1. Introduction
The detection of gravitational waves [1–9] and the direct imaging of supermassive black holes in M87 [10–15] and Sgr A* [16–21] have provided strong evidence for the widespread existence of black holes [22, 23], confirming the predictions of general relativity (GR) as the leading theory of gravity. However, GR faces a significant challenge: the presence of singularities in traditional black hole solutions, where physical quantities diverge, remains unresolved [24, 25]. This issue highlights the limitations of GR in extreme regimes and motivates the search for singularity-free regular black hole solutions. Nonlinear electrodynamics and modified gravity have emerged as promising frameworks, offering models that retain key features like event horizons and asymptotic flatness while avoiding singularities.
The development of regular black hole solutions began with Bardeen's pioneering work in 1968 [26], which introduced the first singularity-free model through a phenomenological modification of the metric. Although the Bardeen black hole demonstrated that singularities could be avoided without sacrificing essential properties, its initial lack of connection to fundamental physics limited its applicability. This gap was bridged when the Bardeen solution was shown to arise from Einstein's equations coupled to nonlinear electrodynamics [27], with a magnetic charge deforming the Schwarzschild spacetime. This breakthrough inspired the development of other regular black hole models, including the Hayward [28], Simpson−Visser [29], and Ayón–Beato–García (ABG) black holes [30]. Among these, the ABG black hole stands out as a significant advancement, coupling general relativity with nonlinear electrodynamics to eliminate the central singularity and provide a clear physical mechanism through the nonlinear electromagnetic field. Subsequent studies have explored rotating solutions [31], circular orbits [32, 33], the weak energy condition [34], quasinormal modes [35], thermodynamics [36], strong lensing and shadow properties [37], and particle collisions near the horizon [38], further solidifying the ABG black hole as a key framework for understanding regular black hole solutions.
Despite these advancements, the motion of celestial bodies in ABG black hole spacetimes has received limited attention. In this work, we focus on the motion of test particles in ABG black hole spacetimes, aiming to uncover the dynamical characteristics influenced by the nonlinear electromagnetic field and test the spacetime through precessing orbits. Precessing orbits, such as the advance of Mercury's perihelion, have been crucial in validating GR [39] and understanding spacetime properties. In GR, the post-Newtonian (PN) approximation is commonly used to study precessing motion, with analytical formulas accounting for first and higher-order corrections proportional to the mass [40–49] and spin [50–59] of the central body. Recent studies have explored the 2PN precessing motion around the Bardeen black hole [60] and the 3PN solution for quasi-Keplerian motion around regular black holes with an asymptotically Minkowski core [61]. Precessing motion has also proven effective in testing alternative gravity theories using planets orbiting the Sun [62–70], exoplanets [71–75], binary pulsars [76–84], and stars orbiting Sgr A* [85–94].
Inspired by the regular ABG black hole [30], we use this model to describe a black hole whose deviation from the Schwarzschild spacetime is characterized by the electric charge parameter q. We derive the 2PN solution for the quasi-Keplerian motion of a test particle in this spacetime, including its relativistic periastron advance. The solution is formulated in terms of the particle's orbital energy and angular momentum, as well as the mass and the electric charge of the ABG black hole.
The paper is organized as follows. Section 2 introduces the metric and action of an ABG black hole. Section 3 gives the 2PN metric of the ABG black hole, the corresponding Lagrangian, and the orbital energy and angular momentum. Section 4 presents the detailed derivation of the 2PN solution for quasi-Keplerian motion. A summary is provided in Section 5 .
2. Metric and action of an ABG black hole
The ABG spherically symmetric black hole metric defined by Schwarzschild coordinates is given by [30]
$\begin{eqnarray}{\rm{d}}{s}^{2}=-f(r){\rm{d}}{t}^{2}+f{(r)}^{-1}{\rm{d}}{r}^{2}+{r}^{2}{\rm{d}}{\theta }^{2}+{r}^{2}\sin {\theta }^{2}{\rm{d}}{\phi }^{2},\end{eqnarray}$
with $\begin{eqnarray}f(r)=1-\frac{2m{r}^{2}}{{({r}^{2}+{q}^{2})}^{\frac{3}{2}}}+\frac{{q}^{2}{r}^{2}}{{({r}^{2}+{q}^{2})}^{2}},\end{eqnarray}$
and the associated electric field $\begin{eqnarray}\begin{array}{l}{F}_{tr}(r)=E(r)\\ =\,q\,{r}^{4}\left[\frac{{r}^{2}-5{q}^{2}}{{({r}^{2}+{q}^{2})}^{4}}+\frac{15}{2}\frac{m}{{({r}^{2}+{q}^{2})}^{7/2}}\right],\end{array}\end{eqnarray}$
here m denotes the standard gravitational mass parameter, while q corresponds to the electric charge parameter measured in units of m. The gravitational constant and the speed of light in vacuum are set as 1.The line element (1 ) is nonsingular static solution of the Einstein nonlinear electrodynamic field equations
$\begin{eqnarray}\begin{array}{rcl}{G}_{\mu \nu } & = & 8\pi {T}_{\mu \nu }=8\pi [{{ \mathcal L }}_{F}{F}_{\mu \eta }{F}_{\nu }^{\eta }-{ \mathcal L }{g}_{\mu \nu }],\\ {{ \mathcal L }}_{F} & = & \frac{\partial { \mathcal L }}{\partial F},\end{array}\end{eqnarray}$
which satisfies the action functional $\begin{eqnarray}S=\int \sqrt{-g}\,{{\rm{d}}}^{4}x\left[\frac{1}{16\pi }R-\frac{1}{4\pi }{ \mathcal L }(F)\right],\,\end{eqnarray}$
where R is the Ricci scalar and ${ \mathcal L }$ is a functional of F ≡ (1/4)FμνFμν.3. The 2PN Lagrangian, energy, angular momentum in ABG spacetime
First, converting the metric in the spherical coordinates given by equation (1 ) into that in Cartesian coordinates via
$\begin{eqnarray}\begin{array}{rcl}{x}^{0} & = & t,\,\,\,{x}^{1}=r\sin \theta \cos \phi ,\\ {x}^{2} & = & r\sin \theta \sin \phi ,\\ {x}^{3} & = & r\cos \theta ,\end{array}\end{eqnarray}$
we obtain $\begin{eqnarray}\begin{array}{rcl}{\rm{d}}{s}^{2} & = & -\left[1\,-\,\frac{2m{r}^{2}}{{({r}^{2}+{q}^{2})}^{\frac{3}{2}}}\,+\,\frac{{q}^{2}{r}^{2}}{{({r}^{2}+{q}^{2})}^{2}}\right]{\rm{d}}{t}^{2}\\ & & +\left\{{\left[1\,-\,\frac{2m{r}^{2}}{{({r}^{2}+{q}^{2})}^{\frac{3}{2}}}\,+\,\frac{{q}^{2}{r}^{2}}{{({r}^{2}+{q}^{2})}^{2}}\right]}^{-1}\,-\,1\right\}\\ & & \times {\left(\frac{{\boldsymbol{x}}\,\cdot \,{\rm{d}}{\boldsymbol{x}}}{r}\right)}^{2}\,+\,{\rm{d}}{{\boldsymbol{x}}}^{2},\end{array}\end{eqnarray}$
where x = (x0, x1, x2, x3).According to the theory of the PN approximation, this metric can be expanded into the power of $\frac{m}{r}$ with r = ∣x∣. Keeping all terms to the 2PN order for the dynamics of the massive particle, we have
$\begin{eqnarray}{g}_{00}=-1+\frac{2m}{r}-\frac{{q}^{2}}{{r}^{2}}-\frac{3m{q}^{2}}{{r}^{3}},\,\end{eqnarray}$
$\begin{eqnarray}{g}_{0i}=0,\,\end{eqnarray}$
$\begin{eqnarray}{g}_{ij}={\delta }_{ij}+\left[\frac{2m}{r}+\frac{4{m}^{2}}{{r}^{2}}\left(1-\frac{{q}^{2}}{4{m}^{2}}\right)\right]\frac{{x}^{i}{x}^{j}}{{r}^{2}},\end{eqnarray}$
with i and j running from 1 to 3.The relation between the coordinate time t and the proper time τ in the 2PN approximations can be written as [95]8 )–(10 ) into the Lagrangian, we obtain
$\begin{eqnarray}\begin{array}{rcl}{\left(\frac{{\rm{d}}\tau }{{\rm{d}}t}\right)}^{2} & = & -{g}_{\mu \nu }\left(\frac{{\rm{d}}{x}^{\mu }}{{\rm{d}}t}\right)\left(\frac{{\rm{d}}{x}^{\nu }}{{\rm{d}}t}\right)\\ & & =1\,-\,{{\boldsymbol{v}}}^{2}\,-\,{g}_{00}^{(2)}\,-\,{g}_{00}^{(4)}\,-\,{g}_{00}^{(6)}\\ & & -{g}_{ij}^{(2)}{v}^{i}{v}^{j}\,-\,{g}_{ij}^{(4)}{v}^{i}{v}^{j},\end{array}\end{eqnarray}$
where the superscript number in brackets denotes the PN order. The corresponding Lagrangian can be obtained as follows [95]: $\begin{eqnarray}\begin{array}{rcl}{ \mathcal L } & = & 1-\frac{{\rm{d}}\tau }{{\rm{d}}t}\\ & & =\frac{1}{2}\left({{\boldsymbol{v}}}^{2}+{g}_{00}^{(2)}\right)+\frac{1}{2}\left({g}_{00}^{(4)}+{g}_{ij}^{(2)}{v}^{i}{v}^{j}\right)\\ & & +\frac{1}{8}\left[{{\boldsymbol{v}}}^{4}+2{{\boldsymbol{v}}}^{2}{g}_{00}^{(2)}+{({g}_{00}^{(2)})}^{2}\right]\\ & & +\,\,\frac{1}{2}\left({g}_{00}^{(6)}+{g}_{ij}^{(4)}{v}^{i}{v}^{j}\right)\\ & & +\frac{1}{4}\left({{\boldsymbol{v}}}^{2}+{g}_{00}^{(2)}\right)\left({g}_{00}^{(4)}+{g}_{ij}^{(2)}{v}^{i}{v}^{j}\right)\\ & & +\,\,\frac{1}{16}\left[{{\boldsymbol{v}}}^{6}+{\left({g}_{00}^{(2)}\right)}^{3}+3{{\boldsymbol{v}}}^{4}{g}_{00}^{(2)}+3{{\boldsymbol{v}}}^{2}{\left({g}_{00}^{(2)}\right)}^{2}\right].\end{array}\end{eqnarray}$
Substituting equations ( $\begin{eqnarray}\begin{array}{rcl}{ \mathcal L } & = & \frac{1}{2}{{\boldsymbol{v}}}^{2}+\frac{m}{r}+\frac{1}{8}{{\boldsymbol{v}}}^{4}+\frac{1}{2}\frac{m}{r}{{\boldsymbol{v}}}^{2}\\ & & +\frac{1}{2}\frac{{m}^{2}}{{r}^{2}}\left(1-\frac{{q}^{2}}{{m}^{2}}\right)+\frac{m{({\boldsymbol{v}}\,\cdot \,{\boldsymbol{x}})}^{2}}{{r}^{3}}\\ & & +\frac{1}{2}\frac{{m}^{3}}{{r}^{3}}\left(1-\frac{4{q}^{2}}{{m}^{2}}\right)\\ & & +\frac{1}{16}{{\boldsymbol{v}}}^{6}+\frac{3}{4}\frac{{m}^{2}}{{r}^{2}}{{\boldsymbol{v}}}^{2}\left(1-\frac{{q}^{2}}{3{m}^{2}}\right)\\ & & +\frac{3}{8}\frac{m}{r}{{\boldsymbol{v}}}^{4}+\frac{3{m}^{2}{({\boldsymbol{v}}\,\cdot \,{\boldsymbol{x}})}^{2}}{{r}^{4}}\left(1-\frac{{q}^{2}}{6{m}^{2}}\right)\\ & & +\frac{m{({\boldsymbol{v}}\,\cdot \,{\boldsymbol{x}})}^{2}{{\boldsymbol{v}}}^{2}}{2{r}^{3}},\end{array}\end{eqnarray}$
where v denotes the the test particle's velocity.According to this Lagrangian, the energy ${ \mathcal E }$ and the angular momentum ${ \mathcal J }$ of the equatorial motion can be obtained as
$\begin{eqnarray}\begin{array}{rcl}{ \mathcal E } & \equiv & {\boldsymbol{v}}\cdot \frac{\partial { \mathcal L }}{\partial {\boldsymbol{v}}}-{ \mathcal L }=\frac{1}{2}{{\boldsymbol{v}}}^{2}-\frac{m}{r}\\ & & +\frac{3}{8}{{\boldsymbol{v}}}^{4}+\frac{1}{2}\frac{m}{r}{{\boldsymbol{v}}}^{2}-\frac{1}{2}\frac{{m}^{2}}{{r}^{2}}\left(1-\frac{{q}^{2}}{{m}^{2}}\right)+\frac{m{({\boldsymbol{v}}\,\cdot \,{\boldsymbol{x}})}^{2}}{{r}^{3}}\\ & & -\frac{1}{2}\frac{{m}^{3}}{{r}^{3}}\left(1-\frac{4{q}^{2}}{{m}^{2}}\right)\,\\ & & +\frac{5}{16}{{\boldsymbol{v}}}^{6}+\frac{3}{4}\frac{{m}^{2}}{{r}^{2}}{{\boldsymbol{v}}}^{2}\left(1-\frac{{q}^{2}}{3{m}^{2}}\right)\\ & & +\frac{9}{8}\frac{m}{r}{{\boldsymbol{v}}}^{4}+\frac{3{m}^{2}{({\boldsymbol{v}}\,\cdot \,{\boldsymbol{x}})}^{2}}{{r}^{4}}\left(1-\frac{{q}^{2}}{6{m}^{2}}\right)\\ & & +\frac{3}{2}\frac{m{({\boldsymbol{v}}\,\cdot \,{\boldsymbol{x}})}^{2}{{\boldsymbol{v}}}^{2}}{{r}^{3}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{ \mathcal J } & \equiv & \left|{\boldsymbol{x}}\times \frac{\partial { \mathcal L }}{\partial {\boldsymbol{v}}}\right|=| {\boldsymbol{x}}\,\times \,{\boldsymbol{v}}| \left[1+\frac{1}{2}{{\boldsymbol{v}}}^{2}\right.\\ & & +\frac{m}{r}+\frac{3}{8}{{\boldsymbol{v}}}^{4}+\frac{3}{2}\frac{{m}^{2}}{{r}^{2}}\left(1-\frac{{q}^{2}}{3{m}^{2}}\right)\\ & & \left.+\frac{3}{2}\frac{m}{r}{{\boldsymbol{v}}}^{2}+\frac{m{({\boldsymbol{v}}\,\cdot \,{\boldsymbol{x}})}^{2}}{{r}^{3}}\right].\end{array}\end{eqnarray}$
4. The quasi-Keplerian motion in the 2PN approximation
We deduce the analytical solution for the quasi-Keplerian motion according to the same procedure given by Brumberg [40].
The trajectory of the test particle in the equatorial plane of the ABG black hole can be expressed as
$\begin{eqnarray}{\boldsymbol{x}}=r(\cos \phi \,{{\boldsymbol{e}}}_{x}+\sin \phi \,{{\boldsymbol{e}}}_{y}),\end{eqnarray}$
where φ is the azimuthal angle. ex and ey are the unit vectors of the x-axis and y-axis.By utilizing the relations ${{\boldsymbol{v}}}^{2}=({\dot{r}}^{2}+{r}^{2}{\dot{\phi }}^{2})$ and ${({\boldsymbol{v}}\,\cdot \,{\boldsymbol{x}})}^{2}={r}^{2}{\dot{r}}^{2}$, where the dot denotes the time derivative, we can reformulate the expressions for orbital energy and angular momentum in equations (14 )–(15 ). These new formulations enable us to derive
$\begin{eqnarray}{r}^{4}{\dot{\phi }}^{2}={{ \mathcal J }}^{2}\left[1\,-\,2\,{ \mathcal E }\,-\,\frac{4m}{r}\,+\,3\,{{ \mathcal E }}^{2}\,+\,8\,{ \mathcal E }\frac{m}{r}\,+\,\frac{4{m}^{2}}{{r}^{2}}\left(1\,+\,\frac{{q}^{2}}{2{m}^{2}}\right)\right],\end{eqnarray}$
and $\begin{eqnarray}{\dot{r}}^{2}=A+\frac{B}{r}+\frac{C}{{r}^{2}}+\frac{D}{{r}^{3}}+\frac{E}{{r}^{4}},\end{eqnarray}$
with $\begin{eqnarray}A=2{ \mathcal E }\left(1\,-\,\frac{3}{2}\,{ \mathcal E }\,+\,2\,{{ \mathcal E }}^{2}\right),\,\end{eqnarray}$
$\begin{eqnarray}B=2m(1\,-\,6\,{ \mathcal E }\,+\,9\,{{ \mathcal E }}^{2}),\,\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}C=-{{ \mathcal J }}^{2}\left[1\,-\,2\,{ \mathcal E }\,+\,8\frac{{m}^{2}}{{{ \mathcal J }}^{2}}\left(1\,+\,\frac{{q}^{2}}{8{m}^{2}}\right)\right.\\ \left.\,-24\frac{{m}^{2}{ \mathcal E }}{{{ \mathcal J }}^{2}}\left(1\,+\,\frac{{q}^{2}}{4{m}^{2}}\right)\,+\,3\,{{ \mathcal E }}^{2}\right],\end{array}\,\end{eqnarray}$
$\begin{eqnarray}D=6m{{ \mathcal J }}^{2}\left[1-\,2\,{ \mathcal E }\,+\,\frac{4}{3}\frac{{m}^{2}}{{{ \mathcal J }}^{2}}\left(1\,+\,\frac{5}{8}\frac{{q}^{2}}{{m}^{2}}\right)\right],\end{eqnarray}$
$\begin{eqnarray}E=-12{m}^{2}{{ \mathcal J }}^{2}\left(1\,+\,\frac{{q}^{2}}{4{m}^{2}}\right).\,\end{eqnarray}$
Throughout this paper, the ‘±' signs denote the anti-clockwise motion and the clockwise motion, respectively.Utilizing the relation 17 )–(18 ) into (24 ), we can express the radial equation in the form 25 ) is a fourth-order polynomial in r−1, we can further re-write it as 25 ) and equation (31 ), we have 31 ) that r± = ar(1 ± er) represents the maximal and minimal values for r. Hence, ar and er can be regarded as the semi-major axis and the eccentricity of the quasi-Keplerian orbit. Specifically, ar and er characterize the size and shape of the bound orbital trajectory.
$\begin{eqnarray}{\dot{r}}^{2}={\left[\frac{{\rm{d}}(1/r)}{{\rm{d}}\phi }\right]}^{2}({r}^{4}{\dot{\phi }}^{2}),\end{eqnarray}$
and plugging equations ( $\begin{eqnarray}{\left[\frac{{\rm{d}}(1/r)}{{\rm{d}}\phi }\right]}^{2}={A}^{{\prime} }+\frac{{B}^{{\prime} }}{r}+\frac{{C}^{{\prime} }}{{r}^{2}}+\frac{{D}^{{\prime} }}{{r}^{3}}+\frac{{E}^{{\prime} }}{{r}^{4}},\end{eqnarray}$
with $\begin{eqnarray}{A}^{{\prime} }=\frac{2{ \mathcal E }}{{{ \mathcal J }}^{2}}\left(1\,+\,\frac{1}{2}\,{ \mathcal E }\right),\end{eqnarray}$
$\begin{eqnarray}{B}^{{\prime} }=\frac{2m}{{{ \mathcal J }}^{2}},\,\end{eqnarray}$
$\begin{eqnarray}{C}^{{\prime} }=-\left(1\,+\,\frac{{q}^{2}}{{{ \mathcal J }}^{2}}\right)\,{,}\end{eqnarray}$
$\begin{eqnarray}{D}^{{\prime} }=2m\left(1\,-\,\frac{3}{2}\frac{{q}^{2}}{{{ \mathcal J }}^{2}}\right),\end{eqnarray}$
$\begin{eqnarray}{E}^{{\prime} }=-{q}^{2}.\,\end{eqnarray}$
Since the right hand side of equation ( $\begin{eqnarray}\begin{array}{l}{\left[\frac{{\rm{d}}(1/r)}{{\rm{d}}\phi }\right]}^{2}=\left[\frac{1}{r}\,-\,\frac{1}{{a}_{r}(1+{e}_{r})}\right]\\ \times \left[\frac{1}{{a}_{r}(1-{e}_{r})}\,-\,\frac{1}{r}\right]\left({C}_{1}+\frac{{C}_{2}}{r}+\frac{{C}_{3}}{{r}^{2}}\right).\end{array}\end{eqnarray}$
Comparing the coefficients between equation ( $\begin{eqnarray}{a}_{r}=\frac{m}{-2{ \mathcal E }}\left[1\,+\,\frac{3}{2}{ \mathcal E }\,+\,\frac{1}{4}{{ \mathcal E }}^{2}\,+\,8\frac{{m}^{2}{ \mathcal E }}{{{ \mathcal J }}^{2}}\left(1\,-\,\frac{7}{8}\frac{{q}^{2}}{{m}^{2}}\right)\right],\,\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{e}_{r}^{2}=1\,+\,\frac{2{ \mathcal E }{{ \mathcal J }}^{2}}{{m}^{2}}\,-\,{ \mathcal E }\left[8\left(1\,-\,\frac{{q}^{2}}{4{m}^{2}}\right)\,+\,7\frac{{ \mathcal E }{{ \mathcal J }}^{2}}{{m}^{2}}\right]\\ \,+\,{{ \mathcal E }}^{2}\left[16\frac{{ \mathcal E }{{ \mathcal J }}^{2}}{{M}^{2}}\,-\,20\left(1\,-\,\frac{5}{4}\frac{{Q}^{2}}{{M}^{2}}\right)\,-\,32\frac{{M}^{2}}{{ \mathcal E }{{ \mathcal J }}^{2}}\left(1\,-\,\frac{7}{8}\frac{{q}^{2}}{{m}^{2}}\right)\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{C}_{1} & = & 1\,-\,\frac{4{m}^{2}}{{{ \mathcal J }}^{2}}\left(1\,-\,\frac{{q}^{2}}{4{m}^{2}}\right)\,-\,8\frac{{m}^{2}{ \mathcal E }}{{{ \mathcal J }}^{2}}\left(1\,-\,\frac{{q}^{2}}{4{m}^{2}}\right)\\ & & -\frac{16{m}^{4}}{{{ \mathcal J }}^{4}}\left(1\,-\,\frac{7}{8}\frac{{q}^{2}}{{m}^{2}}\right),\end{array}\,\end{eqnarray}$
$\begin{eqnarray}{C}_{2}=-2m\left(1\,-\,\frac{5}{2}\frac{{q}^{2}}{{{ \mathcal J }}^{2}}\right),\,\end{eqnarray}$
$\begin{eqnarray}{C}_{3}={q}^{2}.\,\end{eqnarray}$
It can be seen from equation (The solution of equation (31 ) can be written as
$\begin{eqnarray}r=\frac{{a}_{r}(1-{e}_{r}^{2})}{1+{e}_{r}\cos f},\end{eqnarray}$
with f being the true anomaly for the quasi-Keplerian orbit and satisfying $\begin{eqnarray}{\left(\frac{{\rm{d}}f}{{\rm{d}}\phi }\right)}^{2}={C}_{1}+\frac{{C}_{2}}{r}+\frac{{C}_{3}}{{r}^{2}}.\end{eqnarray}$
Substituting equations (34 )–(37 ) into equation (38 ), we have
$\begin{eqnarray}\begin{array}{rcl}\frac{{\rm{d}}f}{{\rm{d}}\phi } & = & F\left\{1\,-\,\frac{{m}^{2}}{{{ \mathcal J }}^{2}}\left[1\,+\,2{ \mathcal E }\,+\,10\frac{{m}^{2}}{{{ \mathcal J }}^{2}}\left(1\,-\,\frac{11}{20}\frac{{q}^{2}}{{m}^{2}}\right)\right]{e}_{r}\cos f\right.\\ & & \left.-\frac{{m}^{4}}{4{{ \mathcal J }}^{4}}\left(1\,-\,\frac{{q}^{2}}{{m}^{2}}\right){e}_{r}^{2}\cos 2f\right\},\end{array}\end{eqnarray}$
with $\begin{eqnarray}\begin{array}{l}F=\left[1\,-\,\frac{3{m}^{2}}{{{ \mathcal J }}^{2}}\left(1\,-\,\frac{{q}^{2}}{6{m}^{2}}\right)\,-\,\frac{13}{2}\frac{{m}^{2}{ \mathcal E }}{{{ \mathcal J }}^{2}}\left(1\,-\,\frac{3}{13}\frac{{q}^{2}}{{m}^{2}}\right)\right.\\ \left.-\frac{{m}^{4}}{{{ \mathcal J }}^{4}}\left(\frac{67}{4}\,-\,\frac{51}{4}\frac{{q}^{2}}{{m}^{2}}\,+\,\frac{1}{8}\frac{{q}^{4}}{{m}^{4}}\right)\right].\end{array}\end{eqnarray}$
Making the integration of equation (39 ), we can obtain
$\begin{eqnarray}\begin{array}{l}\phi \left(\frac{2\pi }{{\rm{\Phi }}}\right)=f+\frac{{m}^{2}}{{{ \mathcal J }}^{2}}\left[1\,+\,2{ \mathcal E }\,+\,10\frac{{m}^{2}}{{{ \mathcal J }}^{2}}\left(1\,-\,\frac{11}{20}\frac{{q}^{2}}{{m}^{2}}\right)\right]{e}_{r}\\ \times \sin f\,+\,\frac{3}{8}\frac{{m}^{4}}{{{ \mathcal J }}^{4}}\left(1\,-\,\frac{{q}^{2}}{3{m}^{2}}\right){e}_{r}^{2}\sin 2f,\end{array}\end{eqnarray}$
with $\begin{eqnarray}\begin{array}{l}{\rm{\Phi }}=2\pi \left[1\,+\,3\frac{{m}^{2}}{{{ \mathcal J }}^{2}}\left(1\,-\,\frac{{q}^{2}}{6{m}^{2}}\right)\,+\,\frac{15}{2}\frac{{m}^{2}{ \mathcal E }}{{{ \mathcal J }}^{2}}\left(1\,-\,\frac{{q}^{2}}{5{m}^{2}}\right)\right.\\ \left.+\,\frac{105}{4}\frac{{m}^{4}}{{{ \mathcal J }}^{4}}\left(1\,-\,\frac{3}{5}\frac{{q}^{2}}{{m}^{2}}\,+\,\frac{1}{70}\frac{{q}^{4}}{{m}^{4}}\right)\right].\end{array}\end{eqnarray}$
Finally, we derive the time dependence of the quasi-Keplerian motion. Combining equations (17 ) and (39 )–(40 ), we have
$\begin{eqnarray}\begin{array}{rcl}{r}^{2}\dot{f} & = & { \mathcal J }\left\{1\,-\,{ \mathcal E }\,-\,2\frac{m}{r}\,-\,3\frac{{m}^{2}}{{{ \mathcal J }}^{2}}\left(1\,-\,\frac{{q}^{2}}{6{m}^{2}}\right)\right.\\ & & +{{ \mathcal E }}^{2}\,+\,2{ \mathcal E }\frac{m}{r}\,+\,\frac{{q}^{2}}{{r}^{2}}\,-\,\frac{7}{2}\frac{{m}^{2}{ \mathcal E }}{{{ \mathcal J }}^{2}}\left(1\,-\,\frac{2}{7}\frac{{q}^{2}}{{m}^{2}}\right)\\ & & +6\frac{m}{r}\frac{{m}^{2}}{{{ \mathcal J }}^{2}}\left(1\,-\,\frac{{q}^{2}}{6{m}^{2}}\right)\\ & & -\frac{{m}^{4}}{{{ \mathcal J }}^{4}}\left(\frac{67}{4}\,-\,\frac{51}{4}\frac{{q}^{2}}{{m}^{2}}\,+\,\frac{1}{8}\frac{{q}^{4}}{{m}^{4}}\right)\,-\,\frac{{m}^{2}}{{{ \mathcal J }}^{2}}\\ & & \times \left[1\,+\,{ \mathcal E }-2\frac{m}{r}\,+\,7\frac{{m}^{2}}{{{ \mathcal J }}^{2}}\left(1\,-\,\frac{5}{7}\frac{{q}^{2}}{{m}^{2}}\right)\right]{e}_{r}\cos f\\ & & \left.-\frac{1}{4}\frac{{m}^{4}}{{{ \mathcal J }}^{4}}\left(1\,-\,\frac{{q}^{2}}{{m}^{2}}\right){e}_{r}^{2}\cos 2f\right\}.\end{array}\end{eqnarray}$
Introducing the post-Newtonian eccentric anomaly u by the relations 37 ) in terms of u as
$\begin{eqnarray}\begin{array}{rcl}\sin f & = & \frac{{(1\,-\,{e}_{r}^{2})}^{\frac{1}{2}}\sin u}{1\,-\,{e}_{r}\cos u};\,\,\cos f\,=\,\frac{\cos u\,-\,{e}_{r}}{1\,-\,{e}_{r}\cos u};\\ f & = & 2\,\arctan \left(\sqrt{\frac{1\,+\,{e}_{r}}{1\,-\,{e}_{r}}}\tan \frac{u}{2}\right),\end{array}\end{eqnarray}$
we have $\begin{eqnarray}\frac{{\rm{d}}f}{{\rm{d}}t}=\frac{{(1-{e}_{r}^{2})}^{1/2}}{1-{e}_{r}\cos u}\frac{{\rm{d}}u}{{\rm{d}}t},\end{eqnarray}$
and we can formulate the orbit given in equation ( $\begin{eqnarray}r={a}_{r}(1-{e}_{r}\cos u).\end{eqnarray}$
Integrating equation (43 ) and making use of equations (44 )–(46 ), we can achieve the final piece of the 2PN closed-form solution for the equatorial motion in ABG spacetime.
$\begin{eqnarray}\begin{array}{rcl}t\left(\frac{2\pi }{{{\rm{T}}}_{u}}\right) & = & u-{e}_{t}\sin u\\ & & +\frac{30m\,{{ \mathcal E }}^{2}}{\sqrt{-2\,{ \mathcal E }{{ \mathcal J }}^{2}}}\left(1\,-\,\frac{{q}^{2}}{5{m}^{2}}\right)(f-u),\end{array}\end{eqnarray}$
with Tu being the period for the eccentric anomaly u of the quasi-Keplerian motion $\begin{eqnarray}\begin{array}{rcl}{{\rm{T}}}_{u} & = & \frac{2\pi m}{{(-2{ \mathcal E })}^{\frac{3}{2}}}\left[1\,-\,\frac{15}{4}{ \mathcal E }\,-\,\frac{105}{32}{{ \mathcal E }}^{2}\right.\\ & & \left.+\,\frac{30m{{ \mathcal E }}^{2}}{\sqrt{-2{ \mathcal E }{{ \mathcal J }}^{2}}}\left(1\,-\,\frac{{q}^{2}}{5{m}^{2}}\right)\right],\end{array}\end{eqnarray}$
and et being the time eccentricity $\begin{eqnarray}\begin{array}{l}{e}_{t}={e}_{r}\left[1\,+\,6\,{ \mathcal E }\,+\,27\,{{ \mathcal E }}^{2}\,-\,\frac{30m{{ \mathcal E }}^{2}}{\sqrt{-2{ \mathcal E }{{ \mathcal J }}^{2}}}\right.\\ \,\times \left.\left(1\,-\,\frac{{q}^{2}}{5{m}^{2}}\right)\,+\,8\frac{{m}^{2}{ \mathcal E }}{{{ \mathcal J }}^{2}}\left(1\,-\,\frac{7}{8}\frac{{q}^{2}}{{m}^{2}}\right)\right].\end{array}\end{eqnarray}$
In the literature, one usually replaces the true anomaly f in the formula of quasi-Keplerian equation with another true anomaly $\upsilon$, which requires that $\sin \upsilon $ contribution vanishes at each PN order in $\phi (\frac{2\pi }{{\rm{\Phi }}})$ [44, 52, 96]. Specifically, the true anomaly $\upsilon$ represents the angular position for the test particle along the orbit measured from the pericenter.
Following the same method given in [96], we set 44 ) with the help of equation (50 ), we have [96]41 ) and requiring the $\sin \upsilon $ term to vanish in $\phi (\frac{2\pi }{{\rm{\Phi }}})$, we can obtain
$\begin{eqnarray}\upsilon =2\arctan \left(\sqrt{\frac{1+{e}_{\phi }}{1-{e}_{\phi }}}\tan \frac{u}{2}\right),\end{eqnarray}$
with $\begin{eqnarray}{e}_{\phi }={e}_{r}(1+\epsilon \,{c}_{1}+{\epsilon }^{2}\,{c}_{2}),\end{eqnarray}$
differing from the radial eccentricity er by some 1PN and 2PN level corrections c1 and c2. Here ε only denotes the PN order and does not have any value. Eliminating u in equation ( $\begin{eqnarray}\begin{array}{rcl}f & = & \upsilon +\epsilon \,{c}_{1}\frac{{e}_{r}}{{e}_{r}^{2}\,-\,1}\sin \upsilon \\ & & +{\epsilon }^{2}\,\left[\left({c}_{2}\,-\,{c}_{1}^{2}\frac{{e}_{r}^{2}}{{e}_{r}^{2}\,-\,1}\right)\frac{{e}_{r}}{{e}_{r}^{2}\,-\,1}\sin \upsilon \right.\\ & & \left.+\frac{{c}_{1}^{2}}{4}\frac{{e}_{r}^{2}}{{({e}_{r}^{2}\,-\,1)}^{2}}\sin 2\upsilon \right].\end{array}\end{eqnarray}$
Substituting this result into equation ( $\begin{eqnarray}{c}_{1}=-2{ \mathcal E },\,\end{eqnarray}$
$\begin{eqnarray}{c}_{2}=-18{ \mathcal E }\frac{{m}^{2}}{{{ \mathcal J }}^{2}}\left(1\,-\,\frac{11}{18}\frac{{q}^{2}}{{m}^{2}}\right),\end{eqnarray}$
which leads to $\begin{eqnarray}{e}_{\phi }={e}_{r}\left[1\,-\,2{ \mathcal E }\,-\,18\frac{{m}^{2}{ \mathcal E }}{{{ \mathcal J }}^{2}}\left(1\,-\,\frac{11}{18}\frac{{q}^{2}}{{m}^{2}}\right)\right],\end{eqnarray}$
$\begin{eqnarray}\phi \left(\frac{2\pi }{{\rm{\Phi }}}\right)=\upsilon \,+\,\frac{1}{8}\frac{{m}^{4}}{{{ \mathcal J }}^{4}}\left(1\,-\,\frac{{q}^{2}}{{m}^{2}}\right){e}_{r}^{2}\sin 2\upsilon .\end{eqnarray}$
With the true anomaly $\upsilon$, we can re-express the time dependence of the quasi-Keplerian motion equation (47 ) in the form of
$\begin{eqnarray}\begin{array}{rcl}t\left(\frac{2\pi }{{{\rm{T}}}_{u}}\right) & = & u-{e}_{t}\sin u\\ & & +\frac{30m\,{{ \mathcal E }}^{2}}{\sqrt{-2\,{ \mathcal E }{{ \mathcal J }}^{2}}}\left(1\,-\,\frac{{q}^{2}}{5{m}^{2}}\right)(\upsilon -u).\end{array}\end{eqnarray}$
Notice that all the formulas in this work are valid up to the 2PN accuracy.
The orbital period and perihelion precession are two important quantities in astronomical observations, regarded as a powerful tool for testing gravitational theories [88, 89, 92, 93, 97–101]. Notably, the electric charge parameter q significantly influences these quantities at the 1PN and 2PN order. This effect allows us to distinguish the ABG black hole from the Schwarzschild black hole using periastron advance and orbital period at this post-Newtonian order in our research.58 ) and (59 ) reduced to the 2PN perihelion precession and orbital period in Schwarzschild spacetime [43, 44]. Thus, equations (58 ) and (59 ) provide a potential method to distinguish ABG black holes from others.
$\begin{eqnarray}\begin{array}{rcl}{\rm{\Delta }}\phi & \equiv & {\rm{\Phi }}-2\pi =6\pi \frac{{m}^{2}}{{{ \mathcal J }}^{2}}\left(1\,-\,\frac{{q}^{2}}{6{m}^{2}}\right)\\ & & +15\frac{\pi {m}^{2}{ \mathcal E }}{{{ \mathcal J }}^{2}}\left(1\,-\,\frac{{q}^{2}}{5{m}^{2}}\right)\\ & & +\frac{105}{2}\frac{\pi {m}^{4}}{{{ \mathcal J }}^{4}}\left(1\,-\,\frac{3}{5}\frac{{q}^{2}}{{m}^{2}}\,+\,\frac{1}{70}\frac{{q}^{4}}{{m}^{4}}\right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{{\rm{T}}}_{u} & = & \frac{2\pi m}{{(-2{ \mathcal E })}^{\frac{3}{2}}}\left[1\,-\,\frac{15}{4}{ \mathcal E }\,-\,\frac{105}{32}{{ \mathcal E }}^{2}\right.\\ & & \left.+\frac{30m{{ \mathcal E }}^{2}}{\sqrt{-2{ \mathcal E }{{ \mathcal J }}^{2}}}\left(1\,-\,\frac{{q}^{2}}{5{m}^{2}}\right)\right].\end{array}\end{eqnarray}$
When the electric charge vanishes (q = 0), equations (5. Summary and discussion
We start with the 2PN metric of the regular ABG black hole in the standard coordinates, calculate the corresponding orbital energy and angular momentum of the test particle in the equatorial plane, and then through an iterative method and function fitting method, derive the 2PN solution for the quasi-Keplerian motion in the ABG spacetime. We obtain two slightly different but equivalent formulations in the 2PN approximation. The results are summarized as follows.
The first formulation can be expressed as
$\begin{eqnarray}{\boldsymbol{x}}=r(\cos \phi \,{{\boldsymbol{e}}}_{x}+\sin \phi \,{{\boldsymbol{e}}}_{y}),\end{eqnarray}$
$\begin{eqnarray}r={a}_{r}(1-{e}_{r}\cos u),\,\end{eqnarray}$
$\begin{eqnarray}\phi \left(\frac{2\pi }{{\rm{\Phi }}}\right)=f+{N}_{0}\sin f+{N}_{1}\sin 2f,\,\end{eqnarray}$
$\begin{eqnarray}f=2\arctan \left(\sqrt{\frac{1+{e}_{r}}{1-{e}_{r}}}\tan \frac{u}{2}\right),\,\end{eqnarray}$
$\begin{eqnarray}t\left(\frac{2\pi }{{{\rm{T}}}_{u}}\right)=u-{e}_{t}\sin u+{N}_{2}(f-u),\,\end{eqnarray}$
and the second formulation can be expressed as $\begin{eqnarray}{\boldsymbol{x}}=r(\cos \phi \,{{\boldsymbol{e}}}_{x}+\sin \phi \,{{\boldsymbol{e}}}_{y}),\end{eqnarray}$
$\begin{eqnarray}r={a}_{r}(1-{e}_{r}\cos u),\,\end{eqnarray}$
$\begin{eqnarray}\phi \left(\frac{2\pi }{{\rm{\Phi }}}\right)=\upsilon +{N}_{3}\sin 2\upsilon ,\,\end{eqnarray}$
$\begin{eqnarray}\upsilon =2\arctan \left(\sqrt{\frac{1+{e}_{\phi }}{1-{e}_{\phi }}}\tan \frac{u}{2}\right),\,\end{eqnarray}$
$\begin{eqnarray}t\left(\frac{2\pi }{{{\rm{T}}}_{u}}\right)=u-{e}_{t}\sin u+{N}_{2}(\upsilon -u),\end{eqnarray}$
where $\begin{eqnarray}{a}_{r}=\frac{m}{-2{ \mathcal E }}\left[1\,+\,\frac{3}{2}{ \mathcal E }+\frac{1}{4}{{ \mathcal E }}^{2}\,+\,8\frac{{m}^{2}{ \mathcal E }}{{{ \mathcal J }}^{2}}\left(1\,-\,\frac{7}{8}\frac{{q}^{2}}{{m}^{2}}\right)\right],\,\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{e}_{r}^{2}=1\,+\,\frac{2{ \mathcal E }{{ \mathcal J }}^{2}}{{m}^{2}}\,-\,{ \mathcal E }\left[8\left(1\,-\,\frac{{q}^{2}}{4{m}^{2}}\right)\,+\,7\frac{{ \mathcal E }{{ \mathcal J }}^{2}}{{m}^{2}}\right]\\ \,+\,{{ \mathcal E }}^{2}\left[16\frac{{ \mathcal E }{{ \mathcal J }}^{2}}{{m}^{2}}\,-\,20\left(1\,-\,\frac{5}{4}\frac{{q}^{2}}{{m}^{2}}\right)\,-\,32\frac{{m}^{2}}{{ \mathcal E }{{ \mathcal J }}^{2}}\left(1\,-\,\frac{7}{8}\frac{{q}^{2}}{{m}^{2}}\right)\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{e}_{t} & = & {e}_{r}\left[1\,+\,6\,{ \mathcal E }\,+\,27\,{{ \mathcal E }}^{2}\,-\,\frac{30m{{ \mathcal E }}^{2}}{\sqrt{-2{ \mathcal E }{{ \mathcal J }}^{2}}}\right.\\ & & \times \left.\left(1\,-\,\frac{{q}^{2}}{5{m}^{2}}\right)\,+\,8\frac{{m}^{2}{ \mathcal E }}{{{ \mathcal J }}^{2}}\left(1\,-\,\frac{7}{8}\frac{{q}^{2}}{{m}^{2}}\right)\right],\,\end{array}\end{eqnarray}$
$\begin{eqnarray}{e}_{\phi }={e}_{r}\left[1\,-\,2{ \mathcal E }\,-\,18\frac{{m}^{2}{ \mathcal E }}{{{ \mathcal J }}^{2}}\left(1\,-\,\frac{11}{18}\frac{{q}^{2}}{{m}^{2}}\right)\right],\,\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{\rm{\Phi }} & = & 2\pi \left[1\,+\,3\frac{{m}^{2}}{{{ \mathcal J }}^{2}}\left(1\,-\,\frac{{q}^{2}}{6{m}^{2}}\right)\right.\\ & & +\frac{15}{2}\frac{{m}^{2}{ \mathcal E }}{{{ \mathcal J }}^{2}}\left(1\,-\,\frac{{q}^{2}}{5{m}^{2}}\right)\,+\,\frac{105}{4}\frac{{m}^{4}}{{{ \mathcal J }}^{4}}\\ & & \times \left.\left(1\,-\,\frac{3}{5}\frac{{q}^{2}}{{m}^{2}}\,+\,\frac{1}{70}\frac{{q}^{4}}{{m}^{4}}\right)\right],\,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{N}_{0} & = & \frac{{m}^{2}}{{{ \mathcal J }}^{2}}\left[1\,+\,2{ \mathcal E }\,+\,10\frac{{m}^{2}}{{{ \mathcal J }}^{2}}\left(1\,-\,\frac{11}{20}\frac{{q}^{2}}{{m}^{2}}\right)\right]\\ & & \times {\left(1\,+\,\frac{2{ \mathcal E }{{ \mathcal J }}^{2}}{{m}^{2}}\right)}^{\,\frac{1}{2}},\,\end{array}\end{eqnarray}$
$\begin{eqnarray}{N}_{1}=\frac{3}{8}\frac{{m}^{4}}{{{ \mathcal J }}^{4}}\left(1\,-\,\frac{{q}^{2}}{3{m}^{2}}\right)\left(1+\frac{2{ \mathcal E }{{ \mathcal J }}^{2}}{{m}^{2}}\right),\,\end{eqnarray}$
$\begin{eqnarray}{N}_{2}=\frac{30m\,{{ \mathcal E }}^{2}}{\sqrt{-2\,{ \mathcal E }{{ \mathcal J }}^{2}}}\left(1\,-\,\frac{{q}^{2}}{5{m}^{2}}\right),\,\end{eqnarray}$
$\begin{eqnarray}{N}_{3}=\frac{1}{8}\frac{{m}^{4}}{{{ \mathcal J }}^{4}}\left(1\,-\,\frac{{q}^{2}}{{m}^{2}}\right)\left(1+\frac{2{ \mathcal E }{{ \mathcal J }}^{2}}{{m}^{2}}\right),\,\end{eqnarray}$
$\begin{eqnarray}{{\rm{T}}}_{u}=\frac{2\pi m}{{(-2{ \mathcal E })}^{\frac{3}{2}}}\left[1\,-\,\frac{15}{4}{ \mathcal E }\,-\,\frac{105}{32}{{ \mathcal E }}^{2}\,+\,\frac{30m\,{{ \mathcal E }}^{2}}{\sqrt{-2{ \mathcal E }{{ \mathcal J }}^{2}}}\left(1\,-\,\frac{{q}^{2}}{5{m}^{2}}\right)\right].\end{eqnarray}$
In the formulations, ar, er and u denote the semi-major axis, the eccentricity, the eccentric anomaly of the quasi-Keplerian motion in the PN approximation, respectively. f and $\upsilon$ are two slightly different definitions of the true anomaly. Tu denotes the orbital period, which is independent on the spin up to the 2PN order. The perihelion precession is obtained by Δφ ≡ Φ − 2π. Notice that the effects of the electric charge of the ABG black hole are characterized by the terms containing q2 in the above formulas.The formulas are expressed in terms of the orbital energy and angular momentum, which have direct physical meaning. The achieved solutions can be used in fitting the motion of the test particle as well as the relative motion of the extreme-mass-ratio inspirals under various gravitation theories. The precise monitoring of stellar orbits, such as the S-stars cluster around Sgr A*, by next-generation instruments like the GRAVITY+ instrument on the VLTI or the Thirty Meter Telescope (TMT), is expected to achieve unprecedented astrometric precision.