Covariant formulation of spin optics for electromagnetic waves

Abstract

We develop geometric optics expansion up to the subleading order for circularly polarized electromagnetic waves on curved spacetime. This subleading order geometric optics expansion, in which the conventional eikonal function is modified by inserting a carefully chosen helicity-dependent correction, is called spin optics. We derive the propagation and polarization equations in the spin optics approximation as electromagnetic waves travel in curved spacetime. Polarization-dependent deviation of the light ray trajectory from the geodesic, describing the gravitational spin Hall effect, is observed. We also establish an analogy with the related phenomena (optical Magnus effect) of condensed matter physics.

Appendices

Geometric optics limit

All the results of geometrical optics could be retrieved by taking Eqs. (12)–(14) and substituting $m_1^\beta =0=l_1^\beta$. The Lorenz condition Eq. (12) and the wave equation Eq. (13) in the leading order approximation in $\omega$ reduces to

$$\begin{aligned} l_0^\alpha m_{0 \alpha }= 0= l_0^\alpha l_{0\alpha }. \end{aligned}$$

(A1)

Next, we calculate $\tilde{m}_{0\alpha }j^\alpha$ from Eq. (13) by taking $m_1^{\alpha }=0=l_{1\mu }$ as they are subleading order terms in $\omega$ and thus irrelevant in geometric optics approximation, to obtain

$$\begin{aligned} l^\beta _{0;\beta }+2 \tilde{m}_{0 \alpha } m^\alpha _{0;\beta }l_0^\beta +2 \frac{a_{;\beta }}{a} l_0^\beta =0. \end{aligned}$$

(A2)

Since the term $\tilde{m}_{0 \alpha } m^\alpha _{0;\beta }l_0^\beta$ is purely imaginary and the remaining terms

$$\begin{aligned} l^\beta _{0;\beta }+2 \frac{a_{;\beta }}{a} l_0^\beta , \end{aligned}$$

are purely real, they should be separately zero, thereby giving

$$\begin{aligned} l^\beta _{0;\beta }+2 \frac{a_{;\beta }}{a} l_0^\beta =0, \qquad m^\alpha _{0;\beta }l_0^\beta =0. \end{aligned}$$

(A3)

In the geometric optics approximation, these are the entire set of equations for electromagnetic waves in curved spacetime.

Checking self-duality

To verify that the tetrad $\left( \dot{x}^\alpha , n^\alpha , m^\alpha , \tilde{m}^\alpha \right)$, satisfying Eqs. (62), (63) and (65) in the subleading order, is not a self-dual solution, let us first calculate Eq. (34):

$$\begin{aligned}{} {} \mathcal{Z}_{\alpha \beta } l^\alpha \tilde{m}^\beta& =\frac{i a}{\omega } \left( -\frac{a_{;\alpha }}{a}l_0^\alpha +m_{0\alpha ;\beta }l_0^\alpha \tilde{m}_0^\beta \right) \nonumber \\{} & {} =\frac{i a}{\omega } \left( \frac{1}{2} l^\alpha _{0;\alpha }-m_0^\alpha l_{0 \alpha ;\beta } \tilde{m}_0^\beta \right) =0, \end{aligned}$$

(B1)

where Eq. (A3) is used to obtain this identity. Similarly, Eq. (33) gives

$$\begin{aligned}{} {} \mathcal{Z}_{\alpha \beta } \left( \tilde{m}^\alpha m^\beta - l^\alpha n^\beta \right)& =\frac{i}{\omega }\left( \frac{a_{;\alpha }}{a}m_0^\alpha -m_{0\alpha ;\beta }l_0^\alpha n_0^\beta \right) \nonumber \\{} & {} =\frac{i}{\omega }\left( - m^\alpha _{0; \alpha }- i b_\alpha m_0^\alpha -m_{0\alpha ;\beta }l_0^\alpha n_0^\beta \right) =0, \end{aligned}$$

(B2)

where Eq. (66) is used to arrive at this identity. Finally, Eq. (32) gives

$$\begin{aligned}{} {} \mathcal{Z}_{\alpha \beta } m^\alpha n^\beta & =\frac{i}{\omega }\left( -m_{0\alpha ;\beta }m_0^\beta n_0^\alpha \right) \nonumber \\{} & {} =\frac{i}{\omega }\left( m_0^\alpha n_{0\alpha ;\beta } m_0^\beta \right) \equiv \frac{i}{\omega } \tilde{\lambda }, \end{aligned}$$

(B3)

where $\lambda$ denotes the Newman–Penrose scalar. Hence, Eq. (32) is not satisfied unless $\tilde{\lambda }=0$.