# Kerr-Schild Form of the Exact Metric for a Constantly Moving Kerr Black Hole
and Null Gravitational Deflection

WEIJUN LI  School of Mathematics and Physics,
University of South China, Hengyang, 421001, China  ZHONGWEN FENG  Physics and
Space Science College,
China West Normal University, Nanchong, 637009, China  XIA ZHOU  Physics and
Space Science College,
China West Normal University, Nanchong, 637009, China  XUELING MU  School of
Information Science and Engineering,
Chengdu University, Chengdu, 610106, China  GUANSHENG HE111Corresponding
author. School of Mathematics and Physics,
University of South China, Hengyang, 421001, China
<EMAIL_ADDRESS>
###### Abstract

The exact metric of a moving Kerr black hole with an arbitrary constant
velocity is derived in Kerr-Schild coordinates. We then calculate the null
equatorial gravitational deflection caused by a radially moving Kerr source up
to the second post-Minkowskian order, acting as an application of the weak
field limit of the metric. The bending angle of light is found to be
consistent with the result given in the previous works.

###### keywords:

Gravitation; Kerr-Schild coordinates; exact metric; velocity effect.

PACS numbers: 95.30.Sf, 98.62.Sb

##  1 Introduction

The time dependence of a background field due to the translational motion of a
gravitational system relative to the observer has an effect on the propagation
of electromagnetic signals, which is the so-called velocity effect [1, 2, 3].
Historically, this kinematical effect was usually ignored before the late
stage of the last century, with the consideration of the low-velocity
characteristic of most of the gravitational systems in the universe and the
limited observational accuracy. In 1983, Birkinshaw and Gull [4] found that
the motion of a gravitational lens induces a characteristic two-sided pattern
for the brightness of an isotropic radiation field, which in turn was proposed
to measure the transverse velocity of a cluster of galaxies. Since this
pioneering work, velocity effects have attracted more and more attentions, and
the investigation of the kinematically correctional effects on the propagation
of light or massive particles has been performed extensively (See, for
instance, Refs. 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21,
22, 23, 24, 25, 26, 27). One of the main reasons responsible for its
prosperity lies in the rapid progresses made in the techniques of position,
velocity, and angular measurements, such as the achievement in the technique
of the Very Long Baseline Interferometry (VLBI) [7, 28, 29, 30, 31, 32] and
the ones in the subsequent techniques applied in current high-accuracy
astrometry projects (e.g., the Square Kilometer Array (SKA) Project [33, 34]
and the Gaia mission [35, 36, 37]).

As is well known, the study of the motion effects in gravitational physics
depends fundamentally on the achievement of the metric of a moving
gravitational body, for which two common methods are feasible. One is the
retarded-potential-solution approach [6, 8], where the Liénard-Wiechert
gravitational potential acts as the solution of the Einstein field equations.
The other is the Lorentz boosting technique which is based on the existing
solutions of the field equations [1, 38]. To our knowledge, the first work to
obtain the weak-field time-dependent metric of a slowly moving point mass by
means of the latter method was accomplished in Ref. 1. In 2004, Wucknitz and
Sperhake [2] made use of the Lorentz boosting technique to get the first post-
Minkowskian (PM) metric of a constantly moving body, and investigated the
velocity effects on the leading-order gravitational deflection of light and
massive particles. In fact, probing strong-field gravitational physics also
plays a important role in the examinations of general relativity (GR) and
alternative theories of gravitation [39, 40], which calls for the exact time-
dependent metric of the central body. Recently, on the basis of the Lorentz
transformation method [1], the exact metrics for moving Schwarzschild [41] and
Reissner-Nordström [42] black holes with a constant velocity have been
obtained in harmonic coordinates. However, it is found that the calculations
of the strict metric of a moving Kerr black hole performed in harmonic
coordinates become very complex. This issue has not been dealt with so far,
and is worth the effort, since the case of a black hole with an intrinsic
angular momentum [43, 44] and thus the motion effects related to it is more
realistic than that for a black hole without rotation.

In this paper, we apply the Lorentz boosting technique to derive the exact
metric of a moving Kerr black hole with an arbitrary constant velocity,
serving as our main result. The derivations are performed in Kerr-Schild
coordinates [43, 45], which have two following advantages: (1) the Kerr-Schild
form avoids the coordinate singularities and it may make the calculations more
convenient; (2) it is suitable to compute the physical parameters of the
metric, as demonstrated in Ref. 46. An additional motivation to use Kerr-
Schild coordinates lies in the fact that the exact analytical solution of the
Einstein’s field equation for a moving rotating black hole in standard
coordinates is too lengthy to be applied conveniently, although the metrics of
static or stationary black holes in Boyer-Linquist coordinates are elegant and
have been adopted by lots of work (See, for instance, Refs. 47, 48, 49, 50).

It should be pointed out that the solution derived in this paper may have good
prospect in potential astrophysical applications (e.g., gravitational physics
[38, 51, 52, 53, 54, 55] and spacetime properties [56, 57, 58]). As we know,
rotating black holes have been verified recently by the most direct evidences
[59, 60], while it is widely believed that all existing astrophysical black
holes can be described by the Kerr solution [44]. What’s more, the relative
motions of the source of light, the observer, and the central body are common
in the universe. There is no doubt that moving rotating black holes with a
translational velocity should not be rare. For example, the final black hole
resulted from the merger of two unequal-mass black holes receives a “kick” and
has a recoil velocity and a spin, due to the asymmetric loss of linear
momentum in the gravitational radiation [61]. If the translational velocity of
a kicked rotating black hole is constant or the acceleration of a moving
primordial Kerr source is too small to be taken into account, the
gravitational field of the black hole can be described by our exact metric.

We know the gravitational lensing phenomenon is one of the most classical
tests of various gravitational theories including GR (See, e.g., Refs. 62, 63,
64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82,
83, and references therein). As an application of the weak field limit of our
exact metric in Kerr-Schild form, we calculate the equatorial gravitational
deflection of light due to a radially moving Kerr black hole up to the 2PM
order, in the weak-field and small-angle approximation. For simplicity, the
source of light and the observer are assumed to be static relative to the rest
frame of the background.

The organization of this paper is as follows. Section 2 presents the
derivation of the exact metric of a moving Kerr black hole with a constant
velocity. Section 3 is devoted to calculating the null gravitational
deflection caused by a radially moving Kerr black hole in the second post-
Minkowskian approximation. A summary is given in Section 4. Throughout the
paper, the geometrized units where $G=c=1$ and the metric signature
$(-,~{}+,~{}+,~{}+)$ are used. Conventionally, Greek indices run over
$0,~{}1,~{}2$, and $3$, while Latin indices run over $1,~{}2$, and $3$.

##  2 The Kerr-Schild form for the metric of a moving Kerr source

We adopt the Lorentz boosting technique [1, 2] to achieve the metric of a
constantly moving Kerr black hole. Let
$\\{\bm{e}_{1},~{}\bm{e}_{2},~{}\bm{e}_{3}\\}$ be the orthonormal basis of a
three-dimensional Cartesian coordinate system. We assume the rest Kerr-Schild
frame of the background and the comoving Kerr-Schild frame for the barycenter
of the gravitational source to be $(t,~{}x,~{}y,~{}z)$ and
$(T,~{}X,~{}Y,~{}Z)$, respectively. The Kerr-Schild form of the Kerr metric in
the comoving frame $(T,~{}X,~{}Y,~{}Z)$ can be then written as [43, 45, 84,
85]

$\displaystyle
ds^{2}=-\,dT^{2}+dX^{2}+dY^{2}+dZ^{2}+\frac{2MR^{3}}{R^{4}+a^{2}Z^{2}}$
$\displaystyle\hskip
28.0pt\times\left[dT+\frac{ZdZ}{R}+\frac{R\,(XdX+YdY)-a\,(XdY-
YdX)}{R^{2}+a^{2}}\right]^{2},$ (1)

where $\frac{X^{2}+Y^{2}}{R^{2}+a^{2}}+\frac{Z^{2}}{R^{2}}=1$, and $M$ and
$a\equiv J/M$ denote the rest mass and angular momentum per mass of the
gravitational source, with $\bm{J}(=J\bm{e}_{3})$ being its intrinsic angular
momentum vector along $z$-axis.

In order to obtain the line element in the background’s rest frame
$(t,~{}x,~{}y,~{}z)$, in which the Kerr black hole is moving with an arbitrary
constant velocity $\bm{v}=v_{1}\bm{e}_{1}+v_{2}\bm{e}_{2}+v_{3}\bm{e}_{3}$, we
apply a Lorentz boost to the metric presented in Eq. (1). The Lorentz
transformation between $(t,~{}x,~{}y,~{}z)$ and the comoving frame
$(T,~{}X,~{}Y,~{}Z)$ is given by

$X^{\mu}=\Lambda^{\mu}_{\nu}x^{\nu}~{},$ (2)

and

$\Lambda^{\mu}_{\nu}=\left(\begin{array}[]{cccc}\gamma&-v_{1}\gamma&-v_{2}\gamma&-v_{3}\gamma\vspace*{2pt}\\\
-v_{1}\gamma&1+\frac{v_{1}^{2}(\gamma-1)}{v^{2}}&\frac{v_{1}v_{2}(\gamma-1)}{v^{2}}&\frac{v_{1}v_{3}(\gamma-1)}{v^{2}}\vspace*{2pt}\\\
-v_{2}\gamma&\frac{v_{1}v_{2}(\gamma-1)}{v^{2}}&1+\frac{v_{2}^{2}(\gamma-1)}{v^{2}}&\frac{v_{2}v_{3}(\gamma-1)}{v^{2}}\vspace*{2pt}\\\
-v_{3}\gamma&\frac{v_{1}v_{3}(\gamma-1)}{v^{2}}&\frac{v_{2}v_{3}(\gamma-1)}{v^{2}}&1+\frac{v_{3}^{2}(\gamma-1)}{v^{2}}\vspace*{2pt}\end{array}\right)~{},~{}$
(3)

where $\gamma=(1-v^{2})^{-\scriptstyle\frac{1}{2}}$ is Lorentz factor and
$v^{2}=v_{1}^{2}+v_{2}^{2}+v_{3}^{2}$. The Kerr-Schild form of the exact
metric for the arbitrarily constantly moving Kerr black hole can be obtained
as follows:

$\displaystyle
g_{00}=-1+\frac{2\gamma^{2}MR^{3}}{R^{4}+a^{2}Z^{2}}\\!\left[1-\frac{v_{1}(RX+aY)}{R^{2}+a^{2}}-\frac{v_{2}(RY-
aX)}{R^{2}+a^{2}}-\frac{v_{3}Z}{R}\right]^{2}~{},$ (4) $\displaystyle
g_{0i}=\frac{2\gamma
MR^{3}}{R^{4}+a^{2}Z^{2}}\left[1-\frac{v_{1}(RX+aY)}{R^{2}+a^{2}}-\frac{v_{2}(RY-
aX)}{R^{2}+a^{2}}-\frac{v_{3}Z}{R}\right]$ $\displaystyle\hskip
27.0pt\times\left\\{\frac{(RX\\!+aY)\delta_{1i}}{R^{2}+a^{2}}+\frac{(RY\\!-aX)\delta_{2i}}{R^{2}+a^{2}}+\frac{Z\delta_{3i}}{R}\right.$
$\displaystyle\left.\hskip
25.5pt+\,\frac{v_{i}(\gamma-1)}{v^{2}}\\!\left[\frac{v_{1}(RX+aY)}{R^{2}+a^{2}}+\frac{v_{2}(RY-
aX)}{R^{2}+a^{2}}+\frac{v_{3}Z}{R}\right]\\!-v_{i}\gamma\right\\}~{},$ (5)
$\displaystyle
g_{ij}=\delta_{ij}+\frac{2MR^{3}}{R^{4}+a^{2}Z^{2}}\\!\left\\{\frac{(RX\\!+aY)\delta_{1i}}{R^{2}+a^{2}}+\frac{(RY\\!-aX)\delta_{2i}}{R^{2}+a^{2}}+\frac{Z\delta_{3i}}{R}\right.$
$\displaystyle\left.\hskip
23.0pt+\,\frac{v_{i}(\gamma-1)}{v^{2}}\\!\left[\frac{v_{1}(RX+aY)}{R^{2}+a^{2}}+\frac{v_{2}(RY-
aX)}{R^{2}+a^{2}}+\frac{v_{3}Z}{R}\right]\\!-v_{i}\gamma\right\\}$
$\displaystyle\hskip
24.0pt\times\left\\{\frac{(RX+aY)\delta_{1j}}{R^{2}+a^{2}}+\frac{(RY-
aX)\delta_{2j}}{R^{2}+a^{2}}+\frac{Z\delta_{3j}}{R}\right.$
$\displaystyle\left.\hskip
23.0pt+\,\frac{v_{j}(\gamma-1)}{v^{2}}\\!\left[\frac{v_{1}(RX+aY)}{R^{2}+a^{2}}+\frac{v_{2}(RY-
aX)}{R^{2}+a^{2}}+\frac{v_{3}Z}{R}\right]\\!-v_{j}\gamma\right\\}~{}.~{}~{}~{}~{}~{}~{}$
(6)

In the limit $a\rightarrow 0$, Eqs. (4) - (6) are reduced to the exact metric
of a moving Schwarzschild black hole with an arbitrary constant velocity in
Kerr-Schild coordinates:

$\displaystyle
g_{00}=-1+\frac{2\gamma^{2}M}{R}\left(1-\frac{\bm{v}\cdot\\!\bm{X}}{R}\right)^{2}~{},$
(7) $\displaystyle g_{0i}=\frac{2\gamma
M}{R}\\!\left(\\!1\\!-\\!\frac{\bm{v}\cdot\\!\bm{X}}{R}\\!\right)\\!\\!\left[\frac{X_{i}}{R}\\!-\\!v_{i}\gamma\\!+\\!\frac{v_{i}(\gamma\\!-\\!1)(\bm{v}\cdot\\!\bm{X})}{v^{2}R}\right]~{},~{}~{}~{}~{}$
(8) $\displaystyle
g_{ij}=\delta_{ij}+\frac{2M}{R}\left[\frac{X_{i}}{R}-v_{i}\gamma+\frac{v_{i}(\gamma-1)(\bm{v}\cdot\\!\bm{X})}{v^{2}R}\right]$
$\displaystyle\hskip
23.0pt\times\left[\frac{X_{j}}{R}-v_{j}\gamma+\frac{v_{j}(\gamma-1)(\bm{v}\cdot\\!\bm{X})}{v^{2}R}\right]~{},$
(9)

with $R=\sqrt{X^{2}+Y^{2}+Z^{2}}$ and $\bm{X}=(X,~{}Y,~{}Z)$. It can be seen
that when $\bm{v}=0$, Eqs. (7) - (9) are further simplified to the
Schwarzschild metric presented in Kerr-Schild coordinates in the background’s
rest frame, which reads [85, 86]:

$\displaystyle g_{00}=-1+\frac{2M}{r}~{},$ (10) $\displaystyle
g_{0i}=\frac{2Mx_{i}}{r^{2}}~{},$ (11) $\displaystyle
g_{ij}=\delta_{ij}+\frac{2Mx_{i}x_{j}}{r^{3}}~{},$ (12)

where $r=\sqrt{x^{2}+y^{2}+z^{2}}$.

##  3 Light deflection by a radially moving Kerr source up to the 2PM order

As an application of the weak field limit of the metric given in Eqs. (4) -
(6), we consider the 2PM gravitational deflection of light induced by a moving
Kerr black hole with a radial velocity [2]. Note that for the convenience of
discussion to be followed, we still denote the comoving Kerr-Schild frame of
the radially moving source as $(T,~{}X,~{}Y,~{}Z)$.

###  3.1 The metric in the weak field limit

The metric of a radially moving Kerr source with
$\bm{v}=v_{1}\bm{e}_{1}=v\bm{e}_{1}$ up to the 2PM order in Kerr-Schild
coordinates can be obtained from Eqs. (4) - (6) as follows:

$\displaystyle
g_{00}=-1+\frac{2\gamma^{2}M}{R}\left(1-\frac{vX}{R}\right)\left[1-v\left(\frac{X}{R}+\frac{2aY}{R^{2}}\right)\right]+O(M^{3})~{},$
(13) $\displaystyle g_{0i}=\frac{2\gamma
M}{R}\\!\left\\{\left(1-\frac{vX}{R}-\frac{v\hskip
1.0ptaY}{R^{2}}\right)\\!\left[\frac{X_{i}}{R}+v\gamma\left(\frac{v\gamma}{\gamma+1}\frac{X}{R}-1\right)\delta_{i1}\right]\right.$
$\displaystyle\left.\hskip 24.0pt+\left(1-\frac{vX}{R}\right)\frac{a(\gamma
Y\delta_{i1}-X\delta_{i2})}{R^{2}}\right\\}+O(M^{3})~{},$ (14) $\displaystyle
g_{ij}=\delta_{ij}\\!+\\!\frac{2M}{R}\\!\left\\{\left[\frac{X_{i}}{R}\\!+\\!\frac{a(Y\delta_{i1}\\!-\\!X\delta_{i2})}{R^{2}}\\!+\\!v\gamma\left(\frac{v\gamma}{\gamma+1}\\!\left(\frac{X}{R}\\!+\\!\frac{aY}{R^{2}}\right)\\!-\\!1\right)\delta_{i1}\right]\right.$
$\displaystyle\left.\hskip
24.0pt\times\left[\frac{X_{j}}{R}+\left(\frac{(\gamma-1)X}{R}-v\gamma\right)\delta_{j1}\right]+\left[\frac{X_{i}}{R}+\left(\frac{(\gamma-1)X}{R}-v\gamma\right)\delta_{i1}\right]\right.$
$\displaystyle\left.\hskip 24.0pt\times\frac{a(\gamma
Y\delta_{j1}-X\delta_{j2})}{R^{2}}\right\\}+O(M^{3})~{},~{}~{}~{}~{}~{}~{}$
(15)

for which the Lorentz transformation (2) is reduced to

$\displaystyle T=\gamma(t-vx)~{},$ (16) $\displaystyle X=\gamma(x-vt)~{},$
(17) $\displaystyle Y=y~{},$ (18) $\displaystyle Z=z~{},$ (19)

with $R=\sqrt{\gamma^{2}(x-vt)^{2}+y^{2}+z^{2}}$. The elements of the inverse
of this weak-field metric tensor up to the 1PM order are

$\displaystyle
g^{00}=-1\\!-\\!\frac{2\gamma^{2}M(R\\!-\\!vX)^{2}}{R^{3}}\\!+\\!O(M^{2})~{},~{}~{}~{}~{}~{}~{}g^{11}=1\\!-\\!\frac{2\gamma^{2}M(X\\!-\\!vR)^{2}}{R^{3}}\\!+\\!O(M^{2})~{},~{}~{}~{}~{}~{}~{}~{}~{}$
(20) $\displaystyle g^{22}=1-\frac{2MY^{2}}{R^{3}}+O(M^{2})~{},\hskip
52.63777ptg^{33}=1-\frac{2MZ^{2}}{R^{3}}+O(M^{2})~{},$ (21) $\displaystyle
g^{01}=\frac{2\gamma^{2}M\\!\left[(1\\!+\\!v^{2})RX\\!-\\!v(R^{2}\\!+\\!X^{2})\right]}{R^{3}}\\!+\\!O(M^{2})~{},~{}~{}~{}g^{02}=\frac{2\gamma
M(R\\!-\\!vX)Y}{R^{3}}\\!+\\!O(M^{2})~{},~{}~{}~{}~{}~{}~{}~{}$ (22)
$\displaystyle g^{03}=\frac{2\gamma M(R-vX)Z}{R^{3}}+O(M^{2})~{},\hskip
29.87547ptg^{12}=\frac{2\gamma M(vR-X)Y}{R^{3}}+O(M^{2})~{},~{}~{}~{}~{}$ (23)
$\displaystyle g^{13}=\frac{2\gamma M(vR-X)Z}{R^{3}}+O(M^{2})~{},\hskip
29.87547ptg^{23}=-\frac{2MYZ}{R^{3}}+O(M^{2})~{}.$ (24)

###  3.2 Equatorial equations of motion

Via calculating the Christoffel symbols, the null geodesic equations
constrained to the equatorial plane ($z=\frac{\partial}{\partial z}=0$) of the
lens up to the 2PM order can be obtained in the background’s rest frame:

$\displaystyle
0=\ddot{t}+\\!\gamma^{3}\Bigg{\\{}\\!\frac{M\\!\left\\{2v^{2}(X^{2}\\!-\\!y^{2})R\\!-\\!v\\!\left[R^{2}\\!+\\!v^{2}(X^{2}\\!-\\!2y^{2})\right]\\!X\\!\right\\}}{R^{5}}\\!+\\!\frac{2v^{2}Ma\\!\left[3XR\\!+\\!v(y^{2}\\!-\\!3X^{2})\right]\\!y}{R^{6}}$
$\displaystyle\hskip
8.5359pt+\,\frac{2M^{2}\left[(1+3v^{2})X^{4}-v\,(3+v^{2})X^{3}R+(2+3v^{2})X^{2}y^{2}-3vXy^{2}R+y^{4}\right]}{R^{7}}\Bigg{\\}}\,\dot{t}^{2}$
$\displaystyle\hskip
8.5359pt+\,\gamma^{3}\Bigg{\\{}\frac{M\left[2(X^{2}-y^{2})R-vX(3X^{2}-v^{2}R^{2})\right]}{R^{5}}+\frac{2Ma\left[3XR+v(y^{2}-3X^{2})\right]y}{R^{6}}$
$\displaystyle\hskip
8.5359pt+\frac{2M^{2}\left[(1+3v^{2})X^{4}-v(3\\!+\\!v^{2})X^{3}R+(1\\!+\\!4v^{2})X^{2}y^{2}-v(2\\!+\\!v^{2})Xy^{2}R+v^{2}y^{4}\right]}{R^{7}}\Bigg{\\}}\,\dot{x}^{2}$
$\displaystyle\hskip
8.5359pt+\,2\,\gamma^{3}\Bigg{\\{}\\!\frac{M\\!\left[(1\\!+\\!v^{2})X^{3}\\!+\\!(1\\!-\\!2v^{2})Xy^{2}\\!+\\!2v(y^{2}\\!-\\!X^{2})R\right]}{R^{5}}\\!-\\!\frac{2vMa\\!\left[3XR\\!+\\!v(y^{2}\\!-\\!3X^{2})\right]\\!y}{R^{6}}$
$\displaystyle\hskip
8.5359pt+\frac{2M^{2}\left\\{X\\!\left[R^{2}+v^{2}(3X^{2}+2y^{2})\right]-vR\left[(3+v^{2})X^{2}+y^{2}\right]\right\\}}{R^{6}}\Bigg{\\}}\,\dot{t}\,\dot{x}$
$\displaystyle\hskip
8.5359pt+\frac{2\gamma^{2}M(R^{2}\\!-\\!4vXR\\!+\\!3v^{2}X^{2})\,y}{R^{5}}\,\dot{t}\,\dot{y}+\\!\frac{2\gamma^{2}M\\!\left[4XR\\!-\\!v(4X^{2}\\!+\\!y^{2})\right]\\!y}{R^{5}}\dot{x}\,\dot{y}+O(M^{3})~{},$
(25) $\displaystyle
0=\ddot{x}+\gamma^{3}\Bigg{\\{}\\!\frac{M\\!\left[(1\\!-\\!3v^{2})X^{3}+Xy^{2}+2v^{3}(X^{2}\\!-\\!y^{2})R\right]}{R^{5}}\\!+\\!\frac{2v^{2}Ma\left(y^{2}-3X^{2}+3vXR\right)y}{R^{6}}$
$\displaystyle\hskip
8.5359pt-\frac{2M^{2}\left\\{XR\left[R^{2}+v^{2}(3X^{2}+2y^{2})\right]-vR^{2}\left[(3+v^{2})X^{2}+y^{2}\right]\right\\}}{R^{7}}\Bigg{\\}}\,\dot{t}^{2}$
$\displaystyle\hskip
8.5359pt+\,\gamma^{3}\Bigg{\\{}\frac{M\\!\left[(2\\!-\\!v^{2})Xy^{2}-(1\\!+\\!v^{2})X^{3}+2v(X^{2}\\!-\\!y^{2})R\right]}{R^{5}}\\!+\\!\frac{2Ma\left(y^{2}\\!-\\!3X^{2}+3vXR\right)y}{R^{6}}$
$\displaystyle\hskip
8.5359pt-\frac{2M^{2}\left[XR(X^{2}+3v^{2}R^{2})-vR^{2}(3X^{2}+v^{2}R^{2})\right]}{R^{7}}\Bigg{\\}}\,\dot{x}^{2}+\,2\gamma^{3}$
$\displaystyle\hskip
8.5359pt\times\Bigg{\\{}\\!\frac{vM\\!\left\\{\\!X\\!\left[(1\\!+\\!v^{2})X^{2}\\!-\\!(2\\!-\\!v^{2})y^{2}\right]\\!-\\!2v(X^{2}\\!-\\!y^{2})R\right\\}}{R^{5}}\\!-\\!\frac{2vMa\left(y^{2}\\!-\\!3X^{2}\\!+\\!3vXR\right)y}{R^{6}}$
$\displaystyle\hskip
8.5359pt-\frac{2M^{2}\\!\left[(1\\!+\\!3v^{2})X^{4}\\!-\\!v(3\\!+\\!v^{2})X^{3}R\\!+\\!(1+4v^{2})X^{2}y^{2}\\!-\\!v(2+v^{2})Xy^{2}R\\!+\\!v^{2}y^{4}\right]}{R^{7}}\\!\Bigg{\\}}\,\dot{t}\,\dot{x}$
$\displaystyle\hskip
8.5359pt+\frac{2v\gamma^{2}M(4X^{2}+y^{2}-4vXR)\,y}{R^{5}}\,\dot{t}\,\dot{y}-\frac{2\gamma^{2}M\\!\left[(3+v^{2})X^{2}+v^{2}y^{2}-4vXR\right]y}{R^{5}}\,\dot{x}\,\dot{y}$
$\displaystyle\hskip 8.5359pt+\,O(M^{3})~{},$ (26) $\displaystyle
0=\ddot{y}+\\!\Bigg{\\{}\\!\frac{M\\!\left[X^{2}\\!+\\!(3\gamma^{2}\\!-\\!2)y^{2}\right]\\!y}{R^{5}}\\!-\\!\frac{v\gamma^{2}Ma\\!\left[2R^{3}\\!-\\!2vX(X^{2}\\!-\\!3y^{2})\right]}{R^{6}}\\!-\\!\frac{2\gamma^{2}M^{2}(R\\!-\\!vX)^{2}y}{R^{6}}\\!\Bigg{\\}}\,\dot{t}^{2}$
$\displaystyle\hskip
7.68236pt-\Bigg{[}\frac{M(R^{2}-3\gamma^{2}y^{2})\,y}{R^{5}}-\frac{2\gamma^{2}Ma\,(X^{3}-3Xy^{2}-vR^{3})}{R^{6}}+\frac{2\gamma^{2}M^{2}(X-vR)^{2}\,y}{R^{6}}\Bigg{]}\,\dot{x}^{2}-2\gamma^{2}$
$\displaystyle\hskip
7.68236pt\times\Bigg{\\{}\\!\frac{3vMy^{3}}{R^{5}}\\!-\\!\frac{Ma\\!\left[(1\\!+\\!v^{2})R^{3}\\!+\\!2vX(3y^{2}\\!-\\!X^{2})\right]}{R^{6}}\\!+\\!\frac{2M^{2}\\!\\!\left[(1\\!+\\!v^{2})XR\\!-\\!v(2X^{2}\\!+\\!y^{2})\right]\\!y}{R^{6}}\\!\Bigg{\\}}\,\dot{t}\,\dot{x}$
$\displaystyle\hskip 7.68236pt+\frac{6v\gamma
MXy^{2}}{R^{5}}\,\dot{t}\,\dot{y}\\!-\\!\frac{6\gamma
MXy^{2}}{R^{5}}\,\dot{x}\,\dot{y}+O(M^{3})~{},$ (27)

where $R=\sqrt{X^{2}+Y^{2}}$, and a dot denotes the derivative with respect to
the affine parameter $\xi$ along the geodesic [2, 38]. Since $\dot{y}$ is
related to the gravitational deflection angle by Eq. (28) below and thus
regarded to be of the first order of magnitude, the terms with the factor
$\dot{y}^{2}$ on the right-hand side of Eqs. (25) - (27) belong to third- or
higher-order terms and have been omitted.

###  3.3 2PM gravitational deflection by a radially moving Kerr source

The geometrical diagram for light propagating in the equatorial plane of a
radially moving Kerr black hole is shown in Fig. 1. The spatial coordinates of
the source (denoted as $A$) and observer (denoted as $B$) of light in the
background’s rest Kerr-Schild frame are assumed as $(x_{A},~{}y_{A},~{}0)$ and
$(x_{B},~{}y_{B},~{}0)$, respectively, where $y_{A}\approx-b$, $x_{A}\ll-b$,
and $x_{B}\gg b$, with $b$ being the impact parameter. The spatial coordinates
of the emission and reception events in the comoving frame are given
respectively to be $(X_{A},~{}Y_{A},~{}0)$ and $(X_{B},~{}Y_{B},~{}0)$. The
green line denotes the perturbed propagation path of light coming from
$x\rightarrow-\infty$ with an initial velocity
$\bm{w}|_{x\rightarrow-\infty}=\bm{e}_{1}$.

Figure 1: The geometrical diagram for the equatorial propagation of light in
the gravitational field of a radially moving Kerr black hole. The
gravitational deflection is greatly exaggerated to distinguish the perturbed
path (green) from the unperturbed one (dashed horizontal).

In the following calculations of light bending, we adopt an iterative
technique, and assume the path parameter $\xi$ to have the dimension of
length, as done in Refs. 2, 72. The gravitational deflection angle of a test
particle propagating from $\xi\rightarrow-\infty$ to $\xi\rightarrow+\infty$
is defined by

$\displaystyle\alpha\equiv\arctan\left.\frac{dy}{dx}\right|_{\xi\rightarrow+\infty}-\arctan\left.\frac{dy}{dx}\right|_{\xi\rightarrow-\infty}$
$\displaystyle\hskip
9.0pt=\arctan\left.\frac{\dot{y}}{\dot{x}}\right|_{x\rightarrow+\infty}-\arctan\left.\frac{\dot{y}}{\dot{x}}\right|_{x\rightarrow-\infty}~{},~{}~{}~{}$
(28)

which can also be written approximately as under the scenario where both the
emitter and the observer are far away from the lens with a finite distance
[72]

$\displaystyle\alpha\approx\arctan\left.\frac{\dot{y}}{\dot{x}}\right|_{B}-\arctan\left.\frac{\dot{y}}{\dot{x}}\right|_{A}~{}.$
(29)

Up to the zero order, Eqs. (25) - (27) yield

$\displaystyle\dot{t}=1+O(M)~{},$ (30) $\displaystyle\dot{x}=1+O(M)~{},$ (31)
$\displaystyle\dot{y}=0+O(M)~{},$ (32)

where we have used the boundary conditions
$\dot{t}|_{\xi\rightarrow-\infty}=\dot{t}|_{x\rightarrow-\infty}=1$,
$\dot{x}|_{\xi\rightarrow-\infty}=\dot{x}|_{x\rightarrow-\infty}=1$, and
$\dot{y}|_{\xi\rightarrow-\infty}=\dot{y}|_{x\rightarrow-\infty}=0$. Eq. (31)
gives a 0PM parameter transformation

$dx=\left[1+O(M)\right]d\xi~{}.$ (33)

Moreover, based on the boundary condition
$y|_{\xi\rightarrow-\infty}=y|_{x\rightarrow-\infty}=-b$, Eq. (32) leads to
the zero-order form of $y$ as

$y=-b+O(M)~{}.$ (34)

In addition, the combination of Eqs. (30) - (31) and (17) gives the zero-order
coordinate transformation

$dX=\left[(1-v)\gamma+O(M)\right]dx~{}.$ (35)

The substitution of Eqs. (30) - (35) into Eqs. (25) - (27) yields the explicit
forms of $\dot{t},~{}\dot{x},$ and $\dot{y}$ up to the 1PM order, which read:

$\displaystyle\dot{t}=1+\frac{M\\!\left[2\left(X^{2}+b^{2}\right)+2X\sqrt{X^{2}+b^{2}}+vb^{2}\right]}{(1+v)\left(X^{2}+b^{2}\right)^{\frac{3}{2}}}+\,O(M^{2})~{},$
(36)
$\displaystyle\dot{x}=1+\frac{M\\!\left[b^{2}+2\,v\sqrt{X^{2}+b^{2}}\left(\sqrt{X^{2}+b^{2}}+X\right)\right]}{(1+v)\left(X^{2}+b^{2}\right)^{\frac{3}{2}}}+\,O(M^{2})~{},$
(37) $\displaystyle\dot{y}=\frac{(1-v)\gamma
MX\left(2X^{2}+3b^{2}\right)}{b\left(X^{2}+b^{2}\right)^{\frac{3}{2}}}+O(M^{2})~{}.$
(38)

Based on Eqs. (33) and (35), the integration of Eq. (38) over $\xi$ gives the
1PM form of $y$

$y=-b\left[1-\frac{M(2X^{2}+b^{2})}{b^{2}\sqrt{X^{2}+b^{2}}}\right]+O(M^{2})~{}.$
(39)

Eq. (37) leads to the 1PM parameter transformation

$d\xi=\left\\{1-\frac{M\\!\left[b^{2}+2\,v\sqrt{X^{2}+b^{2}}\left(\sqrt{X^{2}+b^{2}}+X\right)\right]}{(1+v)\left(X^{2}+b^{2}\right)^{\frac{3}{2}}}+O(M^{2})\right\\}dx~{}.$
(40)

With the help of Eqs. (17), (36) and (37), we can also obtain the coordinate
transformation up to the 1PM order

$dx=\frac{1}{(1-v)\gamma}\left\\{1+\frac{vM\\!\left[b^{2}+2X\left(\sqrt{X^{2}+b^{2}}+X\right)\right]}{(1+v)\left(X^{2}+b^{2}\right)^{\frac{3}{2}}}+O(M^{2})\right\\}dX~{}.$
(41)

Substituting Eqs. (36) - (41) into the integration of Eq. (27) over $\xi$, we
get the explicit form of $\dot{y}$ up to the 2PM order as follows:

$\displaystyle\dot{y}=(1-v)^{2}\gamma^{2}\int\Bigg{\\{}\frac{3Mb^{3}}{\left(X^{2}+b^{2}\right)^{\frac{5}{2}}}+\frac{3M^{2}b\left(2X^{2}+b^{2}\right)\left(4b^{2}-X^{2}\right)}{\left(X^{2}+b^{2}\right)^{4}}$
$\displaystyle\hskip
123.76965pt-\frac{2Ma\\!\left[X^{3}-3b^{2}X+(X^{2}+b^{2})^{\frac{3}{2}}\right]}{\left(X^{2}+b^{2}\right)^{3}}\Bigg{\\}}\,d\xi+O(M^{3})$
$\displaystyle\hskip
0.56917pt=(1-v)\gamma\int\Bigg{\\{}\frac{3Mb^{3}}{\left(X^{2}+b^{2}\right)^{\frac{5}{2}}}+\frac{M^{2}b\left(9b^{6}+30b^{4}X^{2}+15b^{2}X^{4}-6X^{6}\right)}{\left(X^{2}+b^{2}\right)^{5}}$
$\displaystyle\hskip
113.81102pt+\,Ma\\!\left[\frac{6b^{2}X-2X^{3}}{\left(X^{2}+b^{2}\right)^{3}}-\frac{2}{\left(X^{2}+b^{2}\right)^{\frac{3}{2}}}\right]\\!\Bigg{\\}}\,dX+O(M^{3})$
$\displaystyle\hskip
0.56917pt=(1-v)\gamma\Bigg{\\{}\frac{M(2X^{2}+3b^{2})X}{b\left(X^{2}+b^{2}\right)^{\frac{3}{2}}}\\!+\\!\frac{3M^{2}\\!\left[5bX^{5}+16b^{3}X^{3}+7b^{5}X+5(X^{2}+b^{2})^{3}\arctan{\frac{X}{b}}\right]}{4b^{2}\left(X^{2}+b^{2}\right)^{3}}$
$\displaystyle\hskip
10.243pt-\frac{Ma\\!\left[2X^{5}+b^{2}X^{2}\left(4X-\sqrt{X^{2}+b^{2}}\right)+b^{4}\left(2X+\sqrt{X^{2}+b^{2}}\right)\right]}{b^{2}\left(X^{2}+b^{2}\right)^{\frac{5}{2}}}\Bigg{\\}}+O(M^{3})~{}.$
(42)

Finally, substituting Eqs. (37) and (42) into Eq. (29), we achieve the
explicit form of the null gravitational deflection angle up to the 2PM order
in the background’s rest Kerr-Schild frame as

$\alpha=(1-v)\gamma\left(\frac{4M}{b}+\frac{15\pi
M^{2}}{4b^{2}}-\frac{4Ma}{b^{2}}\right)+O(M^{3})~{},$ (43)

which matches well with the result obtained in the previous works (See, e.g.,
Refs. 2, 15, 17, 25). Here, the weak-field conditions
$\frac{M}{\sqrt{X_{A}^{2}+b^{2}}}\ll 1$ and
$\frac{M}{\sqrt{X_{B}^{2}+b^{2}}}\ll 1$ have been used in the derivations. The
contribution of the spin term on the right-hand side of Eq. (43) is negative
and positive for the prograde and retrograde motions of the photon relative to
the rotation of the black hole, respectively. Note that the radial velocity
effect on the second-order gravitational deflection of light and its
measurability have been discussed in the previous work [25]. In addition, the
consistency between Eq. (43) and Eq. (20) obtained in harmonic coordinates in
Ref. 72 (along with Eq. (32) in Ref. 25) indicates further the fact that
observable physical quantities are independent on the concrete coordinates one
uses.

At to Eq. (43), a simple numerical example in astrophysical scenarios is given
as follows. Let us consider the 2PM gravitational deflection of light caused
by a kicked Kerr black hole resulted from the merger of two unequal-mass black
holes mentioned above [61]. The constant radial velocity and the spin of the
black hole are assumed as $v=5.8\times 10^{-4}$ (corresponding to a kick
velocity $\sim 175.0$ km/s) and $a=0.5M$, respectively. The impact parameter
is set to be $b=1.0\times 10^{5}M$. Then, the first-order mass-induced,
second-order mass-induced and spin-induced contributions to the total
deflection angle $\alpha$ in Eq. (43) can be evaluated to be about $8.3$ as,
$242.9\,\mu$as, and $41.2\,\mu$as (absolute value), respectively.

##  4 Summary

In this paper, we have worked out the exact metric of a moving Kerr black hole
with an arbitrary constant velocity in Kerr-Schild coordinates via the Lorentz
boosting technique. As an application of the weak field limit of this
solution, the null gravitational deflection in the equatorial plane of a
radially moving Kerr black hole up to the second post-Minkowskian order has
been computed, in accordance with the result presented in the previous works.
Considering the commonness of the relative motion of the source of light, the
observer, and the deflector, this new solution of the Einstein field equations
might be helpful to study the strong- and weak-field physics of a moving
gravitational system.

## Acknowledgments

We would like to thank the referee for his/her valuable comments that have
improved this manuscript. This work was supported by the National Natural
Science Foundation of China (Grant Nos. 11947018 and 11947128).

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