trajectories from those predicted by geometrical optics is weak but not unobservable. First of all, even small angular deviations are observable at large enough distances. Second, as shown shown in [38], weak quantum measurement techniques can be used to detect the spin Hall effect of light, even when the spatial separation between the left-polarized and the right-polarized beams of light is smaller than the wavelength.
Potentially, this work can be naturally extended in two directions. Firstly, the corrected ray equations are yet to be studied more thoroughly, both analytically and numerically. Rigorous numerical investigations are needed to obtain a precise prediction of the effect, in particular for Kerr black holes. Secondly, Maxwell's equations are a proxy to linearized gravity. It is expected that a similar approach can be carried out to obtain an effective pointwise description of a gravitational wave packet, extending the results of [70].
As discussed in [13], the spin Hall effect of light is directly related to the conservation of total angular momentum. For the discussion presented so far, the considered rays carry extrinsic orbital angular momentum, associated with the ray trajectory, and intrinsic spin angular momentum, associated with the polarization. However, it is well known that light can also carry intrinsic orbital angular momentum [2,3,41] (see also [1] and references therein). In principle, the magnitude of the spin Hall effect can be increased by considering optical beams carrying intrinsic orbital angular momentum [8]. The method and ansatz that we have adopted is insufficient to describe this effect. A more realistic and more precise approach involving wave packets, such as Laguerre-Gaussian beams, should be considered. It may be possible to do so using the machinery developed in [17].
A formulation of the special-relativistic dynamics of massless spinning particles and wave packets beyond the geometrical optics limit has been previously reported by Duval and collaborators, cf. [21] for the spin-1/2 case, see also [65]. This analysis relates the modified dynamics to the approach of Souriau [64], making use of so-called spin enslaving. This has been extended to general helicity by Andzejewski et al. [4]. We expect that the Hamiltonian formulation presented here corresponds to a general relativistic version of the models considered in the mentioned papers. This will be considered in a future work.
The extension for general tensor fields on T * M is straightforward. Note that, in contrast to [59, section 3.5], we have the opposite sign for the last term in the definition of the horizontal derivative. This is because our fields, u α (x, p) and v α (x, p), are defined on T * M , and not on T M , as is the case in the reference mentioned before. We can make use of the following properties:

Appendix B. Variation of the action
Here, we derive the Euler-Lagrange equations that correspond to the action where the Lagrangian density is of the following form: A * α (x, ∇S(x)), ∇ µ [A * α (x, ∇S(x))] .

(B.2)
Here, S(x) is an independent field, while A α and A * α cannot be considered independent, since they depend on ∇ µ S. Following Hawking and Ellis [37, p. 65], we define the variation of a field Ψ i as a one-parameter family of fields Ψ i (u, x), with u ∈ (−ε, ε) and x ∈ M . We use the following notation: Note that the derivative with respect to the parameter u commutes with the covariant derivative, so we have: (B.7) Integrating by parts, and assuming the boundary terms vanish, we obtain: (B.8) Since the above equation must be satisfied for all variations ∆S, ∆A α , and ∆A * α , we obtain the following Euler-Lagrange equations: (B.11) Furthermore, equation (B.11) can be simplified by using equations (B.9) and (B.10). Thus, as a final result, we have the following set of Euler-Lagrange equations: (B.12) Note that the fields c 4µ , d 3µ , f 4µ , and g 3µ are undetermined within this approach, but this is not a problem, because they do not affect the Berry curvature.