Polarized Images of Synchrotron Radiations in Curved Spacetime

In flat spacetime, the radiations of a moving charged particle have been well-studied[25]. By using the retarded Green’s function, the gauge potential caused by a charged particle in motion is simply

$$A_{\mu}(x)=4\pi\int d^{4}x^{\prime}G^{\mbox{ret}}_{\mu\alpha}(x-x^{\prime})J^{\alpha}(x^{\prime}),$$

(2.1)

where the moving charged particle induces the current

$$J^{\alpha}(x^{\prime})=e\int d\tau Z^{\alpha}(\tau)\delta^{(4)}(x^{\prime}-z_{0}(\tau)).$$

(2.2)

$Z^{\alpha}$ is the 4-velocity of the particle and $z_{0}(\tau)$ characterize the worldline of the particle. The corresponding gauge potential is the Liénard-Wiechert potential. The corresponding field strength consists of two parts: one part independent of the acceleration is the static field, and the other, depending linearly on the acceleration, is the radiation field.

In curved spacetime, a similar radiation problem has been addressed by DB in [1] to understand the radiation damping and the Equivalence Principle for the charged particle in motion. Next, we would like to give a brief and necessary review of the pioneering work by DB. A key point in DB’s treatment is to develop the theory of bi-tensors connecting the tensor fields at two different spacetime points, $x$ and $z$, to study the covariant Green’s functions. (For a different approach using the vielbein formalism, see [26]) .

We closely follow DB’s conventions. Let $z$ be the point of the source, and $x$ be other points in the spacetime. The letters of the Greek alphabet $\alpha$ to $\kappa$ are assigned to the point $z$, while indices taken from $\lambda$ to $\omega$ are left to the points $x$. For example, $z^{\alpha}$ and $x^{\mu}$ are the coordinates of $z$ and $x$, respectively. $s(x,z)$ is defined as the bi-scalar of the geodesic interval, which denotes the geodesic length connecting $x$ and $z$. We can conclude that $\lim\limits_{z\to x}s(x,z)=0$, and we can see that $s(x,z)=ds(x)$ when considering $z=x+dx$, that is to say, at this moment we can take $s$ as the proper time of the geodesic through the point $x$. Moreover, $s>0$ denotes a spacelike interval, and $s<0$ means the interval is timelike, while $s=0$ defines the light cone. Given a sufficiently large interval, $s$ may change sign. However, if we focus on a small neighborhood of a given source, the geodesic interval is single-valued. Thus the covariant expansions can be performed to identify the asymptotic forms of Green’s functions containing the information of the electromagnetic waves. It turns out to be convenient to introduce the quantity

$\displaystyle\sigma\equiv\pm 1/2s^{2}\,,$

(2.3)

to avoid the ”branch point” problem, which is positive for spacelike intervals and negative for timelike ones.

Furthermore, in addition to the ordinary tensors, which act on the point $z$ or $x$, we need to introduce bi-tensors, of which some indices refer to the point $z$ and the other ones refer to the point $x$. For example, we have a bi-vector $T_{\mu\alpha}$, in which $\mu$ refers to the point $x$ and $\alpha$ refers to the point $z$. Then, it is useful to introduce the bi-vector of geodesic parallel displacement, denoted by $\bar{g}_{\mu\alpha}(x,z)$, which plays the role of transforming the given bi-tensor into a new bi-tensor all of whose indices refer to the same point. The definition of $\bar{g}_{\mu\alpha}$ is given by the following equations

$\displaystyle\nabla^{\nu}\sigma\nabla_{\nu}\bar{g}_{\mu\alpha}=0,\quad\nabla^{\gamma}\sigma\nabla_{\gamma}\bar{g}_{\mu\alpha}=0,\quad\lim\limits_{x\to z}\bar{g}_{\mu}^{~{}\alpha}=\delta_{\mu}^{~{}\alpha}.$

For a local vector $V_{\mu}$ at the point $x$, it can be generated from $V_{\alpha}$ at the point $z$ by parallel displacement along the geodesic from $z$ to $x$, that is, $V_{\mu}=\bar{g}_{\mu}^{~{}\alpha}V_{\alpha}$. Similarly, one can easily apply the geodesic parallel displacement to local tensors of arbitrary order. For our work, we will not show more details of bi-tensors, which can be found in [1].

Since we are interested in electromagnetic radiations, we next turn to the solutions of the covariant vector wave equations, which take the form

$\displaystyle\nabla^{\nu}\nabla_{\nu}A_{\mu}+R_{\mu}^{~{}\nu}A_{\nu}=J_{\mu}\,,$

(2.4)

in the Lorenz gauge $\nabla_{\mu}A^{\mu}=0$. Green’s functions for the vector wave functions can be obtained from the Hadamard’s ”elementary solution” [31]

$\displaystyle G^{(1)}_{\mu\alpha}=(2\pi)^{-2}\mathopen{}\mathclose{{}\left(u_{\mu\alpha}\sigma^{-1}+v_{\mu\alpha}\log\sigma+w_{\mu\alpha}}\right)$

(2.5)

by moving into the complex plane, where, $u_{\mu\alpha},v_{\mu\alpha},w_{\mu\alpha}$ are bi-vectors. As a result, the “symmetric” Green’s function $\bar{G}_{\mu\alpha}(x,z)$ can be found in this form[1]

$\displaystyle\bar{G}_{\mu\alpha}(x,z)=(8\pi)^{-1}[u_{\mu\alpha}\delta(\sigma)-v_{\mu\alpha}\theta(-\sigma)]$

(2.6)

where $\delta(\sigma)$ is the $\delta$-function and $\theta$ is the Heaviside function with $u_{\mu\alpha}\propto\bar{g}_{\mu\alpha}$ and $v_{\mu\alpha}$ being determined by solving the source free equation $\nabla^{\nu}\nabla_{\nu}G^{(1)}_{\mu\alpha}+R_{\mu}^{~{}\nu}G^{(1)}_{\nu\alpha}$=0. The term $u_{\mu\alpha}\delta(\sigma)$ shows that a sharp pulse of electromagnetic radiation travels along the null geodesics between the source $z$ and the field $x$, with its polarization vector being parallel-transported. The other term $v_{\mu\alpha}\theta(-\sigma)$ in the Green’s function, often referred to as the ”tail” term, describes the electromagnetic radiations produced by the source whose geodesic intervals $s(z,x)$ are timelike. It shows that the sharp pulse can develop a ”tail” due to the scattering with the ”bump” in curved spacetime. Then the retarded Green’s function that we desire for the vector wave equation can be formally expressed as

$\displaystyle G^{\text{ret}}_{\mu\alpha}(x,z)=2\Theta[\Sigma(x),z]\bar{G}_{\mu\alpha}(x,z),$

(2.7)

where $z$ is the source point, $\Sigma(x)$ is an arbitrary spacelike hypersurface containing the field point $x$, and $\Theta[\Sigma(x),z]=1-\Theta[z,\Sigma(x)]=1$ when $z$ lies to the past of $\Sigma(x)$ and $\Theta[\Sigma(x),z]=0$ when $z$ lies to the future. In Fig. 1, we show the radiations received at a field point $x$ on $\Sigma(x)$.

The Green’s functions $\bar{G}_{\mu\alpha}(x,z)$ cannot be solved precisely in generic curved spacetimes. Nevertheless, we can do covariant expansions of $\bar{G}_{\mu\alpha}(x,z)$ around $z$ to obtain the electromagnetic fields generated from the source in the very near region of point $z$. In particular, since we are interested in the synchrotron radiations emitted from non-geodesic charged particles, we only need to include the contributions from the history of the particle at and before the point $z_{0}$ to determine the electromagnetic field at point $x$ in Fig. 1. As shown in Fig. 1, $Z^{\alpha}$ is the tangent vector of the world line of the source, and the proper time is set to be $\tau$. For any $\tau$, one always has a certain spacelike hypersurface $\Sigma$ whose normal vector is $Z^{\alpha}$. Then the induced metric on $\Sigma$ can be defined as $h^{\alpha\beta}=g^{\alpha\beta}+Z^{\alpha}Z^{\beta}$. At the proper time $\tau_{\Sigma}$, the source arrives at the point $z$. We introduce a fixed field point $x$ which has a small spatial distance $\epsilon$ deviated from $z_{\Sigma}$ on the hypersurface $\Sigma$, and in terms of the induced metric, we can have the following relation:

$\displaystyle h^{\alpha\beta}\nabla_{\alpha}\sigma\nabla_{\beta}\sigma=\epsilon^{2}+\mathcal{O}(\epsilon^{3})\,.$

(2.8)

This relation tells us that the quantities like $\nabla_{\mu}\sigma$ are of order $\mathcal{O}(\epsilon)$. For each geodesic interval $s$ connecting the source point $z$ and the field point $x$, following the results in [1], we can have the expansions of the bi-vectors:

	$\displaystyle u_{\mu\alpha}$	$\displaystyle=$	$\displaystyle\mathopen{}\mathclose{{}\left[1-\frac{1}{12}R^{\beta\gamma}\nabla_{\beta}\sigma\nabla_{\gamma}\sigma+\mathcal{O}(\epsilon^{3})}\right]\bar{g}_{\mu\alpha},$
	$\displaystyle v_{\mu\alpha}$	$\displaystyle=$	$\displaystyle-\frac{1}{2}\bar{g}^{\beta}_{~{}\mu}\mathopen{}\mathclose{{}\left(R_{\alpha\beta}-\frac{1}{6}g_{\alpha\beta}R}\right)+\mathcal{O}(\epsilon)\,,$		(2.9)

and

	$\displaystyle\nabla_{\beta}\nabla_{\alpha}\sigma$	$\displaystyle=$	$\displaystyle g_{\alpha\beta}+\frac{1}{3}R_{\alpha~{}\beta}^{~{}\gamma~{}\delta}\nabla_{\gamma}\sigma\nabla_{\delta}\sigma+\mathcal{O}(\epsilon^{3})\,,$
	$\displaystyle\nabla_{\beta}\bar{g}_{\mu\alpha}$	$\displaystyle=$	$\displaystyle-\frac{1}{2}R_{\delta\alpha\beta}^{~{}~{}~{}\gamma}\nabla_{\gamma}\sigma+\mathcal{O}(\epsilon^{2})\,,$
	$\displaystyle\nabla_{\beta}u_{\mu\alpha}$	$\displaystyle=$	$\displaystyle\mathopen{}\mathclose{{}\left(-\frac{1}{2}\bar{g}^{\delta}_{~{}\mu}R_{\delta\alpha\beta}^{~{}~{}~{}\gamma}-\frac{1}{6}\bar{g}_{\mu\alpha}R^{\gamma}_{~{}\beta}}\right)\nabla_{\gamma}\sigma+\mathcal{O}(\epsilon^{2}).$

Here we emphasize that the so-called higher order terms $\mathcal{O}(\epsilon^{2})$ and $\mathcal{O}(\epsilon^{3})$ mean the related terms involve two and three covariant derivatives of $\sigma$, respectively. Thus, the above expansions are valid for any geodesic interval $s$, even for the geodesic intervals on the light cone with $s=0$.

With the expansions of Green’s functions, the retarded gauge potential of a point charged particle reads

	$\displaystyle A^{\text{ret}}_{\mu}$	$\displaystyle=$	$\displaystyle 4\pi e\int_{-\infty}^{+\infty}G^{\text{ret}}_{\mu\alpha}Z^{\alpha}d\tau$
		$\displaystyle=$	$\displaystyle-e\mathopen{}\mathclose{{}\left[u_{\mu\alpha}Z^{\alpha}(\nabla_{\beta}{\color[rgb]{1,0,0}\sigma}Z^{\beta})^{-1}}\right]_{\tau=\tau_{0}}+e\int_{-\infty}^{\tau_{0}}v_{\mu\alpha}Z^{\alpha}d\tau$

where we have defined $\tau_{0}$ as the proper time of the point particle at the point $z_{0}$ in Fig. 1, and $e$ is the particle’s charge. The corresponding electromagnetic field strength tensor can be obtained by straightforward differentiation

	$\displaystyle F^{\text{ret}}_{\mu\nu}$	$\displaystyle=$	$\displaystyle 2e[Z^{\alpha}\nabla_{[\mu}\sigma u_{\nu]\alpha}(Z^{\beta}Z^{\gamma}\nabla_{\beta}\nabla_{\gamma}\sigma)(D_{\tau}\sigma)^{-3}$		(2.10)
			$\displaystyle-(Z^{\alpha}D_{\tau}\nabla_{[\mu}\sigma u_{\nu]\alpha}+D_{\tau}Z^{\alpha}\nabla_{[\mu}\sigma u_{\nu]\alpha})(D_{\tau}\sigma)^{-2}$
			$\displaystyle+Z^{\alpha}(\nabla_{[\mu}u_{\nu]\alpha}+\nabla_{[\mu}\sigma v_{\nu]\alpha})(D_{\tau}\sigma)^{-1}]_{\tau=\tau_{0}}$
			$\displaystyle-2e\int_{-\infty}^{\tau_{0}}d\tau Z^{\alpha}\nabla_{[\mu}v_{\nu]\alpha}$

where we have introduced the derivative operator along the vector $Z^{\alpha}$, that is, $D_{\tau}\equiv Z^{\beta}\nabla_{\beta}$. In Eq. (2.10), the leading terms of $F^{\text{ret}}_{\mu\nu}$ are of order $\mathcal{O}(\epsilon^{-2})$, and they are, in fact, from the Coulomb potential, which would give a renormalization of the mass of the point particle but won’t contribute to the radiations. The subleading terms being of order $\mathcal{O}(\epsilon^{-1})$ from the lightlike part would contribute to the radiations mostly. The synchrotron radiations originating from the gravity involving curvature tensors are of order $\mathcal{O}(\epsilon^{0})$ both in the lightlike part and in the ”tail” part. The radiations from the complicated ”tail” term involve the integrations over the whole history of the particle, and they won’t be observed if we only keep them for a limited time interval. Moreover, such radiations are sub-leading but could be significant in a strong gravitational field.

Next, we turn to find out the electromagnetic radiations and synchrotron radiations generated by a non-geodesic point particle acted upon by the Lorentz force and extract the corresponding electric component as the polarization vector, as well as its strength.

Note that the geodesic interval between the point $z_{0}$ and the point $x$ is on the light cone; thus, we are allowed to define $\nabla_{\alpha}\sigma\equiv\epsilon k_{\alpha}$ at the point $z_{0}$, where $k_{\alpha}k^{\alpha}=0$ is a null vector along the light cone. Then, it’s easy to conclude that at the point $x$, we have $\nabla_{\mu}\sigma=-\bar{g}_{\mu}^{~{}\alpha}\nabla_{\alpha}\sigma=-\epsilon k_{\mu}$. After some calculations, the electromagnetic field tensor in Eq. (2.10) can be expanded and rewritten as

$\displaystyle F^{\text{ret}}_{\mu\nu}=\sum_{d=-2}^{\infty}F^{(d)}_{\mu\nu}\epsilon^{d}+(\text{tail part})\,,$

(2.11)

where

	$\displaystyle F^{(-2)}_{\mu\nu}$	$\displaystyle=$	$\displaystyle 2e\mathopen{}\mathclose{{}\left[Z^{\alpha}k_{[\mu}\bar{g}_{\nu]\alpha}(k_{\eta}Z^{\eta})^{-3}}\right]_{\tau=\tau_{0}}\,$		(2.12)
	$\displaystyle F^{(-1)}_{\mu\nu}$	$\displaystyle=$	$\displaystyle 4e\mathopen{}\mathclose{{}\left[k_{[\nu}\bar{g}_{\mu]\alpha}(k_{\beta}D_{\tau}Z^{[\beta}Z^{\alpha]})(k_{\eta}Z^{\eta})^{-3}}\right]_{\tau=\tau_{0}}\,$		(2.13)
	$\displaystyle F^{(0)}_{\mu\nu}$	$\displaystyle=$	$\displaystyle 2e\bigg{\{}k_{[\nu}\bar{g}_{\mu]\alpha}Z^{\alpha}\mathopen{}\mathclose{{}\left(\frac{1}{3}R_{\gamma~{}\beta}^{~{}\theta~{}\delta}k_{\theta}k_{\delta}Z^{\beta}Z^{\gamma}+\frac{1}{12}R^{\beta\gamma}k_{\beta}k_{\gamma}}\right)(k_{\eta}Z^{\eta})^{-3}$		(2.14)
			$\displaystyle+\mathopen{}\mathclose{{}\left(\frac{1}{2}k_{[\nu}\bar{g}_{\mu]}^{~{}\delta}R_{\delta\alpha\beta}^{~{}~{}~{}\gamma}+\frac{1}{6}k_{[\nu}\bar{g}_{\mu]\alpha}R_{\beta}^{~{}\gamma}}\right)k_{\gamma}Z^{\alpha}Z^{\beta}(k_{\eta}Z^{\eta})^{-2}$
			$\displaystyle+\frac{1}{6}\bar{g}^{\theta}_{~{}[\nu}\bar{g}_{\mu]\alpha}R_{\theta~{}\beta}^{~{}\delta~{}\gamma}k_{\gamma}k_{\delta}Z^{\alpha}Z^{\beta}(k_{\eta}Z^{\eta})^{-2}$
			$\displaystyle-Z^{\alpha}\bigg{[}k_{\gamma}\mathopen{}\mathclose{{}\left(\frac{1}{2}\bar{g}_{~{}[\nu}^{\beta}\bar{g}_{\mu]}^{~{}\delta}R_{\delta\alpha\beta}^{~{}~{}~{}\gamma}+\frac{1}{6}\bar{g}_{~{}[\nu}^{\beta}\bar{g}_{\mu]\alpha}R_{\beta}^{~{}\gamma}}\right)$
			$\displaystyle+\frac{1}{2}k_{[\nu}\bar{g}_{\mu]}^{~{}\beta}\mathopen{}\mathclose{{}\left(R_{\alpha\beta}-\frac{1}{6}g_{\alpha\beta}R}\right)\bigg{]}(k_{\eta}Z^{\eta})^{-1}\bigg{\}}_{\tau=\tau_{0}}$

In Eq. (2.11), the electromagnetic radiations of charged particles involve a tail term which gets contribution only from the spacetime curvature. Due to the presence of a tail term, the EEP is invalid for the electromagnetic radiations of charged particles in curved spacetime. Nevertheless, as the leading order of the tail part is of $\mathcal{O}(\epsilon^{0})$, if we focus on the photons with high enough frequency, the tail part can be dropped and only $F^{(-2)}_{\mu\nu}$ and $F^{(-1)}_{\mu\nu}$ terms survive. As a result, the EEP is valid for electromagnetic radiations of charged particles under the approximation of geometrical optics. Our results are consistent with Eq. (23) in QW’s paper [27].

Moreover, the term $F^{(-2)}_{\mu\nu}$ corresponds to the electromagnetic field generated by the Coulomb potential. And a nonvanishing $F^{(-1)}_{\mu\nu}$ would give a non-trivial electromagnetic radiations dependent on the $4$-acceleration of charged particles $D_{\tau}Z^{\alpha}$. In other words, if the charged particles move along geodesic trajectories, this term would vanish, so we focus on the particles in non-geodesic motions in the following discussion.

When charged particles of mass $m$ are only impacted by a magnetic field

$\displaystyle D_{\tau}Z^{\alpha}=\frac{e}{m}F^{\alpha}_{~{}\beta}Z^{\beta}\,,$

(2.15)

electromagnetic radiations are the so-called synchrotron radiations, which have an excellent property of polarization encoding the information of the source and the magnetic field. Therefore, it is fascinating to have a careful study of synchrotron radiations.

The electric vector can be extracted from the electric components in $F^{(-1)}_{\mu\nu}$ via a timelike tetrad

$\displaystyle E^{\mu}\equiv\frac{1}{\epsilon}F^{(-1)\mu\nu}\xi_{\nu}\,,$

(2.16)

where $\xi_{\nu}$ is the timelike component of a local Minkowski frame that can be chosen arbitrarily. The form of $F^{(-1)}_{\mu\nu}$ of a general electromagnetic wave can always be rewritten as

$\displaystyle F^{(-1)}_{\mu\nu}\equiv k_{\mu}\mathcal{A}_{\nu}-k_{\nu}\mathcal{A}_{\mu}\,,$

(2.17)

where $\mathcal{A}_{\mu}$ can be read from the expression (2.13)

$\displaystyle\mathcal{A}_{\mu}=-4e\mathopen{}\mathclose{{}\left[\bar{g}_{\mu\alpha}(k_{\beta}D_{\tau}Z^{[\beta}Z^{\alpha]})(k_{\eta}Z^{\eta})^{-3}}\right]_{\tau=\tau_{0}}.$

(2.18)

Taking into account the conditions, $k\cdot k=0$ and $k\cdot\mathcal{A}=0$, we obtain the luminosity of the synchrotron radiations emitted by the source $Z^{\alpha}$:

$\displaystyle L=4\pi\epsilon^{2}E^{\mu}E_{\mu}=4\pi(k_{\rho}Z^{\rho})^{2}(\mathcal{A}_{\sigma}\mathcal{A}^{\sigma})\,,$

(2.19)

Plunging the definition of $k_{\alpha}$ into the Eq. (2.8) we have $k_{\alpha}Z^{\alpha}=1$. Combining with this relation, we substitute the expression of $\mathcal{A}_{\mu}$ into $L$ and finally find the luminosity

$\displaystyle L=4\pi|k_{\beta}D_{\tau}Z^{\beta}Z^{\alpha}-D_{\tau}Z^{\alpha}|^{2}_{\tau=\tau_{0}}.$

(2.20)

Next, we move to the polarization vector of the synchrotron radiations. Because the polarization vector $f^{\mu}\propto E^{\mu}$ should be normalized and transverse along the null geodesic, we have $f^{\mu}f_{\mu}=1\,,f^{\mu}k_{\mu}=0$. Note that a general polarization vector of the electromagnetic wave has four components but only one physical degree of freedom. The reason is that apart from the above two constraints, gauge freedom exists. We can see that the physical information of the polarization vector remains unchanged if one chooses different observers. Considering the fact that $\xi^{\mu}$ can be regarded as the $4$-velocity of an observer, we change $\xi$ to $\xi^{\prime}$, it is not hard to check that the variation $\delta f^{\mu}\equiv f^{\prime\mu}-f^{\mu}$ satisfies $\delta f^{\mu}\delta f_{\mu}=0\,,\delta f^{\mu}k_{\mu}=0$, which means that it is proportional to the $4$-momentum of radiation, i.e., $\delta f^{\mu}\propto k^{\mu}$. In this sense, $f^{\prime\mu}$ is indistinguishable from $f^{\mu}$: they differ only by a gauge transformation proportional to $k^{\mu}$. Moreover, from Eq. (2.17), we can see that

$\displaystyle f^{\mu}\sim F^{(-1)\mu\nu}\xi_{\nu}=(\mathcal{A}_{\nu}\xi^{\nu})k_{\mu}-(k_{\nu}\xi^{\nu})\mathcal{A}_{\mu}\,.$

(2.21)

The first term $(\mathcal{A}_{\nu}\xi^{\nu})k_{\mu}\propto k_{\mu}$ is just a gauge term that can be dropped, and therefore without loss of generality, the polarization vector can be taken as

$\displaystyle f_{\mu}=\frac{\mathcal{A}_{\mu}}{\sqrt{\mathcal{A}^{2}}}=N^{-1}\mathopen{}\mathclose{{}\left[\bar{g}_{\mu\alpha}(k_{\beta}D_{\tau}Z^{[\beta}Z^{\alpha]})}\right]_{\tau=\tau_{0}}\,,$

(2.22)

where we have normalized the vector appropriately by a factor $N^{-1}$. Here we would like to stress that one can recover the term $(\mathcal{A}_{\nu}\xi^{\nu})k_{\mu}\propto k_{\mu}$ in Eq. (2.21) from the expression in Eq. (2.22) under a certain observer denoted by $\xi_{\mu}$ by requiring

$\displaystyle f^{\mu}\xi_{\mu}=0\,,$

(2.23)

which can be easily derived by the definition of polarization vector $f^{\mu}$.

The formulas (2.20) and (2.22) are the main results of our work. They may reduce to those in flat spacetime, according to the EEP. More importantly, they can be applied to explore the polarized structure of synchrotron radiations in curved spacetime without resorting coordinate transformations.

In this work, the source of our interest is composed of the large number of electrons moving in the region near a Schwarzschild black hole. Since the Schwarzschild black hole is bathed in a uniform magnetic field, the equation of motion of electrons must include a Lorentz force term[34]. For such an equation, one cannot obtain a general analytical solution. To simplify our discussions, we will constrain the charged particles’ motion to the equatorial plane vertical to the uniform magnetic field and consider their circular orbits. Thus, the tangent vector of the world line of the source takes

$\displaystyle Z^{\alpha}=(\dot{t}_{s},\dot{r}_{s},0,\dot{\phi}_{s})\,,$

(3.3)

where we have set $\theta_{s}=\pi/2,r_{s}=\text{const}$ with the subscript $s$ signifying the source, and the $\cdot$ means the derivative with respect to the proper time $\tau$. Because the energy $E=-mZ_{t}$ and angular momentum $L=mZ_{\phi}+qA_{\phi}$ are conserved along the world line of the source, where $m$ and $q$ are the mass and the charge of the source, respectively, we can obtain

$\displaystyle\dot{r}_{s}^{2}=\mathcal{R}(r_{s})\equiv\mathcal{E}^{2}-\mathopen{}\mathclose{{}\left(1-\frac{2M}{r_{s}}}\right)+\mathopen{}\mathclose{{}\left[1+r_{s}^{2}\mathopen{}\mathclose{{}\left(\frac{\mathcal{L}}{r_{s}^{2}}-\mathcal{B}}\right)^{2}}\right]=0\,,$

(3.4)

from the normalization condition of $4$-velocity, that is, $Z^{\alpha}Z_{\alpha}=-1$. Note that in Eq. (3.4), we reparameterize $B,E,L$ as

	$\displaystyle\mathcal{B}$	$\displaystyle\equiv$	$\displaystyle\frac{qB}{2m}\,,$
	$\displaystyle\mathcal{E}$	$\displaystyle\equiv$	$\displaystyle E/m=\dot{t}_{s}\mathopen{}\mathclose{{}\left(1-\frac{2M}{r_{s}}}\right)\,,$
	$\displaystyle\mathcal{L}$	$\displaystyle\equiv$	$\displaystyle L/m=\dot{\phi}_{s}r_{s}^{2}+\mathcal{B}r_{s}^{2}\,.$		(3.5)

In our convention, $\mathcal{B}$ is always set to be positive, as a negative $\mathcal{B}$ system is equivalent to a positive $\mathcal{B}$ system via a reflection transformation. Thus, a positive $\mathcal{L}$ corresponds to a prograde orbit and a negative one to a retrograde orbit, see Fig. 2. For circular orbits, one further has $\mathcal{R}^{\prime}(r_{s})=0$ corresponding to $\ddot{r}_{s}=0$, with “ $\prime$ ” the derivative with respect to $r$, which gives

$\displaystyle r_{s}\mathopen{}\mathclose{{}\left(\frac{\mathcal{L}}{r_{s}^{2}}-\mathcal{B}}\right)\mathopen{}\mathclose{{}\left[\frac{\mathcal{L}}{r_{s}^{2}}\mathopen{}\mathclose{{}\left(1-\frac{3M}{r_{s}}}\right)+\mathcal{B}\mathopen{}\mathclose{{}\left(1-\frac{M}{r_{s}}}\right)}\right]-\frac{M}{r_{s}^{2}}=0\,.$

(3.6)

Note that $\mathcal{E}$ does not appear in Eq. (3.6). Thus we can use this equation to identify the reparameterized angular momentum $\mathcal{L}$ for given $r_{s}$ and $\mathcal{B}$. Then, plunging $\mathcal{L}$, $\mathcal{B}$ and $r_{s}$ in Eq. (3.4), one can obtain the reparameterized energy $\mathcal{E}$. Then the $4$-vector $Z^{\alpha}$ can be completely determined. The initial conditions we need to know are the source orbit’s radius and the magnetic field’s strength around the black hole.

In addition, only some of the timelike circular orbits are stable. For example, for a vacuum Schwarzschild black hole, the timelike circular orbits inside $r=6M$ are unstable, and the critical radius is known as the innermost stable circular orbit (ISCO). In the following, we would like to focus on the situation in the radial position of the source located from $r_{s}=6M$ to $r_{s}=10M$.

Up to now, we have known the information on the orbits of the source and the polarization vector of synchrotron radiations caused by the source. However, since our final task is to find out the information on the polarization of synchrotron radiations at the position of an observer, which is generally placed at infinity, we have to figure out the propagation of electromagnetic waves from the source to the observer. For simplicity, we only consider high-frequency emissions such that we can use the geometric optics approximation to deal with the propagation problem. That is to say, the emissions that spread away from the source go along null geodesics in spacetime.

Null geodesics in Schwarzschild black hole spacetime have been studied very clearly[35]. Here we only introduce some basic physical quantities and useful formulas. In our convention, we use $k^{\mu}$ to denote the $4$-momentum of photons. Even though there is spherical symmetry in the spacetime of a Schwarzschild black hole such that the source can be fixed at $\theta_{s}=\pi/2$, the observers are not necessarily on the equatorial plane. Thus $k^{\theta}$ cannot be set to zero in general. Along the null geodesics, there are three conserved quantities, the energy $\omega$ and the angular-momentum $l$ and Carter constant $Q^{2}$,

$\displaystyle\omega=-k_{t}\,,\quad l=k_{\phi}\,,\quad Q^{2}=k_{\theta}^{2}+\frac{k_{\phi}^{2}}{\sin^{2}\theta}.$

(3.7)

Then, from $k^{\mu}k_{\mu}=0$, one can find

$\displaystyle k^{t}/\omega=\mathopen{}\mathclose{{}\left(1-\frac{2M}{r}}\right)^{-1}\,,\,k^{r}/\omega=\pm\frac{\sqrt{R(r)}}{r^{2}}\,,\,k^{\theta}/\omega=\pm\frac{\sqrt{\Theta(\theta)}}{r^{2}}\,,\,k^{\phi}/\omega=\frac{\lambda}{r^{2}\sin^{2}\theta}\,,$

(3.8)

where,

	$\displaystyle R(r)$	$\displaystyle=$	$\displaystyle r^{4}-(r^{2}-2Mr)\rho^{2}\,,$		(3.9)
	$\displaystyle\Theta(\theta)$	$\displaystyle=$	$\displaystyle\rho^{2}-\frac{\lambda^{2}}{\sin^{2}\theta}\,$		(3.10)

with the impact parameters $\lambda\equiv l/\omega$ and $\rho^{2}\equiv Q^{2}/\omega^{2}$. Next, consider an observer with the coordinate $(t_{o},r_{o},\theta_{o},\phi_{o})$. The position of the photons reaching the eyes of the observer can be described in terms of celestial coordinates, which are usually chosen as

	$\displaystyle\alpha$	$\displaystyle=$	$\displaystyle-\frac{\lambda}{\sin\theta_{o}}\,,$
	$\displaystyle\beta$	$\displaystyle=$	$\displaystyle\pm_{o}\sqrt{\Theta(\theta_{o})}\,,$		(3.11)

where $\pm_{o}=\text{sign}(k^{\theta}_{o})$ denotes the sign of $k^{\theta}$ in the observer’s frame. Alternatively, for convenience, one can use the polar coordinates $(\rho,\varphi)$ instead of the Cartesian coordinates $(\alpha,\beta)$. The two coordinate frames are related by

$\displaystyle\rho=\sqrt{\alpha^{2}+\beta^{2}}=\frac{Q}{\omega}\,,\quad\quad\tan\varphi=\frac{\beta}{\alpha}\,.$

(3.12)

Then, considering the null geodesics connecting the source $(t_{s},r_{s},\pi/2,\phi_{s})$ to the observer $(t_{o},r_{o},\theta_{o},\phi_{o})$, one integrate Eq. (3.8) and has

$\displaystyle I_{r}=G_{\theta}\,,\quad\Delta\phi=\phi_{o}-\phi_{s}=\lambda G_{\phi}\,,$

(3.13)

where

$\displaystyle I_{r}=\fint^{r_{o}}_{r_{s}}\frac{dr}{\pm_{r}\sqrt{R(r)}}\,,\quad G_{\theta}=\fint_{\pi/2}^{\theta_{o}}\frac{d\theta}{\pm_{\theta}\sqrt{\Theta(\theta)}}\,,\quad G_{\phi}=\fint_{\pi/2}^{\theta_{o}}\frac{\csc^{2}\theta d\theta}{\pm_{\theta}\sqrt{\Theta(\theta)}}$

(3.14)

with the notation $\fint$ indicating that these integrals are path integrals along the photon trajectories. And the symbols $\pm_{r}=\text{sign}(k^{r})$ and $\pm_{\theta}=\text{sign}(k^{\theta})$ indicate the sign of $k^{r}$ and $k^{\theta}$, which switch at radial and angular turning points, respectively. Let $\bar{m}$ be the number of times light rays cross the equatorial plane from the source to the observer. One can find

$\displaystyle G_{\theta}=\frac{1}{\rho}\arccos\mathopen{}\mathclose{{}\left(-\frac{\sin\varphi}{\sqrt{\sin^{2}\varphi+\cot^{2}\theta_{o}}}}\right)+\frac{\bar{m}\pi}{\rho}\,.$

(3.15)

Here and hereafter, we set $M=1$ for simplicity and without loss of generality. In addition, different values of $\bar{m}$ correspond to the $(\bar{m}+1)\text{th}$ image on the observers’ screen. In this work, we would like to focus on the primary image and secondary image of the source. Now, from Eqs. (3.2-3.15), one can completely obtain the image of the source on the observer’s screen, given the spacetime coordinates of the source and the observer.

In this subsection, we study polarization information transmission along null geodesics connecting the source and the observer. Note that, for a spacetime of Petrov type-D [16], there is a conserved quantity along the geodesics for a parallel-transported and transverse vector, dubbed as the Penrose-Walker(PW) constant, which is typically written as

$\displaystyle\kappa=2r[(k^{\mu}\hat{l}_{\mu})(f^{\nu}\hat{n}_{\nu})-(k^{\mu}\hat{m}_{\mu})(f^{\nu}\hat{\bar{m}}_{\nu})]\equiv\omega\mathopen{}\mathclose{{}\left(\kappa_{1}+i\kappa_{2}}\right)\,,$

(3.16)

for a Schwarzschild black hole spacetime. In the above formula, $r$ is the radial coordinate introduced in the metric (3.1), $k^{\mu}$ is the $4$-momentum of photons, $f^{\mu}$ is the polarization vector, $\kappa_{1,2}$ are rescaled real part and imaginary part of PW constant, and $\{\hat{l}^{\mu},\hat{n}^{\mu},\hat{m}^{\mu},\hat{\bar{m}}^{\mu}\}$ are the Newman-Penrose tetrads taking the form

	$\displaystyle\hat{l}$	$\displaystyle=$	$\displaystyle\frac{1}{\sqrt{2(1-\frac{2}{r})}}\mathopen{}\mathclose{{}\left[\partial_{t}+(1-\frac{2}{r})\partial_{r}}\right]\,,$
	$\displaystyle\hat{n}$	$\displaystyle=$	$\displaystyle\frac{1}{\sqrt{2(1-\frac{2}{r})}}\mathopen{}\mathclose{{}\left[\partial_{t}-(1-\frac{2}{r})\partial_{r}}\right]\,,$
	$\displaystyle\hat{m}$	$\displaystyle=$	$\displaystyle\frac{1}{\sqrt{2}r}\mathopen{}\mathclose{{}\left[\partial_{\theta}+\frac{i}{\sin\theta}\partial_{\phi}}\right]\,,$
	$\displaystyle\hat{\bar{m}}$	$\displaystyle=$	$\displaystyle\frac{1}{\sqrt{2}r}\mathopen{}\mathclose{{}\left[\partial_{\theta}-\frac{i}{\sin\theta}\partial_{\phi}}\right]\,.$		(3.17)

From Eq. (3.16), we can see that all the information of polarization is encoded in the PW constant at the source, thus considering the light rays connecting the source and the observer, we can decode the information of polarization at the observer. On the screen of observers, the polarization vector can be read from the PW constant as

	$\displaystyle\vec{\mathcal{E}}$	$\displaystyle=$	$\displaystyle(\mathcal{E}_{\alpha},\mathcal{E}_{\beta})\,,$		(3.18)
		$\displaystyle=$	$\displaystyle\frac{1}{\rho^{2}}(\beta\kappa_{2}+\alpha\kappa_{1},\beta\kappa_{1}-\alpha\kappa_{2}).$

In this subsection, we turn our attention to the calculations of the image fluxes. Since the source in this work is a charged particle with a finite size, Eq. (2.20) cannot be directly used to give the total flux of the image on the screen. The formalism for dealing with such a problem was developed in [36] for an extremal black hole and was generalized to include the non-extreme case in [37]. In this paper, we closely follow the method presented in [37], which has been applied to calculate the observational signature for near-extremal black holes in the modified theory in [38, 39].

Firstly, we introduce the Newtonian flux, which is related to the luminosity by

$\displaystyle F_{N}=\frac{L}{4\pi r_{o}^{2}},$

(3.19)

where $r_{o}$ is the distance between the observer and the source. The distance $r_{o}$ can be large, $r_{o}\gg 1$. Then, we assume that the source has a minimal proper radius $b\ll 1$ emitting with the intensity $I_{s}$ in the rest frame of the source. The relation between the intensity and the luminosity is given by

$\displaystyle I_{s}=\frac{L}{4\pi^{2}b^{2}}.$

(3.20)

Taking the gravitational effect into account, we find the fluxes of the image

$\displaystyle F_{o}=\oiint\frac{d\alpha d\beta}{r_{o}^{2}}g^{4}I_{s}=\frac{\mathcal{A}}{\pi b^{2}}g^{4}F_{N},$

(3.21)

where $g=\nu_{o}/\nu_{s}$ is the redshift factor and $\mathcal{A}=\oiint d\alpha d\beta$ is the area of the image.

To determine the area of the image $\mathcal{A}$, we would like to introduce a local Minkowski coordinate system $(T,X,Y,Z)$ in the neighborhood of the source so that we have

$\displaystyle Te_{(t)}+Xe_{(r)}+Ye_{(\phi)}-Ze_{(\theta)}=(x^{\mu}-x_{\star}^{\mu})\partial_{\mu},$

(3.22)

where $x^{\mu}_{\star}$ are the position coordinates of the source and $e_{(i)}$ are the tetrad of the source. In this work we choose $e_{(i)}$ to be

$\displaystyle e_{(t)}=\gamma\frac{\partial_{t}+\Omega_{s}\partial_{\phi}}{\sqrt{-g_{tt}}},\quad e_{(r)}=\frac{\partial_{r}}{\sqrt{g_{rr}}},\quad e_{(\theta)}=\frac{\partial_{\theta}}{\sqrt{g_{\theta\theta}}},\quad e_{(\phi)}=\gamma\mathopen{}\mathclose{{}\left(v_{s}\frac{\partial_{t}}{\sqrt{-g_{tt}}}+\frac{\partial_{\phi}}{\sqrt{g_{\phi\phi}}}}\right),$

(3.23)

with $\Omega_{s}=\frac{d\phi}{dt}$ being the angular velocity of the source and

$\displaystyle\gamma=\frac{1}{\sqrt{1-v_{s}^{2}}},\qquad v_{s}=\sqrt{\frac{g_{\phi\phi}}{-g_{tt}}}\Omega_{s}.$

(3.24)

Using the terminology introduced in [37], we call the surface $T=X=0$ the ”source screen” and denote by $(Y_{s},Z_{s})$ the position of the intersection of a light ray with the source screen. Then the area of the image becomes

$\displaystyle\mathcal{A}=\oiint d\alpha d\beta=\mathopen{}\mathclose{{}\left|\frac{\partial(\alpha,\beta)}{\partial(Y_{s},Z_{s})}}\right|\oiint dY_{s}dZ_{s}\,,$

(3.25)

where $\mathopen{}\mathclose{{}\left|\frac{\partial(\alpha,\beta)}{\partial(Y_{s},Z_{s})}}\right|$ is the Jacobian determinant between $(\alpha,\beta)$ and $(Y_{s},Z_{s})$. In addition, the projection of the hemisphere of the source onto the screen is an ellipse with an area

$\displaystyle\oiint dY_{s}dZ_{s}=\frac{\pi b^{2}}{|\hat{k}\cdot\hat{X}|},$

(3.26)

where the unit vector $\hat{k}$ is given by

$\displaystyle\hat{k}=\frac{1}{k^{(t)}}\mathopen{}\mathclose{{}\left(k^{(r)}\hat{X}+k^{(\phi)}\hat{Y}-k^{(\theta)}\hat{Z}}\right).$

(3.27)

Then we can read the area of the image

$\displaystyle\mathcal{A}=\frac{\pi b^{2}}{|\hat{k}\cdot\hat{X}|}\mathopen{}\mathclose{{}\left|\frac{\partial(\alpha,\beta)}{\partial(Y_{s},Z_{s})}}\right|\,,$

(3.28)

and find that the image flux takes the form

$\displaystyle F_{o}=\frac{g^{4}L}{4\pi r_{o}^{2}|\hat{k}\cdot\hat{X}|}\mathopen{}\mathclose{{}\left|\frac{\partial(\alpha,\beta)}{\partial(Y_{s},Z_{s})}}\right|\,.$

(3.29)

Here we omit the detail of the computation of the Jacobian determinant $\mathopen{}\mathclose{{}\left|\frac{\partial(\alpha,\beta)}{\partial(Y_{s},Z_{s})}}\right|$, and suggest the interested readers find the details in [37].

Hereto, we have completed all the theoretical preparations and are ready to apply our model to some specific situations.