If the velocity is relativistic, and we do not restrict ourselves to a linear approximation, then the gravitational field of an isolated mass is best described by a metric known as a boosted Schwarzschild solution. It is simply the Schwarzschild metric written in a reference frame where this mass is moving with a constant velocity in a flat background.
Note, that since there is no point masses in general relativity, this metric either describes the spacetime outside of a moving extended body or a moving black hole. This last interpretation makes this metric quite useful in the field of numerical relativity where such solutions serve as initial data for numerically simulating black hole collisions. For example, the paper:
- Matzner, R. A., Huq, M. F., & Shoemaker, D. (1998). Initial data and coordinates for multiple black hole systems. Physical Review D, 59(2), 024015, arXiv:gr-qc/9805023.
contains the explicitly written forms for such metrics, and so the following equations are mostly lifted from there.
We start by noting that Schwarzchild metric could be written in Kerr–Schild form:
\begin{equation}
ds^{2} = \eta_{\mu \nu} dx^{\mu} dx^{\nu} + 2H(x^{\alpha}) l_{\mu}
l_{\nu} dx^{\mu} dx^{\nu}
\label{1}
\end{equation}
where $\eta_{\mu \nu}$ is the usual flat space Minkowski metric, $H$ is a scalar
function of position and time and $l_{\mu}$ is a (position dependent) null vector (null both in the background, and in the full metric),
$$
\eta^{\mu \nu} l_{\mu} l_{\nu} = g^{\mu \nu} l_{\mu} l_{\nu} = 0,
$$
For Schwarzschild metric specifically, the function $H = M/r$, where $M$ is the mass of the particle and $r=(x^2+y^2+z^2)^{1/2}$ is a normal expression for radial coordinate. The null vector is
\begin{equation}
l_{\mu} = \left(1, \frac{x }{r}, \frac{y}{r}, \frac{z}{r}\right).
\end{equation}
Transformation into a “boosted” reference frame consist in applying the usual (Minkowski space) Lorentz transformation to the cartesian coordinates $x^{\mu}=\{t, x, y, z\}$ which also straightforwardly transforms $H$ and $l_{\mu}$:
\begin{equation}
\begin{array}{rcl}
x'^{\beta} & = & \Lambda_{\alpha}^{\beta} x^{\alpha}
\\
H(x^{\alpha})
& \rightarrow & H(\Lambda^{-1 \alpha}\,_{\beta} x'^{\beta})
\\
l'_{\delta}
& = & \Lambda^{\gamma}\,_{\delta}
l_{\gamma}(\Lambda^{-1 \, \alpha}\,_{\beta} x'^{\beta})
\\
g'_{\mu \nu}
& = & \eta_{\mu \nu} + 2H l'_{\mu} l'_{\nu}
\end{array}
\end{equation}
Note, that since the new vector $l'_\mu$ is also null, the metric $g'_{\mu \nu}$ also has Kerr–Schild form.
For example, for the motion along the $z$-direction ($v \equiv v^z$), with new coordinates $(x, y, z, t)$ we have
\begin{equation}
\begin{array}{lcl}
r^{2} & = & x^{2} + y^{2} + \gamma^{2}(z - v t)^{2} \\
l_{t} & = & \gamma(1 - v \gamma (z - v t)/r) \\
l_{x} & = & x/r \\
l_{y} & = & y/r \\
l_{z} & = & \gamma(\gamma (z - v t)/r - v) \\
H & = & M/r \\
\end{array}
\end{equation}
where $\gamma = 1/\sqrt{(1-v^2)}$.
As an illustration here is a slice through this spacetime (also from the abovementioned paper). The lapse scalar and shift vector plotted could be viewed as analogues of scalar and vector potential.
[![Image from gr-qc/9805023][1]][1]
Figure caption:
Contours of lapse and shift vector $\beta_{i}$ for a boosted
Schwarzschild (Eddington-Finkelstein) black hole. This is a cut
through the equator; the hole is boosted at 0.5 upward in the
figure. The shift is smaller ahead of the hole than behind, which
allows the black hole to move through the computational grid. The
ovoid figure is the horizon (distorted in these coordinates). The
cardioid curves are contours of constant lapse $\alpha$, with values from
the topmost contour down of 0.91, 0.86, 0.78, 0.71, 0.64, 0.58,
0.52. Along the axis of motion the lapse for this $v=0.5$ case is
$\sqrt{3}/2$ at the leading point and $1/2$ at the trailing point.
[1]: https://i.sstatic.net/ScMJw.png
