Analytical Derivation of Second-Order Deflection in Equatorial Plane of a Radially Moving Kerr-Newman Black Hole

The 2PM equations of motion for test particles in the field of a longitudinally and constantly moving KN black hole can be calculated directly from the corresponding metric [20], which can be obtained from the harmonic KN metric [21] via a Lorentz boosting. Let $\{\bm{e}_{1},~{}\bm{e}_{2},~{}\bm{e}_{3}\}$ denote the orthonormal basis of a three-dimensional Cartesian coordinate system. We assume the rest frame of the background spacetime and the comoving frame for the barycentre of the gravitational source to be $(t,~{}x,~{}y,~{}z)$ and $(X_{0},~{}X_{1},~{}X_{2},~{}X_{3})$, respectively. The 2PM equations of motion for test particles propagating in the equatorial plane of a moving KN source with a constant radial velocity $\bm{v}=v\bm{e}_{1}$ can be written as follows [19]

Here, $\gamma=(1-v^{2})^{-\scriptstyle\frac{1}{2}}$ is the Lorentz factor and a dot denotes derivative with respect to the affine parameter $\xi$ which describes the trajectory [9, 22]. $M$, $Q$, and $\bm{J}(=\!J\bm{e}_{3})$ are the rest mass, electrical charge, and angular momentum vector (along positive $X_{3}$ axis) of the gravitational source, respectively. $\Phi\equiv-\frac{M}{R}=-\frac{M}{\sqrt{X_{1}^{2}+X_{2}^{2}-a^{2}}}$ denotes the Newtonian gravitational potential, with $a\!=\!\frac{J}{M}$ being the angular momentum per mass. The relation $M^{2}\geq a^{2}+Q^{2}$ is assumed to avoid naked singularity for the black hole. Notice that Eqs. (1) - (3) correspond to the $t,~{}x,$ and $y\,-$component of the geodesic equations in the background’s rest frame, respectively, and that the residual component is trivial and omitted for the equatorial-plane propagation ($z=\frac{\partial}{\partial z}=0$). The coordinates $X_{0},~{}X_{1},~{}X_{2}$, and $X_{3}$ are related to $t,~{}x,~{}y$, and $z$ by the Lorentz transformation

The schematic diagram for the deflection of test particles caused by a longitudinally moving KN source is shown in Fig. 1. The spatial coordinates of the test-particle source (denoted by $A$ on the source plane) and observer (denoted by $B$ on the observer plane) are assumed as $(x_{A},~{}y_{A},~{}0)$ and $(x_{B},~{}y_{B},~{}0)$ respectively in the background’s rest frame, where $y_{A}\approx-b$, $x_{A}\ll-b$, and $x_{B}\gg b$, with $b$ being the impact parameter. In the comoving frame, the locations of $A$ and $B$ are denoted by $(X_{A},~{}Y_{A},~{}0)$ and $(X_{B},~{}Y_{B},~{}0)$, respectively. The solid blue line represents the perturbed propagation path of a test particle coming from $x=-\infty$ with a relativistic initial velocity $\bm{w}|_{x\rightarrow-\infty}~{}(\approx\bm{w}|_{x\rightarrow x_{A}})~{}=w\bm{e}_{1}~{}$ ($0<w\leq 1$). We assume $v<w$ [9, 19] and that the test particle can pass by the black hole to reach the observer. The gravitational deflection is greatly exaggerated to distinguish between the perturbed and unperturbed (dashed horizontal line) paths.

The gravitational deflection angle for a test particle traveling from $A$ to $B$ is defined as $\alpha\equiv\arctan\left.\frac{dy}{dx}\right|_{B}-\arctan\left.\frac{dy}{dx}\right|_{A}$. In the following, we employ an iterative technique to derive the gravitational deflection angle up to the second order, which can be written as

Up to the zero order, Eqs. (1) - (3) result in

where we have followed Wucknitz and Sperhake’s idea [9] and used the boundary conditions $\dot{t}|_{\xi\rightarrow-\infty}=\dot{t}|_{x\rightarrow-\infty}=\frac{1}{w}$, $\dot{x}|_{\xi\rightarrow-\infty}=\dot{x}|_{x\rightarrow-\infty}=1$, and $\dot{y}|_{\xi\rightarrow-\infty}=\dot{y}|_{x\rightarrow-\infty}=0$. From Eqs. (7) - (8), we can get a zero-order parameter transformation and the zero-order value for $y$

where the boundary condition $y|_{\xi\rightarrow-\infty}=y|_{x\rightarrow-\infty}=-b$ has been adopted.

Substituting Eqs. (6) - (8) and (10) into Eqs. (1) - (3) and integrating the latter equations over $\xi$, we can obtain the analytical first-order forms for $\dot{t}$, $\dot{x}$, and $\dot{y}$ as follows:

where Eq. (9) and the zero-order parameter transformation $dX_{1}\!=\!\left[\gamma\left(1\!-\!\frac{v}{w}\right)+O(M)\right]dx$ [9] have been used, and the integral constants have been determined in the lens’s comoving frame. With the help of Eqs. (9) and (10), we integrate Eq. (13) over $\xi$ and obtain the explicit form for $y$ up to the 1PM order as

We then plug Eqs. (11) - (14) into Eq. (3), ignore the third and higher order terms, and have

in which the zero-order relation $\dot{X}_{1}=\gamma\left(\dot{x}-v\dot{t}\,\right)=\gamma\left(1-\frac{v}{w}\right)$ has been used. In addition, Eq. (12) gives the first-order parameter transformation

Therefore, we can apply Eq. (16) to the integration of Eq. (15) over $\xi$, up to the 2PM order, and obtain

In order to integrate Eq. (17) more conveniently, we perform a coordinate transformation $dX_{1}=\gamma(1-v\dot{t}/\dot{x})dx$ which is to be calculated up to the first post-Minkowskian order. Considering Eqs. (11) - (12), we can get

We then substitute Eq. (18) into Eq. (17) and have

where $C$ denotes the integral constant and we have dropped the third and higher order terms in the derivation.

Finally, substituting Eqs. (12) and (19) into Eq. (5) and considering the conditions $X_{A}\ll-b$ and $X_{B}\gg b$, we can obtain the 2PM gravitational deflection angle of a test particle propagating from $A~{}(X_{A},~{}Y_{A},~{}0)$ to $B~{}(X_{B},~{}Y_{B},~{}0)$ as follow

where the kinematical correctional coefficients $N_{i}~{}(i=1,~{}2,~{}3,~{}4)$ are the functions of $v$ and $w$:

Notice that in Eq. (20) the second-order terms with the factor $\frac{1}{X_{A}}$ or $\frac{1}{X_{B}}$ have been ignored since they are much smaller than the second-order terms with the factor $\frac{1}{b^{2}}$ such as the term $\frac{3N_{2}\left(1+\frac{4}{w^{2}}\right)\pi M^{2}}{4b^{2}}$. The gravitational deflection angle $\alpha$ is a function of $v$ and $w$, therefore, we denote it as $\alpha(v,w)$ from now on.

In the above derivations, we assume that the trajectory parameter $\xi$ has the dimension of length [9]. In fact, we can also assume it to have the dimension of time [22], and the same result can be achieved. The detailed derivations are given in the Appendix.