Observables for the Effect of Gravity on Electromagnetic Polarization

Abstract

Does gravity affect the polarization of electromagnetic radiation in an observable way? The effect of gravity on the observed polarization of a ray of electromagnetic radiation is investigated for an arbitrary 4-dimensional spacetime and radiation with a frequency spectrum within the geometric optics limit and with arbitrary state of polarization. Focusing on effects observable by a single inertial observer, we show how the presence of curvature along the null geodesic of polarized electromagnetic radiation may induce observable changes in the state of polarization. We find a set of scalars that quantify the effect and derive their transport equations. Two of these scalars, the polarization degree and the circular polarization degree, are polarization state observables that are conserved along the radiation geodesic. Four observables that quantify time rate of change of the observed state of polarization are identified. One observable, the polarization axis rotation rate, measures angular motion of the polarization axis. The presence of curvature along the null geodesic of polarized electromagnetic radiation may induce rotation of the polarization axis of the radiation, as measured by an inertial observer. These observables quantify gravitational effects on electromagnetic polarization unambiguously. They are generally sensitive to gravity, which implies that polarized electromagnetic radiation can potentially be used to measure the gravitational fields it traverses. These observables and their corresponding transport equations provide a complete representation of how gravity affects the observed state of polarization of electromagnetic radiation with frequencies above the geometric optics limit. Polarization wiggling is sourced by curvature twist, which is a scalar derived from the Riemann tensor. Curvature twist is closely related to the magnetic part of the Weyl tensor, the second Weyl scalar as well as the rotation of the rest frame geodesic congruence. The results of this paper are valid for any metric theory of gravity.

Appendix

1.1 A Miscellaneous Derivations

1.1.1 A.1 Transverse Covariant Derivative of the Screen Rotator

The covariant derivative of the Levi-Civita tensor vanishes [17, 48]:

$$\begin{aligned} \nabla _{\gamma }\epsilon _{\alpha \beta \mu \nu }=0 \end{aligned}$$

Given a null vector $p^{\mu }$ with an expansion $p^{\mu }=p( u^{\mu }+{\hat{p}}^{\mu })$ in terms of a timelike unit vector $u^{\mu }$ and a spacelike unit-vector ${\hat{p}}^{\mu }$, the transverse covariant derivative of the screen rotator $\epsilon _{\mu \nu }$, as defined by (10), expands as follows:

$$\begin{aligned} \mathcal {D}_{\rho }\epsilon _{\mu \nu }=S_{\mu }^{\lambda }S_{\nu }^{\gamma }\epsilon _{\alpha \beta \lambda \gamma }\nabla _{\rho }u^{\alpha }{\hat{p}}^{\beta }+S_{\mu }^{\lambda }S_{\nu }^{\gamma }\epsilon _{\alpha \beta \lambda \gamma }u^{\alpha }\nabla _{\rho }{\hat{p}}^{\beta } \end{aligned}$$

Consider the first term, $S_{\mu }^{\lambda }S_{\nu }^{\gamma }\epsilon _{\alpha \beta \lambda \gamma }\nabla _{\rho }u^{\alpha }{\hat{p}}^{\beta }$: $S_{\mu }^{\lambda }S_{\nu }^{\gamma }\epsilon _{\alpha \beta \lambda \gamma }\nabla _{\rho }u^{\alpha }{\hat{p}}^{\beta }\ne 0$ if and only if $u_{\alpha }\nabla _{\rho }u^{\alpha }\ne 0$. However,

$$\begin{aligned} u_{\alpha }\nabla _{\rho }u^{\alpha }=\frac{1}{2}\nabla _{\rho }\left( u_{\alpha }u^{\alpha }\right) =0 \end{aligned}$$

since $u^{\alpha }$ is a unit vector. This implies that

$$\begin{aligned} S_{\mu }^{\lambda }S_{\nu }^{\gamma }\epsilon _{\alpha \beta \lambda \gamma }\nabla _{\rho }u^{\alpha }{\hat{p}}^{\beta }=0. \end{aligned}$$

By the same argument,

$$\begin{aligned} S_{\mu }^{\lambda }S_{\nu }^{\gamma }\epsilon _{\alpha \beta \lambda \gamma }u^{\alpha }\nabla _{\rho }{\hat{p}}^{\beta }=0. \end{aligned}$$

Therefore, the transverse covariant derivative of the screen rotator vanishes:

$$\begin{aligned} \mathcal {D}_{\rho }\epsilon _{\mu \nu }=0. \end{aligned}$$

(101)

1.1.2 A.2 The Commutator [p, u]

Let us evaluate the commutator [p, u] for a null vector $p^{\mu }$ and a timelike unit vector $u^{\mu }$. Its definition is

$$\begin{aligned} {\left[ p,u\right] }^{\beta }\equiv \nabla _{p}u^{\beta }-\nabla _{u}p^{\beta }. \end{aligned}$$

(102)

It can be decomposed into transverse and non-transverse terms by using the screen projector:

$$\begin{aligned} \nabla _{p}u^{\beta }-\nabla _{u}p^{\beta }=\mathcal {D}_{p}u^{\beta }-\mathcal {D}_{u}p^{\beta }-u^{\beta }u_{\gamma }\nabla _{p}u^{\gamma }+u^{\beta }u_{\gamma }\nabla _{u}p^{\gamma }+{\hat{p}}^{\beta }{\hat{p}}_{\gamma }\nabla _{p}u^{\gamma }-{\hat{p}}^{\beta }{\hat{p}}_{\gamma }\nabla _{u}p^{\gamma }. \end{aligned}$$

First, we find that the transverse terms vanish:

$$\begin{aligned} \mathcal {D}_{p}u^{\beta }&=S_{\alpha }^{\beta }\mathcal {D}_{p}u^{\alpha }=\mathcal {D}_{p}( S_{\alpha }^{\beta }u^{\alpha }) -u^{\alpha }\mathcal {D}_{p}S_{\alpha }^{\beta }=0\\ \mathcal {D}_{u}p^{\beta }&=S_{\alpha }^{\beta }\mathcal {D}_{u}p^{\alpha }=\mathcal {D}_{u}( S_{\alpha }^{\beta }p^{\alpha }) -p^{\alpha }\mathcal {D}_{u}S_{\alpha }^{\beta }=0. \end{aligned}$$

The remaining terms become

$$\begin{aligned} -u^{\beta }u_{\gamma }\nabla _{p}u^{\gamma }+u^{\beta }u_{\gamma }\nabla _{u}p^{\gamma }+{\hat{p}}^{\beta }{\hat{p}}_{\gamma }\nabla _{p}u^{\gamma }&-{\hat{p}}^{\beta }{\hat{p}}_{\gamma }\nabla _{u}p^{\gamma }\\ =&-\frac{p^{\beta }}{p}\nabla _{u}p+\left( \frac{p^{\beta }}{p}-u^{\beta }\right) {\hat{p}}_{\gamma }\nabla _{p}u^{\gamma }. \end{aligned}$$

Now,

$$\begin{aligned} {\hat{p}}_{\gamma }\nabla _{p}u^{\gamma }=-\frac{\nabla _{p}p}{p}, \end{aligned}$$

where $p\equiv -u_{\alpha }p^{\alpha }$ is the scalar momentum of the null vector. Thus, we find that the commutator [p, u] is

$$\begin{aligned} {\left[ p,u\right] }^{\beta }=-\frac{p^{\beta }}{p}\left( \nabla _{u}p+\frac{\nabla _{p}p}{p}\right) +u^{\beta }\frac{\nabla _{p}p}{p}. \end{aligned}$$

(103)

1.1.3 A.3 Curvature Twist

Using that

$$\begin{aligned} \epsilon ^{\lambda \mu }R_{\mu \alpha \beta \gamma }\epsilon ^{\alpha }=\epsilon ^{\lambda \mu }S_{\mu }^{\rho }S_{\alpha }^{\sigma }\epsilon ^{\alpha }R_{\rho \sigma \beta \gamma }=\frac{1}{2}\epsilon ^{\lambda \mu }( S_{\mu }^{\rho }S_{\alpha }^{\sigma }-S_{\mu }^{\sigma }S_{\alpha }^{\rho }) \epsilon ^{\alpha }R_{\rho \sigma \beta \gamma }, \end{aligned}$$

we get by the use of Proposition 2:

$$\begin{aligned} \epsilon _{\lambda }^{*} \epsilon ^{\lambda \mu }R_{\mu \alpha \beta \gamma }\epsilon ^{\alpha }=\frac{1}{2}\epsilon _{\lambda }^{*}\epsilon ^{\lambda \mu }\epsilon _{\mu \alpha }\epsilon ^{\rho \sigma }R_{\rho \sigma \beta \gamma }\epsilon ^{\alpha }=\frac{1}{2}\epsilon _{\lambda }^{*}( -S_{\alpha }^{\lambda }) \epsilon ^{\alpha }\epsilon ^{\rho \sigma }R_{\rho \sigma \beta \gamma }. \end{aligned}$$

Then, by using that $\epsilon _{\lambda }^{*}S_{\alpha }^{\lambda }\epsilon ^{\alpha }=1$ and applying the decomposition of $p^{\alpha }$ from (5), using the antisymmetry of the Riemann tensor in the two last indices, we obtain the following identity:

$$\begin{aligned} \epsilon _{\gamma }^{*}\epsilon ^{\gamma \mu }R_{\mu \lambda \alpha \beta }p^{\alpha }u^{\beta }\epsilon ^{\lambda }=-\frac{p}{2}\epsilon ^{\rho \sigma }R_{\rho \sigma \beta \gamma }{\hat{p}}^{\beta }u^{\gamma }. \end{aligned}$$

(104)

The Riemann tensor has several index symmetries. Among these are the antisymmetry of the first and last two indices

$$\begin{aligned} R_{\mu \alpha \beta \gamma }=R_{\left[ \mu \alpha \right] \beta \gamma }=R_{\mu \alpha [ \beta \gamma ] }, \end{aligned}$$

(105)

the switching relation

$$\begin{aligned} R_{\mu \alpha \beta \gamma }=R_{\beta \gamma \mu \alpha } \end{aligned}$$

(106)

and the permutation relation, which can be written

$$\begin{aligned} R_{\gamma \beta \rho \sigma }=-R_{\gamma \rho \sigma \beta }-R_{\gamma \sigma \beta \rho }=R_{\gamma \rho \beta \sigma }-R_{\gamma \sigma \beta \rho }. \end{aligned}$$

(107)

The right-hand side of (104) can be rewritten by using the antisymmetry of (105) and the switching relation of (106). Equation (104) then takes the form

$$\begin{aligned} \epsilon _{\gamma }^{*}\epsilon ^{\mu \gamma }R_{\mu \lambda \alpha \beta }p^{\alpha }u^{\beta }\epsilon ^{\lambda }=\frac{p}{2}u^{\gamma }{\hat{p}}^{\beta }\epsilon ^{\rho \sigma }R_{\gamma \beta \rho \sigma }. \end{aligned}$$

(108)

By using the permutation symmetry of (107), we obtain the following identity:

$$\begin{aligned} \epsilon _{\gamma }^{*}\epsilon ^{\mu \gamma }R_{\mu \lambda \alpha \beta }p^{\alpha }u^{\beta }\epsilon ^{\lambda }=p \mathcal {Z}, \end{aligned}$$

(109)

where

$$\begin{aligned} \mathcal {Z}\equiv u^{\gamma }{\hat{p}}^{\beta }\epsilon ^{\rho \sigma }R_{\gamma \rho \beta \sigma }=\frac{1}{2}u^{\gamma }{\hat{p}}^{\beta }\epsilon ^{\rho \sigma }R_{\gamma \beta \rho \sigma } \end{aligned}$$

(110)

is the curvature twist scalar.

1.1.4 A.4 Curvature Twist in Terms of the Weyl Tensor

$\mathcal {Z}$ can be expressed in terms of the Weyl tensor by applying the relationship between the Weyl and Riemann tensors [17]:

$$ R_{\rho \sigma \mu \nu }=C_{\rho \sigma \mu \nu }+\left( g_{\rho [\mu }R_{\left. \nu \right] \sigma }-g_{\sigma [\mu }R_{\left. \nu \right] \rho }\right) -\frac{1}{3}g_{\rho [\mu }g_{\left. \nu \right] \sigma }R, $$

where R is the Ricci scalar. Let us first evaluate the right-hand side of (110) term by term. First,

$$\begin{aligned}&u^{\rho }{\hat{p}}^{\mu }\epsilon ^{\sigma \nu }( g_{\rho [\mu }R_{\left. \nu \right] \sigma }-g_{\sigma [\mu }R_{\left. \nu \right] \rho }) =\\&\qquad \quad \frac{1}{2}\left( u^{\rho }{\hat{p}}^{\mu }g_{\rho \mu }\epsilon ^{\sigma \nu }R_{\nu \sigma }-{\hat{p}}^{\mu }\epsilon ^{\sigma \nu }{u}_{\nu }R_{\mu \sigma }-u^{\rho }\epsilon ^{\sigma \nu }{\hat{p}}_{\sigma }R_{\nu \rho }+u^{\rho }{\hat{p}}^{\mu }\epsilon ^{\sigma \nu }g_{\sigma \nu }R_{\mu \rho }\right) . \end{aligned}$$

All terms on the right-hand side vanish because of the antisymmetry and transversality of the screen rotator, $\epsilon ^{\sigma \nu }$. Next,

$$\begin{aligned} u^{\rho }{\hat{p}}^{\mu }\epsilon ^{\sigma \nu }\frac{1}{3}g_{\rho [\mu }g_{\left. \nu \right] \sigma }R=\frac{R}{6}\left( u^{\rho }{\hat{p}}^{\mu }g_{\rho \mu }\epsilon ^{\sigma \nu }g_{\nu \sigma }-u^{\rho }{\hat{p}}_{\sigma }\epsilon ^{\sigma \nu }g_{\rho \nu }\right) . \end{aligned}$$

Also here, the terms on the right-hand side vanish due to the antisymmetry and transversality of the screen rotator, $\epsilon ^{\sigma \nu }$. Thus, curvature twist can be expressed in terms of the Weyl scalar as follows:

$$\begin{aligned} \mathcal {Z}\equiv u^{\rho }{\hat{p}}^{\mu }\epsilon ^{\sigma \nu }R_{\rho \sigma \mu \nu }=u^{\rho }{\hat{p}}^{\mu }\epsilon ^{\sigma \nu }C_{\rho \sigma \mu \nu }. \end{aligned}$$

(111)

It is then straight forward to prove the alternative expression

$$\begin{aligned} \mathcal {Z}=\frac{1}{2}u^{\gamma }{\hat{p}}^{\beta }\epsilon ^{\rho \sigma }R_{\gamma \beta \rho \sigma }=\frac{1}{2}u^{\gamma }{\hat{p}}^{\beta }\epsilon ^{\rho \sigma }C_{\gamma \beta \rho \sigma }. \end{aligned}$$

(112)

1.2 B Representing the State of Polarization of Electromagnetic Radiation

An arbitrary state of polarization can be defined in terms of the four Stokes parameters. We will now review the polarization representations applied in this paper. It uses the nomenclature and definitions of Section 2 above.

1.2.1 B.1 The Polarization Matrix

The polarization state of a classical field of electromagnetic radiation is completely characterized by the four Stokes parameters [42, 43]. They are measurable quantities that can be expressed in terms of the electric field vector and its projections against two mutually orthogonal unit vectors that both are transverse to the direction of propagation of the radiation, $\hat{p}$. Let $e_{A}^{\mu }, A=1,2$ denote the two transverse unit vectors, the polarization basis. They form a basis for any transverse field. Since the electric field vector $E^{\mu }$ is transverse to the direction of propagation, it can be defined by its projection onto this basis: $E_{A}\equiv e_{A}^{\mu }E_{\mu }$. We can then define the coherency matrix $I_{\textrm{AB}}$ in terms of $E_{A}$ as the expectation value

$$\begin{aligned} I_{\textrm{AB}}\equiv \left\langle E_{A}E_{B}^{*}\right\rangle \end{aligned}$$

(113)

evaluated as a time average over a large number of radiation cylces [42]. We notice that the use of a complex representation of the electric field in (113) is permissible as long as the average is taken over a large number of radiaton cycles, see Born and Wolf for details [42].    $I_{\textrm{AB}}$ is a $2\times 2$ Hermitian matrix. Written in matrix form, it can be expressed in terms of the four Stokes parameters I, Q, U, V as

$$\begin{aligned} I_{\textrm{AB}}=\frac{1}{2}{\left( \begin{array}{cc} I+Q &{} U-i V \\ U+i V &{} I-Q \end{array}\right) }_{\textrm{AB}}. \end{aligned}$$

We define the polarization matrix $P_{\textrm{AB}}$ as the traceless part of the relative coherency matrix $i_{\textrm{AB}}\equiv I_{\textrm{AB}}/I$. Written in terms of the relative Stokes parameters $\mathcal {Q}\equiv Q/I$, $\mathcal {U}\equiv U/I$ and $\mathcal {V}\equiv V/I$, it takes the form

$$\begin{aligned} P_{\textrm{AB}}\equiv i_{\textrm{AB}}-\frac{1}{2}\delta _{\textrm{AB}}=\frac{1}{2}{\left( \begin{array}{cc} \mathcal {Q} &{} \mathcal {U}-i \mathcal {V} \\ \mathcal {U}+i \mathcal {V} &{} -\mathcal {Q} \end{array}\right) }_{\textrm{AB}}. \end{aligned}$$

(114)

The polarization degree $\mathcal {P}\equiv \sqrt{\mathcal {Q}^{2}+\mathcal {U}^{2}+\mathcal {V}^{2}}$ can be expressed in terms of the polarization matrix as

$$\begin{aligned} \mathcal {P}^{2}=2P^{\textrm{BA}}P_{\textrm{AB}}=2P^{\textrm{AB}}P_{\textrm{AB}}^{*}. \end{aligned}$$

(115)

1.2.2 B.2 The Coherency and Polarization Tensors

Define the coherency tensor for an electromagnetic field, similar to the definition of the coherency matrix of (113):

$$\begin{aligned} I_{\mu \nu }\equiv \left\langle E_{\mu }E_{\nu }^{*}\right\rangle . \end{aligned}$$

(116)

The expectation value $\langle E_{\mu }E_{\nu }^{*}\rangle$ of (116) is assumed to be a time average over a large number of radiation cycles. The intensity I is the scalar

$$\begin{aligned} I\equiv g^{\mu \nu }I_{\mu \nu }=S^{\mu \nu }I_{\mu \nu }. \end{aligned}$$

We retain the coherency matrix $I_{\textrm{AB}}$ of (113) by projecting $I_{\mu \nu }$ with the polarization basis $e_{A}^{\mu }$:

$$\begin{aligned} I_{\textrm{AB}}=e_{A}^{\mu }e_{B}^{\nu }I_{\mu \nu }. \end{aligned}$$

In the geometric optics limit, the amplitude $\mathcal {E}^{\mu }\equiv i a p \epsilon ^{\mu }$ of a plane electromagnetic wave can be assumed constant over a cycle. This implies that the Stokes parameters, expressed in terms of time-averaged squares of the electric field, $\langle E_{A}E_{B}^{*}\rangle$ are additive when adding up contributions from multiple wave components. Therefore, the Stokes parameters of an electromagnetic field expressed as a composition of plane waves can be computed by summing up the Stokes parameters of each component [42]. This makes the Stokes parameters useful for characterizing any radiation field of arbitrary composition and arbitrary degree of polarization. Hence, the coherency tensor $I_{\mu \nu }$ of (116) is a convenient covariant and gauge invariant field representation of any electromagnetic radiation field.

Define the relative coherency tensor $i_{\mu \nu }$ as

$$\begin{aligned} i_{\mu \nu }\equiv \frac{I_{\mu \nu }}{I}. \end{aligned}$$

For a single plane wave with polarization vector $\epsilon ^{\mu }$, the relative coherency tensor is

$$\begin{aligned} i_{\mu \nu }^{\textrm{pw}}( \epsilon ) =\epsilon _{\mu }\epsilon _{\nu }^{*}. \end{aligned}$$

Following the definition of the polarization matrix in (114), the polarization tensor $P_{\mu \nu }$ is defined as

$$\begin{aligned} P_{\mu \nu }=i_{\mu \nu }-\frac{1}{2}S_{\mu \nu }. \end{aligned}$$

(117)

The polarization degree $\mathcal {P}$ can be expressed in terms of the polarization tensor as

$$\begin{aligned} \mathcal {P}^{2}=2P^{\nu \mu }P_{\mu \nu }=2P^{\mu \nu }P_{\mu \nu }^{*}. \end{aligned}$$

(118)

1.2.3 B.3 Stokes Parameters

From (114), the polarization matrix $P_{\textrm{AB}}$ can be expanded in terms of the Pauli matrices $\sigma _{a},a=1,2,3$. The polarization matrix then expands as

$$\begin{aligned} P_{\textrm{AB}}=\frac{1}{2}\mathcal {S}_{a}\sigma _{\textrm{AB}}^{a}, \end{aligned}$$

where the relative Stokes parameters are $\mathcal {S}_{1}=\mathcal {U}, \mathcal {S}_{2}=\mathcal {V},\mathcal {S}_{3}=\mathcal {Q}$. Conversely, the relative Stokes parameters can be found by contracting the polarization matrix with the Pauli matrices:

$$\begin{aligned} \mathcal {S}_{a}=\sigma _{a}^{\textrm{AB}}P_{\textrm{AB}}. \end{aligned}$$

Then, by selecting a transverse polarization basis $e_{\mu }^{A}, A=1,2$, we can define a Pauli tensor basis

$$\begin{aligned} \sigma _{\mu \nu }^{a}=\sigma _{\textrm{AB}}^{a}e_{\mu }^{A}e_{\nu }^{B}. \end{aligned}$$

Based on the relationship between Pauli matrices, this basis satisfies

$$\begin{aligned} {\left( \sigma ^{a}\sigma ^{b}\right) }_{\mu \nu }=\delta ^{\textrm{ab}}S_{\mu \nu }+i \varepsilon ^{\textrm{abc}}\sigma _{\mu \nu }^{c}. \end{aligned}$$

This allows us to expand the polarization tensor in the Pauli tensor basis as

$$\begin{aligned} P_{\mu \nu }=\frac{1}{2}\mathcal {S}_{a}\sigma _{\mu \nu }^{a}, \end{aligned}$$

Conversely, the relative Stokes parameters can be found by contracting the polarization tensor with the Pauli tensors:

$$\begin{aligned} \mathcal {S}_{a}=\sigma _{a}^{\mu \nu }P_{\mu \nu }. \end{aligned}$$

Let $s^{q}, q=Q,U$, denote the two symmetric Pauli matrices, with $s^{Q}\equiv \sigma ^{3}$ and $s^{U}\equiv \sigma ^{1}$. Defining $\varepsilon ^{\textrm{QU}}=-\varepsilon ^{\textrm{UQ}}=\varepsilon ^{312}=1$, the relationship between the two symmetric Pauli matrices can be written

$$\begin{aligned} {\left( s^{p}s^{q}\right) }_{\mu \nu }=\delta ^{\textrm{pq}}S_{\mu \nu }+ \varepsilon ^{\textrm{pq}}\epsilon _{\mu \nu }, \end{aligned}$$

and the antisymmetric Pauli matrix $\sigma ^{2}$ can be expressed as

$$ \sigma _{\mu \nu }^{2}=-i \epsilon _{\mu \nu }. $$

The polarization tensor then takes a form that separates the symmetric and antisymmetric parts:

$$\begin{aligned} P_{\mu \nu }=-\frac{i}{2} \mathcal {V} \epsilon _{\mu \nu }+\frac{1}{2}S_{q}s_{\mu \nu }^{q}, q=Q,U. \end{aligned}$$

(119)

Equation (119) can be inverted to obtain the circular polarization degree in terms of the polarization tensor:

$$\begin{aligned} \mathcal {V}=i \epsilon ^{\mu \nu }P_{\mu \nu }. \end{aligned}$$

(120)