Faraday effect of light caused by plane gravitational wave

Abstract

A gravitational field can cause a rotation of the polarisation vector of light. This phenomenon is known as the gravitational Faraday effect. We study the gravitational Faraday effect of linearly polarised light propagating in the gravitational field of a weak plane gravitational wave (GW) with “$+$", “$\times$", and elliptical polarisation modes. The corresponding gravitational Faraday rotation angle is proportional to the GW amplitude and to the squared distance traveled by the light and inversely proportional to the GW squared wavelength. The Faraday rotation is maximal if the light propagates along directions perpendicular to the GW propagation and tilted by $\pi /4$ to the directions of its polarisation. There is no a gravitational Faraday rotation when light and a GW propagate along the same directions, or when light propagates along directions of a GW polarisation. Helicity of an elliptically polarised GW gives cubic order contribution to the Faraday rotation.

1 Introduction

The Faraday effect of light is a magneto-optical phenomenon, which is a rotation of the polarisation vector of linearly-polarised light propagating in a transparent material in the presence of a magnetic field along the light propagation [1]. There is an analogous effect, the gravitational Faraday effect of light, which is a rotation of the polarisation vector of a linearly-polarised electromagnetic wave (light) propagating in a gravitational field of non-zero angular momentum. Such a gravitational field can be created by a rotating black hole, whose angular momentum causes “dragging" of inertial frames, which results in a rotation of the polarisation vector. This phenomenon can be observed in local frames which do not rotate relative to a rest frame in the asymptotically flat region. Study and observation of the gravitational Faraday effect of light in stationary gravitational fields can be found in many works [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]. Note that there is a dual to the gravitational Faraday effect, the gravitational spin-Hall effect [17,18,19,20,21,22,23,24,25,26,27,28].

It is known that a gravitational wave (GW) carries an angular momentum. Thus, space-time of a GW can also cause an inertial frame dragging, and, as a result, the gravitational Faraday effect. This phenomenon was already studied in a linearised gravitational field [29] and in a strong gravitational field of a plane-fronted GW interacting with a linearly polarised electromagnetic shock wave [30,31,32]. Here we study the gravitational Faraday effect in arbitrary gravitational fields and consider a gravitational Faraday rotation caused by a weak plane GW of “$+$", “$\times$", and elliptical polarisation modes.

This paper is organised as follows. In Section II we briefly review the laws of geometric optics in a curved space-time and describe the gravitational Faraday effect of light in the formalism presented in [19] and adopted here to linearly polarised light. In Section III we discuss fields of observers used to detect the gravitational Faraday effect. Section IV contains a study of the gravitational Faraday effect of light caused by a plane GW. We summarise our results in the last section. Here we use a system of units in which $G=c=1$ and conventions adopted in the book [33], in particular, the space-time metric signature is $+2$.

2 Gravitational Faraday effect of light

Consider an electromagnetic wave (light) propagating in a curved space-time background. If the light is highly monochromatic over some space-time regions and its reduced wavelength (wavelength/$2\pi$) measured in a local Lorentz frame is much less than inhomogeneities in the light and the typical radius of the space-time curvature¹ [34]., we can use an asymptotic short-wave (geometric optics) approximation [33]. For example, for an astrophysical black hole of mass $M=\kappa M_{\odot }$, where $\kappa \sim (1, 10^{11})$, the geometric optics approximation implies that at the vicinity of the black hole horizon $\lambda \ll 10^{4}\kappa \,\text {m}$, which is satisfied even for very long radio waves. By using this approximation, we consider the light as a plane-fronted monochromatic wave whose propagation can be described by laws of geometric optics²:

$$\begin{aligned} k^{\alpha }k_{\alpha }=0&\, ,\hspace{0.25cm}&k^{\beta }k^{\alpha }_{\,\,;\beta }=0, \end{aligned}$$

(1)

$$\begin{aligned} p^{\alpha }p_{\alpha }=1&\, ,\hspace{0.25cm}&p^{\alpha }k_{\alpha }=0\, ,\hspace{0.25cm}k^{\beta }p^{\alpha }_{\,\,;\beta }=0. \end{aligned}$$

(2)

Here and in what follows, the semicolon stands for the covariant derivative associated with the space-time metric $g_{\alpha \beta }$, $k^{\alpha }=dx^{\alpha }/d\lambda$ is null wave vector tangent to a light ray $\Gamma \!\!:x^{\alpha }=x^{\alpha }(\lambda )$, where $\lambda$ is affine parameter of the ray, and $p^{\alpha }$ is a unit linear polarisation vector. These laws imply that light rays are space-time null geodesics (1) and the polarisation vector is orthogonal to a light ray and parallel propagated along it (2).

Define at some point of a space-time a null vector $k^{\alpha }$ and a unit space-like vector $p^{\alpha }$ orthogonal to it, then the propagation laws ensure that these vectors will preserve their norms and remain orthogonal at any point of the corresponding null geodesic. However, the polarisation vector $p^{\alpha }$ is not defined uniquely. As follows from the geometric optics equations, one can add to the polarisation vector a multiple of the wave vector without affecting the laws,

$$\begin{aligned} p^{\alpha }\rightarrow p^{\alpha }+\kappa k^{\alpha }. \end{aligned}$$

(3)

Here $\kappa$ is a constant. Thus, to specify orientation of $p^{\alpha }$ at the initial point of a null geodesic, we have to fix the gauge $\kappa$. We shall return to this problem later in this section (see Eq. (10) below).

Our goal is to study the evolution of a polarisation vector of light along spatial trajectory of the light ray. This is done by an observer-defined local decomposition of a space-time into space and time, which is known as the space-time threading approach.³ Consider a family of infinitely many observers filling a three-dimensional space like a continuous medium. World lines of these observers form a congruence of integral curves⁴ of a time-like future-directed unit vector field $u^\alpha =u^\alpha (x^\beta )$, $u^\alpha u_\alpha =-1$. Such a field, which we shall call here field of observers, represents space-time frame of reference at any point of arbitrary (dynamic) gravitational field [37]. We shall discuss field of observers in the next section. Here we assume that such a field is defined, i.e. at every point $x^{\alpha }$ of a space-time (or at least in its domain which contains the light ray) we have a unique time-like unit vector $u^{\alpha }$, which belongs to the field.

Fig. 1

World lines of observers in a space-time with metric $g_{\alpha \beta }$. The doted line represents world line of a light ray $\Gamma$ with affine parameter $\lambda$ and tangent null vector $k^{\alpha }$. Observers with $u^{\alpha }_{1}$ and $u^{\alpha }_{2}$ detect the light (events 1 and 2) propagating in the spatial directions $n_{1}^{\alpha }$ and $n_{2}^{\alpha }$ defined in their local subspaces $\Sigma _{u1}$ and $\Sigma _{u2}$

Each of the observers defines a local frame of reference, a right-handed orthonormal Lorentz frame:

$$\begin{aligned} \{e_{(0)}^{\alpha }= & {} u^{\alpha },\, e_{(a)}^{\alpha };a=1,2,3\},\nonumber \\ u^{\alpha }u_{\alpha }=-1,{} & {} e_{(a)}^{\alpha }u_{\alpha }=0\, ,\hspace{0.25cm}e_{(a)}^{\alpha }e_{(b)}^{\beta }g_{\alpha \beta }=\delta _{(a)(b)}. \end{aligned}$$

(4)

Here $\delta _{(a)(b)}$ is the three-dimensional Kronecker tensor. Let $\Sigma _{u}$ be three-dimensional local subspace of the tangent space defined at every event on observer’s world line and orthogonal to $u^{\alpha }$. A vector from the tangent space can be projected into the subspace $\Sigma _{u}$ by the projection operator

$$\begin{aligned} h^{\alpha }_{\,\,\,\beta }=\delta ^{\alpha }_{\,\,\,\beta }+u^{\alpha }u_{\beta }, \end{aligned}$$

(5)

where $\delta ^{\alpha }_{\,\,\,\beta }$ is the four-dimensional Kronecker tensor, and

$$\begin{aligned} h^{\alpha }_{\,\,\,\beta }h^{\beta }_{\,\,\,\gamma }=h^{\alpha }_{\,\,\,\gamma }. \end{aligned}$$

(6)

Applying the projection operator to a null vector $k^{\alpha }$ tangent to a light ray we define a unit space-like vector $n^{\alpha }$,

$$\begin{aligned} h^{\alpha }_{\,\,\,\beta }k^{\beta }=k^{\alpha }-\omega \,u^{\alpha }=\omega \,n^{\alpha }\, ,\hspace{0.5cm}\omega =-k_{\alpha }u^{\alpha }, \end{aligned}$$

(7)

which is orthogonal to $u^{\alpha }$ and which defines spatial direction of a light ray at some point O on $\Gamma$. Here $\omega >0$ is the angular frequency of the light measured by a local observer with $u^{\alpha }$ at O. This gives us a local decomposition (see Fig. 1)

$$\begin{aligned} k^{\alpha }=\omega (u^{\alpha }+n^{\alpha }). \end{aligned}$$

(8)

If we substitute (8) into the propagation equation (1), we derive a propagation equation for the unit vector $n^{\alpha }$,

$$\begin{aligned} n^{\beta }n^{\alpha }_{\,\,;\beta }=(u^{\alpha }+n^{\alpha })(n_{\gamma }u^{\beta }u^{\gamma }_{\,\,;\beta }+n_{\gamma }n^{\beta }u^{\gamma }_{\,\,;\beta })-u^{\beta }u^{\alpha }_{\,\,;\beta }-u^{\beta }n^{\alpha }_{\,\,;\beta }-n^{\beta }u^{\alpha }_{\,\,;\beta }, \end{aligned}$$

(9)

which depends on the observers field $u^{\alpha }$. It is expected, for the light frequency $\omega$ and its local spatial direction $n^{\alpha }$ are observer-dependent quantities, which is a manifestation of the Doppler and aberration effects. For a given smooth observers field $u^{\alpha }$, the light frequency $\omega$ and the unit vector $n^{\alpha }$ defined by the observer at O and propagated along the light ray coincide with the frequency and the unit vector defined by another observer from the field at the next point on the light ray.

Let us now consider a polarisation vector $p^{\alpha }$. To define $p^{\alpha }$ uniquely at initial point O on a light ray $\Gamma$, we have to fix the gauge $\kappa$ in (3). We require that at the point O

$$\begin{aligned} p^{\alpha }u_{\alpha }=0. \end{aligned}$$

(10)

Then, according to the second relation in (2) and the expression (8), we also have at O

$$\begin{aligned} p^{\alpha }n_{\alpha }=0. \end{aligned}$$

(11)

Thus, the polarisation vector lies in a local two-dimensional subspace perpendicular to $u^{\alpha }$ and $n^{\alpha }$. If we orient the local frame (4) at O so that $e_{(3)}^{\alpha }=n^{\alpha }$, then we can expand the polarisation vector in a local two-dimensional basis $\{e_{(1)}^{\alpha },\, e_{(2)}^{\alpha }\}$ as follows:

$$\begin{aligned} p^{\alpha }=\cos \phi \, e_{(1)}^{\alpha }+\sin \phi \, e_{(2)}^{\alpha }. \end{aligned}$$

(12)

Fig. 2

Spatial trajectory of a light ray parametrised by its proper length $\ell$. Direction of the light propagation is indicated by the unit vector $n^{\alpha }$. The local basis $\{e_{(1)}^{\alpha },\, e_{(2)}^{\alpha }\}$ spans two-dimensional plane $\pi$ orthogonal to $n^{\alpha }$. Orientation of the polarisation vector $p^{\alpha }$ lying in the plane is defined by the angle $\phi$ measured from $e_{(1)}^{\alpha }$ in the right-hand direction

Here, by convention, the angle $\phi$ is measured from $e_{(1)}^{\alpha }$ in the right-hand direction when viewed along the light ray trajectory (see Fig. 2). It defines orientation of the polarisation vector. We shall call this angle polarisation phase.

Our goal is to calculate how polarisation phase changes along a light ray. To compute this change we have to define a transport law for the basis $\{e_{(1)}^{\alpha },\, e_{(2)}^{\alpha }\}$ along a light ray. For example, if the basis is transported arbitrarily, the polarisation phase change can take an arbitrary value, or the expansion (12) may not hold at any other point on the light ray. Thus, in order to find the polarisation phase change due to gravitational field, we need to know how to transport properly the basis along a light ray. This problem also arises in the measurement of polarisation in the magneto-optical Faraday effect, where we have to align properly optic axes of a polariser and an analyser. This is done by transporting the analyser from the initial point, where its axis is aligned with the polariser axis, to the final point along light trajectory in such a way that it does not rotate with respect to the trajectory during an infinitesimally small step of the transport. Such a transport is called Fermi-Walker transport and is defined by vanishing Fermi derivative of a unit vector indicating direction of the analyser optical axis along the unit vector tangent to the light trajectory (see Eq. (14) below) [38].⁵

To define a transport law for the basis $\{e_{(1)}^{\alpha },\, e_{(2)}^{\alpha }\}$ along a light ray we shall use the decomposition (8) and express the basis transport as follows:

$$\begin{aligned} k^{\beta }e^{\alpha }_{(a);\beta }=\omega \left( u^{\beta }e^{\alpha }_{(a);\beta }+n^{\beta }e^{\alpha }_{(a);\beta }\right) , \end{aligned}$$

(13)

where $a=1,2$. Thus, we need to define the transport law along observer’s world line and along the related spatial trajectory of the light ray.

We have discussed above how the basis should be transported along a ray trajectory. Namely, its spatial Fermi-Walker derivative along tangent to the trajectory unit vector should vanish, that implies

$$\begin{aligned} h^{\alpha }_{\,\,\,\gamma }n^{\beta }e^{\gamma }_{(a);\beta }=h^{\alpha }_{\,\,\,\gamma }n^{\beta }n^{\gamma }_{\,\,;\beta }n_{\delta }e^{\,\,\,\delta }_{(a)}-n^{\alpha }e_{(a)\gamma }n^{\beta }n^{\gamma }_{\,\,;\beta }. \end{aligned}$$

(14)

Here we employed the relation $e_{(a)\alpha }h^{\alpha }_{\,\,\,\gamma }=e_{(a)\gamma }$. Following (8) and (13) and recalling that $n^{\alpha }=e_{(3)}^{\alpha }$, and therefore $n_{\alpha }e^{\alpha }_{(a)}=0$, the expression above reduces to

$$\begin{aligned} h^{\alpha }_{\,\,\,\gamma }n^{\beta }e^{\gamma }_{(a);\beta }=-n^{\alpha }e_{(a)\gamma }n^{\beta }n^{\gamma }_{\,\,;\beta }. \end{aligned}$$

(15)

Contracting it with $e_{(b)\alpha }$, $b=1,2$ and recalling that $e_{(b)\alpha }n^{\alpha }=0$, we derive

$$\begin{aligned} e_{(b)\alpha }n^{\beta }e^{\alpha }_{(a);\beta }=0. \end{aligned}$$

(16)

Note that the orthogonality condition $n_{\alpha }e^{\alpha }_{(a)}=0$ is preserved by the Fermi-Walker transport.

Let us now define a transport of the frame vectors $e^{\alpha }_{(a)}$ in (4) along world lines of the field of observers $u^\alpha$. We consider a general congruence of non-inertial observers with 4-acceleration $a^{\alpha }=u^{\beta }u^{\alpha }_{\,\,;\beta }$ and angular velocity

$$\begin{aligned} \omega ^{\alpha }=\tfrac{1}{2}\varepsilon ^{\alpha \beta \gamma \delta }u_{\beta }u_{\gamma ;\delta }, \end{aligned}$$

(17)

which is also called the vorticity vector of the observers congruence [39]. Here $\varepsilon _{\alpha \beta \gamma \delta }$ is the Levi-Civita pseudo tensor, which for the observer-adapted frame (4) is normalised as

$$\begin{aligned} \varepsilon _{\alpha \beta \gamma \delta }e_{(0)}^{\alpha }e_{(1)}^{\beta }e_{(2)}^{\gamma }e_{(3)}^{\delta }=+1. \end{aligned}$$

(18)

Then the transport law along world line of an observer from the congruence is (see §13.6 in [33])

$$\begin{aligned} u^{\beta }e^{\alpha }_{(a);\beta }=(u^{\alpha }a_{\beta }-a^{\alpha }u_{\beta })e_{(a)}^{\beta }-\varepsilon ^{\alpha }_{\,\,\,\beta \gamma \delta }u^{\beta }\omega ^{\gamma }e_{(a)}^{\delta }. \end{aligned}$$

(19)

This transport law ensures that the frame (4) remains orthonormal and the basis vectors $e^{\alpha }_{(a)}$ preserve their orientation, when transported along the world line. It means that the basis vectors corotate with the congruence.

Using the transport laws (14) and (19) together with the expressions (8) and (9), one can verify that the tetrad (4) remains orthonormal, when it is transported along a light ray. As a result, the propagation law (2) for the polarisation vector $p^{\alpha }$ implies that the conditions (10), (11), and the expansion (12) hold along a light ray, and the basis $\{e_{(1)}^{\alpha },\, e_{(2)}^{\alpha }\}$ preserves its spatial orientation.

Now we are ready to calculate the polarisation phase change along a light ray. Using the third expression in (2) and the expression (12) we derive

$$\begin{aligned} k^{\alpha }\phi _{,\alpha }=e_{(1)\alpha }k^{\beta }e^{\alpha }_{(2);\beta }. \end{aligned}$$

(20)

Then, according to the expressions (13), (16), and (19), this expression reduces to

$$\begin{aligned} k^{\alpha }\phi _{,\alpha }=-\omega \,\varepsilon _{\alpha \beta \gamma \delta }e_{(1)}^{\alpha }u^{\beta }\omega ^{\gamma }e_{(2)}^{\delta }. \end{aligned}$$

(21)

It follows from the expression (18) where $e_{(0)}^{\alpha }=u^{\alpha }$ and $e_{(3)}^{\alpha }=n^{\alpha }$ that

$$\begin{aligned} \varepsilon _{\alpha \beta \gamma \delta }u^{\alpha }e_{(1)}^{\beta }e_{(2)}^{\gamma }=n_{\delta }. \end{aligned}$$

(22)

Then, applying (22) to (21) we derive

$$\begin{aligned} k^{\alpha }\phi _{,\alpha }=-\omega \,\omega _{\alpha }n^{\alpha }. \end{aligned}$$

(23)

Finally, the decomposition (8) and the expression (17) give

$$\begin{aligned} k^{\alpha }\phi _{,\alpha }\equiv \frac{d\phi }{d\lambda }=-\omega _{\alpha }k^{\alpha }. \end{aligned}$$

(24)

This result allows us to compute change of a linear polarisation phase $\phi$ along a null ray $\Gamma \!\!:x^{\alpha }=x^{\alpha }(\lambda )$,

$$\begin{aligned} \Delta \phi =-\int _{\Gamma }\omega _{\alpha }k^{\alpha }d\lambda , \end{aligned}$$

(25)

which is known as the gravitational Faraday effect of light. Note that this effect depends on field of observers via its vorticity $\omega ^{\alpha }$.

3 Fields of observers

A field of observers $u^{\alpha }$ plays a crucial role in the gravitational Faraday effect. To illustrate this, consider first inertial observers whose world lines are time-like geodesics. If their vorticity vanishes at some point on a geodesic, then it is zero along whole the geodesic [39]. In this case, according to (25), such observers do not detect the gravitational Faraday rotation. On the other side, one can consider a field of non-inertial observers with non-zero vorticity in a flat space-time. Such observers detect the gravitational Faraday rotation, which depends on their kinematics. These examples, as well as a brief account given in the previous section, imply that the gravitational Faraday rotation, like the Doppler and the aberration effects, is an observer-dependent phenomenon. It means that given a light ray propagating in a curved space-time, observers belonging to two different time-like congruences will measure different polarisation phase change between two points on the light ray. The difference is because these observers transport their basis vectors differently, in accordance with their vorticity, along the light ray. This difference is also natural for a time-dependent gravitational field, where different fields of observers are not physically equivalent and, as a result, observed by them motion and other physical phenomena are different.

In some gravitational fields there is a preferred time-like congruence, which can be used to define a specific field of observers. For example, in a stationary space-time there is a time-like Killing vector field. Thus, it is natural to consider a field of observers whose four-velocity vector is proportional to the time-like Killing vector (see [17, 18]). Similarly, for space-times with conformal Killing vector field one can also consider observers field whose four-velocity vector is proportional to the time-like conformal Killing vector [11]. This case is equivalent to the previous one, for source-free Maxwell equations are conformally invariant in a four-dimensional space-time. Another example is a class of space-times admitting shear-free time-like vector field [40, 41]. In such space-times, which are often of the cosmological nature, one can define shear-free field of observers. Yet another example is a weak gravitational field. In such a case, there is a flat space-time background with respect to which a weak gravitational field is defined and a field of observers represents local inertial frames. We shall use such a field in the next section. Here, for an illustrative purpose and for a comparison with other works we apply the general expression (25) to two important cases: the linearised theory of gravity and the Kerr space-time.

3.1 Linearised theory of gravity

$$\begin{aligned} g_{\alpha \beta }\approx \eta _{\alpha \beta }+h_{\alpha \beta }\, ,\hspace{0.5cm}|h_{\alpha \beta }|\ll 1, \end{aligned}$$

(26)

where $\eta _{\alpha \beta }=\text {diag}(-1,1,1,1)$ is the Minkowski metric. In what follows, calculations are done up to first order in $h_{\alpha \beta }$ and its derivatives. We consider a field of observers which form in the flat background space-time an inertial latticework,

$$\begin{aligned} u^{\alpha }\approx \left( 1+\tfrac{1}{2}h_{00}\right) \delta ^{\alpha }_{0}\, ,\hspace{0.25cm}u_{\alpha }\approx -\left( 1-\tfrac{1}{2}h_{00}\right) \delta ^{0}_{\alpha }+h_{0i}\delta ^{i}_{\alpha }. \end{aligned}$$

(27)

Vorticity (17) of this field is

$$\begin{aligned} \omega ^{\alpha }\approx -\tfrac{1}{2}e^{ijk}h_{0j,k}\delta ^{\alpha }_{i}, \end{aligned}$$

(28)

where $e_{ijk}$ is the three-dimensional Levi-Civita pseudo tensor, $e_{123}=+1$. Thus, the expression (25) takes the form,

$$\begin{aligned} \Delta \phi \approx \tfrac{1}{2}\int _{\Gamma }e_{ijk}k^{i}h_{0j,k}d\lambda . \end{aligned}$$

(29)

For the metric outside a weakly-gravitating body (see Ch. 19 in [33]),

$$\begin{aligned} \Delta \phi \approx \int _{\Gamma }B_{i}k^{i}d\lambda , \end{aligned}$$

(30)

where

$$\begin{aligned} B^{i}=3x^{j}S_{j}\frac{x^{i}}{r^5}-\frac{S^{i}}{r^{3}}\, ,\hspace{0.5cm}r=\sqrt{x_{i}x^{i}}, \end{aligned}$$

(31)

is gravitomagnetic field and $S^{i}$ is the body intrinsic angular momentum. The gravitomagnetic field is equal to the angular velocity of precession of fixed gyroscopes. Thus, this results illustrates a relation between the gravitational Faraday effect and “dragging of inertial frames." The integral (30) was also considered in earlier works, e.g. [3, 29, 42] and for both the electromagnetic and gravitational plane waves in [43]. It represents the weak gravitational field limit of the gravitational Faraday rotation of light due to a Kerr black hole [17].

3.2 Kerr space-time

The Kerr space-time metric can be written in Boyer-Lindquist coordinates as follows:

$$\begin{aligned} ds^2 = -h(dt-g_{\phi }d\phi )^2+\frac{\Sigma }{\Delta }dr^2+\Sigma d\theta ^2+\frac{\Delta }{h}\sin ^2\theta d\phi ^2, \end{aligned}$$

(32)

where

$$\begin{aligned} h= & {} (\Delta -a^2\sin ^2\theta )/\Sigma \, ,\hspace{0.5cm}g_{\phi }=-\frac{2aMr}{\Sigma h}\sin ^2\theta ,\nonumber \\ \Sigma= & {} r^2+a^2\cos ^2\theta \, ,\hspace{0.5cm}\Delta =r^2-2Mr+a^2. \end{aligned}$$

(33)

This metric represents a Kerr black hole of mass M and intrinsic angular momentum $S=aM$, with $0\le |a|\le M$. The space-time (32) is stationary. Thus, we consider time-like Killing vector field of observers,

$$\begin{aligned} u^{\alpha }=\frac{\delta ^{\alpha }_{0}}{\sqrt{h}}\, ,\hspace{0.5cm}u_{\alpha }=-\sqrt{h}\left( \delta ^{0}_{\alpha }-g_{\phi }\delta ^{\phi }_{\alpha }\right) . \end{aligned}$$

(34)

Note that here $h>0$, i.e. our field of observers is defined outside the black hole ergosphere. We can compute the vorticity (17) of this field as follows:

$$\begin{aligned} \omega ^{\alpha }=-\frac{aM}{\Sigma ^3h}\left( 2r\Delta \cos \theta \delta ^{\alpha }_{r}+(r^2-a^2\cos ^2\theta )\sin \theta \delta ^{\alpha }_{\theta }\right) . \end{aligned}$$

(35)

The expression (25) takes the following form:

$$\begin{aligned} \Delta \phi =\int _{\Gamma }\frac{aM}{\Sigma ^2h}\left( 2r\cos \theta dr+(r^2-a^2\cos ^2\theta )\sin \theta d\theta \right) . \end{aligned}$$

(36)

Except for the sign convention, it coincides with the expression for the gravitational Faraday rotation due to a Kerr black hole derived in [17] by means of different approach. Thus we see that for light propagating in the equatorial plane, $\theta =\pi /2$, there is no gravitational Faraday rotation. This is a generalisation of the result analysed in [42] in the weak gravitational field approximation. In this case it reduces to the expressions (30) and (31) above with $S=aM$ and takes the following form:

$$\begin{aligned} \Delta \phi =\int _{\Gamma }\frac{S}{r^3}\left( 2\cos \theta dr+r\sin \theta d\theta \right) . \end{aligned}$$

(37)

For light propagating along the rotation axis from $r=R$ to $r\rightarrow \infty$ this expression gives $\Delta \phi =\pm S/R^2$, with $'+'$ for $\theta =0$ and $'-'$ for $\theta = \pi$, the result derived in [42].

4 Plane gravitational wave

GW’s detections from coalescing compact binaries opened up a new era of observational astronomy and cosmology. Up to date, there are about 90 GW’s detections and there will be way more in the future. Sources of the currently detected GW’s have the luminosity distance in the range from about 40 to about 5300 Mpc. The detected GW’s are well modelled by a plane GW solution to the linearised Einstein field equations. In the transverse-traceless (TT) gauge the space-time metric of a weak monochromatic plane GW propagating in the z-direction has the following form:

$$\begin{aligned} ds^2=-dt^2+(1+h_{+})dx^2+2h_{\times }dxdy+(1-h_{+})dy^2+dz^2, \end{aligned}$$

(38)

where

$$\begin{aligned} h_{+}=\Re \{A_{+}e^{-i\Omega (t -z)}\}\, ,\hspace{0.5cm}h_{\times }=\Re \{A_{\times }e^{-i\Omega (t -z)}\}. \end{aligned}$$

(39)

Here $A_{+}$ and $A_{\times }$ are constant complex amplitudes⁶ of the “$+$" and “$\times$" polarisation modes and $\Omega$ is their constant real frequency. The TT gauge corresponds to a global Lorentz frame $u^{\alpha }=\delta ^{\alpha }_{0}$ with the space-time coordinates $\{x^{0}=t,\,x^{1}=x,\, x^{2}=y,\,x^{3}=z\}$. World line of the frame is a time-like geodesic, as well as any time-like world line with constant spatial coordinates $x^{i}$.

In the global Lorentz frame all spatial points are equivalent, i.e. there is no selected “origin". According to the formalism presented in the previous sections, we have to select an inertial observer who can detect light in accordance with the decomposition (8). Such an observer represents naturally an inertial proper frame. In the vicinity of the proper frame a space-time metric can be presented in the Fermi normal coordinates (see p. 332 in [33] and for details [44]),

$$\begin{aligned} ds^2= & {} -\left( 1+R_{\hat{0}\hat{i}\hat{0}\hat{j}}x^{\hat{i}}x^{\hat{j}}\right) d\tau ^2-\tfrac{4}{3}R_{\hat{0}\hat{i}\hat{k}\hat{j}}x^{\hat{i}}x^{\hat{j}}d\tau dx^{\hat{k}}\nonumber \\{} & {} +\left( \delta _{\hat{k}\hat{l}}-\tfrac{1}{3}R_{\hat{k}\hat{i}\hat{l}\hat{j}}x^{\hat{i}}x^{\hat{j}}\right) dx^{\hat{k}}dx^{\hat{l}}+\mathcal{O}(|x^{\hat{i}}|^3)dx^{\hat{\alpha }}dx^{\hat{\beta }}, \end{aligned}$$

(40)

where the components of the Riemann tensor $R_{\hat{\alpha }\hat{\beta }\hat{\gamma }\hat{\delta }}$ are evaluated on the observer’s world line, $x^{\hat{0}}=\tau$, $x^{\hat{i}}=0$, and $\tau$ is the proper time, that is $R_{\hat{\alpha }\hat{\beta }\hat{\gamma }\hat{\delta }}$ depend on $\tau$ only. To construct the Fermi normal coordinates we consider a time-like geodesic with $u^{\alpha }=\delta ^{\alpha }_{0}=dx^{\alpha }/d\tau$ and choose an orthonormal tetrad parallel transported along it. The tetrad components are computed in accordance with the linearised theory of gravity, $g_{\alpha \beta }\approx \eta _{\alpha \beta }+h_{\alpha \beta }$, with $|h_{\alpha \beta }|\ll 1$, i.e. up to first order in the gravitational field modes (39). In the TT frame they are the following:

$$\begin{aligned} e^{\alpha }_{\hat{0}}= & {} (1,0,0,0)\, ,\hspace{0.25cm}e^{\alpha }_{\hat{1}}=\left( 0,1-\tfrac{1}{2}h_{+},-\tfrac{1}{2}h_{\times },0\right) ,\nonumber \\ e^{\alpha }_{\hat{2}}= & {} \left( 0,-\tfrac{1}{2}h_{\times },1+\tfrac{1}{2}h_{+},0\right) \, ,\hspace{0.25cm}e^{\alpha }_{\hat{3}}=(0,0,0,1). \end{aligned}$$

(41)

The Riemann tensor components in the Fermi frame are evaluated by the tensor transformation law,

$$\begin{aligned} R_{\hat{\alpha }\hat{\beta }\hat{\gamma }\hat{\delta }}=R_{\alpha \beta \gamma \delta }e^{\alpha }_{\hat{\alpha }}e^{\beta }_{\hat{\beta }}e^{\gamma }_{\hat{\gamma }}e^{\delta }_{\hat{\delta }}. \end{aligned}$$

(42)

In the linearised theory we have (see, e.g. p. 438 in [33])

$$\begin{aligned} R_{\alpha \beta \gamma \delta }=\frac{1}{2}(h_{\alpha \delta ,\beta \gamma }+h_{\beta \gamma ,\alpha \delta }-h_{\alpha \gamma ,\beta \delta }-h_{\beta \delta ,\alpha \gamma }). \end{aligned}$$

(43)

Therefore, the Riemann tensor components in the Fermi frame and in the TT frame are the same to first order in $h_{ij}$, that is $R_{\hat{\alpha }\hat{\beta }\hat{\gamma }\hat{\delta }}=R_{\alpha \beta \gamma \delta }+\mathcal{O}(h^2)$. According to the expressions (38) and (39), we derive

$$\begin{aligned} R_{0101}= & {} -R_{0202}=-\tfrac{1}{2}h_{+,tt}\, ,\hspace{0.25cm}R_{0102}=-\tfrac{1}{2}h_{\times ,tt},\nonumber \\ R_{0113}= & {} -R_{0223}=\tfrac{1}{2}h_{+,tz}\, ,\hspace{0.25cm}R_{0123}=R_{0213}=\tfrac{1}{2}h_{\times ,tz},\\ R_{1313}= & {} -R_{2323}=-\tfrac{1}{2}h_{+,zz}\, ,\hspace{0.25cm}R_{1323}=-\tfrac{1}{2}h_{\times ,zz}.\nonumber \end{aligned}$$

(44)

These components are evaluated at $x^{\hat{i}}=0$, in accordance with the proper frame metric construction (40). Let us also note that the quadratic approximation in (40) is valid if the proper distance $\ell =|x^{\hat{i}}|$ normal to the geodesic is such that

$$\begin{aligned} \ell \ll \text {min}\left\{ \frac{1}{|R_{\hat{\alpha }\hat{\beta }\hat{\gamma }\hat{\delta }}|^{1/2}}\, ,\hspace{0.25cm}\frac{|R_{\hat{\alpha }\hat{\beta }\hat{\gamma }\hat{\delta }}|}{|R_{\hat{\alpha }\hat{\beta }\hat{\gamma }\hat{\delta },\hat{\sigma }}|}\right\} . \end{aligned}$$

(45)

This condition implies that the spatial geodesics in the vicinity of the proper frame (40) do not intersect.⁷ Thus, according to the expressions (39) and (44), we should have

(46)

where is the reduced GW wavelength, and

(47)

The condition (46) is generally satisfied for sufficiently small amplitude, which is true for a linearised theory of gravity.

In accordance with the theory presented in Section II, we now consider a congruence of non-inertial observers $x^{\hat{i}}=const\ne 0$ in the vicinity of the proper frame

$$\begin{aligned} u^{\hat{\alpha }}=\delta ^{\hat{\alpha }}_{\hat{0}}. \end{aligned}$$

(48)

Note that each observer from the congruence is equivalent to the inertial observer (48). The corresponding field of observers 4-velocity computed in accordance with the quadratic metric expansion (40) is

$$\begin{aligned} u^{\hat{\alpha }}=\left( 1-\tfrac{1}{2}R_{\hat{0}\hat{i}\hat{0}\hat{j}}x^{\hat{i}}x^{\hat{j}}\right) \delta ^{\hat{\alpha }}_{\hat{0}}+\mathcal{O}(|x^{\hat{i}}|^3), \end{aligned}$$

(49)

and its covariant form reads

$$\begin{aligned} u_{\hat{\alpha }}=-\left( 1+\tfrac{1}{2}R_{\hat{0}\hat{i}\hat{0}\hat{j}}x^{\hat{i}}x^{\hat{j}}\right) \delta ^{\hat{0}}_{\hat{\alpha }}-\tfrac{2}{3}R_{\hat{0}\hat{i}\hat{k}\hat{j}}x^{\hat{i}}x^{\hat{j}}\delta ^{\hat{k}}_{\hat{\alpha }}+\mathcal{O}(|x^{\hat{i}}|^3). \end{aligned}$$

(50)

Then the corresponding vorticity is

$$\begin{aligned} \omega ^{\hat{\alpha }}=-e^{\hat{i}\hat{j}\hat{k}}R_{\hat{0}\hat{j}\hat{k}\hat{l}}x^{\hat{l}}\delta ^{\hat{\alpha }}_{\hat{i}}+\mathcal{O}(|x^{\hat{i}}|^2). \end{aligned}$$

(51)

Here we have used the Bianchi identity $R_{\hat{0}[\hat{j}\hat{k}\hat{l}]}=0$ and $e^{\hat{i}\hat{j}\hat{k}}$ is the three-dimensional Levi-Civita (pseudo) tensor, $e^{\hat{1}\hat{2}\hat{3}}=e_{\hat{1}\hat{2}\hat{3}}=+1$. To compute the polarisation phase change (25) we need to know wave vector $k^{\hat{\alpha }}$ of a null geodesic of the space-time (40). The polarisation phase change consistent with the metric (40) and the vorticity (51) should be computed up to first order in $x^{\hat{i}}$. This implies that $k^{\hat{\alpha }}$ should be computed in flat (Minkowski) metric. Accordingly, we have

$$\begin{aligned} k^{\hat{\alpha }}_{0}=(\omega _{0},k^{\hat{i}}_{0})=const\, ,\hspace{0.5cm}\Gamma \!\!:x^{\hat{\alpha }}(\lambda )=k^{\hat{\alpha }}_{0}\lambda . \end{aligned}$$

(52)

This is a null geodesic passing through the proper frame origin at $\lambda =0$. Here $\omega _{0}$ is the light frequency and $k^{\hat{i}}_{0}$ is the three-dimensional wave vector, such that $|k^{\hat{i}}|=\omega _{0}$, measured at the origin. Then, the polarisation phase change (25) is computed as follows:

$$\begin{aligned} \Delta \phi =\int _{0}^{\lambda }e_{\hat{i}}^{\,\,\,\hat{j}\hat{k}}R_{\hat{0}\hat{j}\hat{k}\hat{l}}[\tau (\lambda )]x^{\hat{l}}(\lambda )k^{\hat{i}}_{0}d\lambda . \end{aligned}$$

(53)

Note that a similar expression was derived in [29], where the authors studied different optical phenomena in a weak linearised gravitational field. Their approach to the gravitational Faraday effect is embedded into a general framework and requires detailed calculations restricted to the leading-order approximation. However, this work represents a general formalism resulting in the expression (25), which is applicable to a general gravitational field.

In what follows, we consider first “$+$" and “$\times$" polarisation modes separately, and then discuss their superposition, that is elliptically polarised modes. We define direction of a light ray $k^i_{0}$ in the proper frame by spherical angular coordinates $\varphi _0\in [0, 2\pi )$ and $\theta _0\in [0, \pi ]$,

$$\begin{aligned} k^{\hat{1}}=\omega _{0}\sin \theta _0\cos \varphi _0\, ,\hspace{0.25cm}k^{\hat{2}}=\omega _{0}\sin \theta _0\sin \varphi _0\, ,\hspace{0.25cm}k^{\hat{3}}=\omega _{0}\cos \theta _0\, ,\hspace{0.5cm}\omega _{0}=|k^{\hat{i}}|. \nonumber \\ \end{aligned}$$

(54)

In accordance with the proper frame metric expansion (40), the polarisation phase change (53) should be computed up to $|x^{\hat{i}}|^2$ order. The expression (52) implies

$$\begin{aligned} |x^{\hat{i}}|=\omega _{0}\lambda =c\tau =\ell , \end{aligned}$$

(55)

where $\tau$ is the time measured in the proper frame, $c=1$ is the speed of light, and $\ell$ is the proper distance of the light propagation from the proper frame origin. Thus, the polarisation phase change computations should be accurate to $\ell ^2$ order.

4.1 “$+$" polarisation

We can now compute the polarisation phase change (53) due to a plane GW with “$+$" polarisation. Using the expressions above we derive

(56)

Here, according to the condition (47), we should have . This expression implies that the polarisation change is maximal if the light propagates along directions perpendicular to the GW propagation, $\theta _0=\pi /2$, in the directions of maximal vorticity, $\varphi _{0}=(2n+1)\pi /4$, $n=0,1,2,3$, that is tilted by $\pi /4$ to the directions of its polarisation (see Figure 35.2 on p. 953 in [33]).

4.2 “$\times$" polarisation

The polarisation phase evolution due to a plane GW with “$\times$" polarisation is

(57)

And according to the condition (47), . This expression implies that the polarisation change is maximal if the light propagates along directions perpendicular to the GW propagation, $\theta _0=\pi /2$, in the directions of maximal vorticity, $\varphi _{0}=n\pi /2$, $n=0,1,2,3$, that is tilted by $\pi /4$ to the directions of its polarisation (see Figure 35.2 on p. 953 in [33]).

4.3 Elliptical polarisation

Elliptical polarisation mode can be presented as a superposition of the “$+$" and “$\times$" polarisation modes with different amplitudes,

$$\begin{aligned} h_{\sigma }=\Re \{(A_{+}+i\sigma A_{\times })e^{-i\Omega (t -z)}\}. \end{aligned}$$

(58)

Here $\sigma =\pm 1$ and $+$(−) sign stands for the right-(left-) handed polarisation, when viewed from the source along the wave propagation. The polarisation phase evolution due to an elliptically polarised plane GW is

(59)

We see that the helicity contribution to the gravitational Faraday rotation is of cubic order, which is higher than our proper frame expansion (40).

5 Discussion

Here we studied the gravitational Faraday effect of light due to a weak plane GW with “$+$", “$\times$", and elliptical polarisation modes. We computed the polarisation phase in the proper frame (40), which is valid for , where $\ell$ is distance from the proper frame origin. The derived expressions show that the gravitational Faraday rotation is proportional to the GW amplitude. It is maximal for light propagating along directions perpendicular to the GW propagation and tilted by $\pi /4$ to the directions of its polarisation, where vorticity of the field of observers is maximal. There is no a gravitational Faraday rotation when light and a GW propagate along the same directions, or when light propagates along directions of a GW polarisation. Considering GW of elliptical polarisation we found that the GW helicity $\sigma =\pm 1$ contribution to the Faraday rotation is cubic order in . Note that the leading order behaviour can be inferred from a general consideration and dimensional analysis. The magnitude of the gravitational Faraday rotation is proportional to the space-time curvature. This implies that it should be proportional to a GW amplitude and inversely proportional to its squared wavelength. Because the angle of rotation is a dimensionless quantity (more precisely, it is measured in radians) and because it is an integral quantity, it should be proportional to the squared distance traveled by light. And finally, according to the expression (25), it is proportional to the vorticity of field of observers (see Figure 35.2 on p. 953 in [33]), which justifies the trigonometric functions in the derived expressions.

The formalism presented here allows to take into account higher-order space-time curvature terms, by considering the space-time metric and field of observers vorticity perturbatively. For example, the perturbative form of the space-time metric (40) was constructed in [47]. Also, note that in the linearised theory an arbitrary GW can be resolved into a superposition of plane waves and using the formalism presented here one can compute the net gravitational Faraday effect of such a wave. It would also be interesting to analyse the polarisation phase change for observed GW’s. Let us also note that by using the general expression (25), one can study the gravitational Faraday rotation in strong gravity regime and study this effect due to colliding [31, 32] and twisted [48] GW’s in full nonlinear regime. To conclude, we remark that given the conditions (46) and (47) satisfied, one could use the pulsar timing arrays (PTA) to detect the gravitational Faraday effect due to ultralong gravitational waves [49,50,51,52,53,54,55,56,57,58].