1. Introduction
The detection of gravitational waves [1,2] and the direct imaging of supermassive black holes at the centers of both galaxy M87 [3] and galaxy Sgr A* [4] underscore the widespread presence of black holes in the Universe and affirm the validity of General Relativity (GR). According to the predictions of general relativity [5], black holes, such as Schwarzschild, Reissner-Nordström and Kerr black holes, feature two distinct types of singularities: a coordinate singularity known as the event horizon and an essential singularity located at r=0. The essential singularity causes curvature invariants to diverge and renders geodesics incomplete [6,7]. The event horizon and central singularity of black holes introduce specific challenges, including issues of information loss and the limitations of GR. Black holes characterized by the presence of an event horizon but the absence of an essential singularity are referred to as regular black holes, where curvature invariants remain finite throughout. To circumvent spacetime singularities, various approaches have been proposed, including quantum pressure-induced bouncing [8-10], considering a regular core [11-19], and constructing quasi-black holes [20-23].
The first solution for the regular black hole that features an event horizon but without an essential singularity was introduced by Bardeen [11]. Subsequently, Ayón-Beato and García [12] demonstrated that the Bardeen black hole serves an exact solution of GR in conjunction with nonlinear electrodynamics, providing a modification of Reissner-Nordström black hole solution. Since then, numerous other regular black hole solutions have been proposed, including the Hayward black hole [13], and the Simpson-Visser black hole [14] among others. Most of these solutions extend Bardeen's original concept, utilizing nonlinear electrodynamics as the source. Notably, most regular black holes exhibit a core that is asymptotically de Sitter [11-14], characterized by a constant positive curvature. An exception is the regular black hole with an asymptotically Minkowski core [18], which considerably simplifies the physics in the deep core region. Various properties of these regular black holes have been extensively studied, including thermodynamic stability [24-28], gravitational lensing [29-33], shadows [34-36], quasinormal modes [36-38], bound orbits [39,40], and the structure of timelike and null geodesics [41-44].
The powerful gravitational field of black holes significantly influences their surrounding environment. A key aspect of this influence is the motion of test particles near black holes. Understanding the dynamics of these particles is essential, as it not only elucidates the nature of black hole spacetimes, but also serves as a critical test of general relativity. The post-Newtonian (PN) approximations are widely utilized to calculate the motion of objects in the analysis of strong gravitational fields, leading to the development of various analytical solutions that incorporate different PN effects, such as those related to mass [45-54] and spin [55-63]. The 2PN precessing motion around the regular Bardeen black hole has been studied [64]. Additionally, we have obtained the 3PN solution for the quasi-Keplerian motion of a test particle around the regular black hole with an asymptotically Minkowski core [65]. On the other hand, the precessing motion have been proved to be an effective tool for testing alternative gravity theories by the planets orbiting the Sun [66-74], by the exoplanets around other stars [75-79], by the binary pulsars [80-88], and by the stars orbiting Sgr A* [89-98].
In this work, we derive the trajectories of test particles in the regular Simpson-Visser spacetime [14], which extends the black-bounce-Schwarzschild solution by incorporating charge. This charged extension is interpreted as a manifestation of standard Maxwell electromagnetism coupled with an anisotropic fluid. Although astrophysical objects are generally considered to be electrically neutral due to rapid charge neutralization by surrounding plasma, several mechanisms may allow a black hole to acquire charge. These include the accumulation of charged matter, induction from rotation in an external magnetic field [99], or the inheritance of charge from a collapsing progenitor [100]. In this work, we focus on bound orbits around the Simpson-Visser black-bounce spacetime, with particular attention to precessing and periodic motions. Our analysis centers on the impact of the bounce parameter on these orbits, especially the precession of the orbital periastron. Then, we will apply this model to the precessing motion of OJ 287 and determine the upper limits of the dimensionless bounce parameter as a/m=3.45 ± 0.01, where m represents the black hole's mass. Our analysis reveals that, compared with the bound given by the periastron advance of star S2 [97], our bound on a/m is reduced by one order of magnitude. This research aims to deepen our understanding of test particle dynamics in strong gravitational fields and provide potential observational signatures for probing such exotic spacetimes.
The structure of this study is organized as follows. In section 2 , we introduce the metric and action of the Simpson-Visser spacetime. In section 3 , we provide a detailed derivation of the 2PN solution for the quasi-Keplerian motion of the test particle and derives its advance of periastron, including the effects of the bounce parameter a. In section 4 , we estimate the upper bounds for a based on the detection of the precession of OJ 287. Finally, we conclude with a summary and discussion in section 5 .
2. The metric and action of a Simpson-Visser black hole
The spacetime of a Simpson-Visser black hole is described by the following expression [14]
$\begin{eqnarray}\begin{array}{rcl}{\rm{d}}{s}^{2} & =& -\left(\right.1-\frac{2m}{\sqrt{{r}^{2}+{a}^{2}}}\left)\right.{\rm{d}}{t}^{2}+\left(\right.1-\frac{2m}{\sqrt{{r}^{2}+{a}^{2}}}{\left)\right.}^{-1}\\ & & \times {\rm{d}}{r}^{2}+({r}^{2}+{a}^{2})({\rm{d}}{\theta }^{2}+\sin {\theta }^{2}{\rm{d}}{\phi }^{2}),\,\end{array}\end{eqnarray}$
where, a is the bounce parameter. By adjusting parameter a, assuming without loss of generality that a≥0, and following the analysis in [14], the metric can represent the following scenarios:(i) The Schwarzschild spacetime when a=0;
(ii) A regular black hole with a one-way spacetime throat for 0 < a < 2m;
(iii) A one-way wormhole with a null throat when a=2m, as discussed in [101];
In [111], it was shown that the metric in equation (1 ) arises as an exact solution to Einstein's field equations when coupled minimally to a self-interacting phantom scalar field φ(x), and supplemented by a nonlinear electrodynamics field characterized by the tensor Fμν. The corresponding action is provided in [111]2 ) is given as follows 1 ) represents an exact solution to the Einstein equations, corresponding to the action described in equation (2 ).
$\begin{eqnarray}S=\int \sqrt{-g}\,{{\rm{d}}}^{4}r\left[\right.R+2\epsilon \,{g}^{\mu \nu }{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi -{ \mathcal L }(F)-2V(\varphi )\left]\right.,\,\end{eqnarray}$
with ε=± 1, where ${ \mathcal L }(F)$ denotes the gauge-invariant Lagrangian density with F=FμνFμν. The Einstein field equations obtained from the equation ( $\begin{eqnarray}{G}_{\mu }^{\nu }=-{T}_{\mu }^{\nu }[F]-{T}_{\mu }^{\nu }[\varphi ],\end{eqnarray}$
where ${T}_{\mu }^{\nu }[F]$ and ${T}_{\mu }^{\nu }[\phi ]$ are the stress-energy tensors of the electromagnetic and scalar fields: $\begin{eqnarray}{T}_{\mu }^{\nu }[F]=-2\frac{{\rm{d}}{ \mathcal L }}{{\rm{d}}F}{F}_{\mu \sigma }{F}^{\nu \sigma }+\frac{1}{2}{\delta }_{\mu }^{\nu }{ \mathcal L }(F),\end{eqnarray}$
$\begin{eqnarray}{T}_{\mu }^{\nu }[\varphi ]=2\epsilon \,{\partial }_{\mu }\varphi {\partial }^{\nu }\varphi -{\delta }_{\mu }^{\nu }[\epsilon \,{g}^{\rho \sigma }{\partial }_{\rho }\varphi {\partial }_{\sigma }\varphi -V(\varphi )].\end{eqnarray}$
The forms of ${ \mathcal L }(F)$, φ and V(φ) are derived and presented in [111], where it is explicitly demonstrated that equation (3. 2PN post-Newtonian motion in Simpson-Visser spacetime
By transforming equation (1 ) from spherical coordinates into Cartesian coordinates using
$\begin{eqnarray}\begin{array}{rcl}{x}^{0} & =& t,\,\,\,{x}^{1}=r\sin \theta \cos \phi ,\\ {x}^{2} & =& r\sin \theta \sin \phi ,\,\,\,{x}^{3}=r\cos \theta ,\end{array}\end{eqnarray}$
we obtain $\begin{eqnarray}\begin{array}{rcl}{\rm{d}}{s}^{2} & =& -\left(\right.1-\frac{2m}{\sqrt{{r}^{2}+{a}^{2}}}\left)\right.{\rm{d}}{t}^{2}+\left[\right.\left(\right.1-\frac{2m}{\sqrt{{r}^{2}+{a}^{2}}}{\left)\right.}^{-1}-\left(\right.1+\frac{{a}^{2}}{{r}^{2}}\left)\right.\left]\right.\\ & & \times \left(\right.\frac{{\boldsymbol{x}}\cdot {\rm{d}}{\boldsymbol{x}}}{r}{\left)\right.}^{2}+\left(\right.1+\frac{{a}^{2}}{{r}^{2}}\left)\right.{\rm{d}}{{\boldsymbol{x}}}^{2},\end{array}\end{eqnarray}$
where x=(x0, x1, x2, x3).By applying the PN approximation and expanding the metric in powers of m/r, where r=∣x∣, the 2PN metric can be obtained as follows
$\begin{eqnarray}{g}_{00}=-1+\frac{2m}{r}-\frac{m{a}^{2}}{{r}^{3}},\end{eqnarray}$
$\begin{eqnarray}{g}_{0i}=0,\end{eqnarray}$
$\begin{eqnarray}{g}_{ij}=\left(1+\frac{{a}^{2}}{{r}^{2}}\right){\delta }_{ij}+\left[\frac{2m}{r}+\frac{4{m}^{2}}{{r}^{2}}\left(1-\frac{{a}^{2}}{4{m}^{2}}\right)\right]\frac{{x}^{i}{x}^{j}}{{r}^{2}},\end{eqnarray}$
where the indices i and j range from 1 to 3.Then, we can calculate the Lagrangian of the test particle as
$\begin{eqnarray}\begin{array}{rcl}{ \mathcal L } & =& \frac{1}{2}{{\boldsymbol{v}}}^{2}+\frac{m}{r}+\frac{1}{8}{{\boldsymbol{v}}}^{4}+\frac{1}{2}\frac{m}{r}{{\boldsymbol{v}}}^{2}+\frac{1}{2}\frac{{m}^{2}}{{r}^{2}}\\ & & +\frac{m{({\boldsymbol{v}}\cdot {\boldsymbol{x}})}^{2}}{{r}^{3}}+\frac{1}{2}\frac{{m}^{3}}{{r}^{3}}\left(1-\frac{{a}^{2}}{{m}^{2}}\right)+\frac{1}{16}{{\boldsymbol{v}}}^{6}\\ & & +\frac{3}{4}\frac{{m}^{2}}{{r}^{2}}{{\boldsymbol{v}}}^{2}\left(1+\frac{2}{3}\frac{{a}^{2}}{{m}^{2}}\right)\\ & & +\frac{3{m}^{2}{({\boldsymbol{v}}\cdot {\boldsymbol{x}})}^{2}}{{r}^{4}}\left(1-\frac{{a}^{2}}{6{m}^{2}}\right)+\frac{3}{8}\frac{m}{r}{{\boldsymbol{v}}}^{4}+\frac{m{({\boldsymbol{v}}\cdot {\boldsymbol{x}})}^{2}{{\boldsymbol{v}}}^{2}}{2{r}^{3}},\end{array}\end{eqnarray}$
where v is the velocity for the test particle. Then we can obtain the energy ${ \mathcal E }$ and the angular momentum ${ \mathcal J }$ as $\begin{eqnarray}\begin{array}{rcl}{ \mathcal E } & =& \frac{1}{2}{{\boldsymbol{v}}}^{2}-\frac{m}{r}+\frac{3}{8}{{\boldsymbol{v}}}^{4}+\frac{1}{2}\frac{m}{r}{{\boldsymbol{v}}}^{2}-\frac{1}{2}\frac{{m}^{2}}{{r}^{2}}\\ & & +\frac{m{({\boldsymbol{v}}\cdot {\boldsymbol{x}})}^{2}}{{r}^{3}}-\frac{1}{2}\frac{{m}^{3}}{{r}^{3}}\left(1-\frac{{a}^{2}}{{m}^{2}}\right)+\frac{5}{16}{{\boldsymbol{v}}}^{6}\\ & & +\frac{3}{4}\frac{{m}^{2}}{{r}^{2}}{{\boldsymbol{v}}}^{2}\left(1+\frac{2}{3}\frac{{a}^{2}}{{m}^{2}}\right)+\frac{9}{8}\frac{m}{r}{{\boldsymbol{v}}}^{4}\\ & & +\frac{3{m}^{2}{({\boldsymbol{v}}\cdot {\boldsymbol{x}})}^{2}}{{r}^{4}}\left(1-\frac{{a}^{2}}{6{m}^{2}}\right)+\frac{3}{2}\frac{m{({\boldsymbol{v}}\cdot {\boldsymbol{x}})}^{2}{{\boldsymbol{v}}}^{2}}{{r}^{3}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{ \mathcal J } & =& | {\boldsymbol{x}}\times {\boldsymbol{v}}| \left[1+\frac{1}{2}{{\boldsymbol{v}}}^{2}+\frac{m}{r}+\frac{3}{8}{{\boldsymbol{v}}}^{4}+\frac{3}{2}\frac{{m}^{2}}{{r}^{2}}\right.\\ & & \left.\times \left(1+\frac{2}{3}\frac{{a}^{2}}{{m}^{2}}\right)+\frac{3}{2}\frac{m}{r}{{\boldsymbol{v}}}^{2}+\frac{m{({\boldsymbol{v}}\cdot {\boldsymbol{x}})}^{2}}{{r}^{3}}\right].\end{array}\end{eqnarray}$
Next, we will derive the post-Newtonian motion following the same approach as Brumberg [45]. Owing to spherical symmetry, the motion can be confined to the equatorial plane. The origin is placed at the position of the regular Simpson-Visser black hole. The particle's trajectory in the gravitational field is given by 12 ) and (13 ) as 25 ) is a fourth-order polynomial in r-1, it can be further reformulated as 25 ) and (26 ), we can obtain 26 ) shows that r+=ar(1+er) represents the aphelion of the orbit, and r-=ar(1-er) is the perihelion of the orbit. Hence, ar and er can be regarded as the semi-major axis and eccentricity.
$\begin{eqnarray}{\boldsymbol{x}}=r(\cos \phi \,{{\boldsymbol{e}}}_{x}+\sin \phi \,{{\boldsymbol{e}}}_{y}),\end{eqnarray}$
By rewriting ${ \mathcal E }$ and ${ \mathcal J }$ from equations ( $\begin{eqnarray}\begin{array}{rcl}{ \mathcal E } & =& \frac{1}{2}({\dot{r}}^{2}+{r}^{2}{\dot{\phi }}^{2})-\frac{m}{r}+\frac{3}{8}{({\dot{r}}^{2}+{r}^{2}{\dot{\phi }}^{2})}^{2}\\ & & +\frac{1}{2}\frac{m}{r}({\dot{r}}^{2}+{r}^{2}{\dot{\phi }}^{2})-\frac{{m}^{2}}{2{r}^{2}}+\frac{m}{r}{\dot{r}}^{2}\\ & & -\frac{{m}^{3}}{2{r}^{3}}\left(\right.1-\frac{{a}^{2}}{{m}^{2}}\left)\right.+\frac{5}{16}{({\dot{r}}^{2}+{r}^{2}{\dot{\phi }}^{2})}^{3}\\ & & +\frac{3}{4}\frac{{m}^{2}}{{r}^{2}}({\dot{r}}^{2}+{r}^{2}{\dot{\phi }}^{2})\left(\right.1+\frac{2}{3}\frac{{a}^{2}}{{m}^{2}}\left)\right.\\ & & +\frac{9}{8}\frac{m}{r}{({\dot{r}}^{2}+{r}^{2}{\dot{\phi }}^{2})}^{2}+\frac{3{m}^{2}}{{r}^{2}}{\dot{r}}^{2}\\ & & \times \left(\right.1-\frac{{a}^{2}}{6{m}^{2}}\left)\right.+\frac{3}{2}\frac{m}{r}{\dot{r}}^{2}({\dot{r}}^{2}+{r}^{2}{\dot{\phi }}^{2}),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{ \mathcal J } & =& {r}^{2}\dot{\phi }\left[\right.1+\frac{1}{2}({\dot{r}}^{2}+{r}^{2}{\dot{\phi }}^{2})+\frac{m}{r}+\frac{3}{8}{({\dot{r}}^{2}+{r}^{2}{\dot{\phi }}^{2})}^{2}\\ & & +\frac{3}{2}\frac{m}{r}({\dot{r}}^{2}+{r}^{2}{\dot{\phi }}^{2})+\frac{3}{2}\frac{{m}^{2}}{{r}^{2}}\left(\right.1+\frac{2}{3}\frac{{a}^{2}}{{m}^{2}}\left)\right.+\frac{m}{r}{\dot{r}}^{2}\left]\right.,\end{array}\end{eqnarray}$
where ‘ · ' indicates differentiation with respect to the coordinate time. From these two equations, we can obtain $\begin{eqnarray}\begin{array}{rcl}{r}^{4}{\dot{\phi }}^{2} & =& {{ \mathcal J }}^{2}\left[1-2\,{ \mathcal E }-\frac{4m}{r}+3\,{{ \mathcal E }}^{2}+\frac{4{m}^{2}}{{r}^{2}}\left(1-\frac{{a}^{2}}{2{m}^{2}}\right)\right.\\ & & \left.+8\,{ \mathcal E }\frac{m}{r}\right],\end{array}\end{eqnarray}$
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 $\begin{eqnarray}A=2{ \mathcal E }\left(1-\frac{3}{2}\,{ \mathcal E }+2\,{{ \mathcal E }}^{2}\right),\end{eqnarray}$
$\begin{eqnarray}B=2m\left(\right.1-6\,{ \mathcal E }+9\,{{ \mathcal E }}^{2}\left)\right.,\end{eqnarray}$
$\begin{eqnarray}C=-{{ \mathcal J }}^{2}\left(\right.1-2\,{ \mathcal E }+\frac{8{m}^{2}}{{{ \mathcal J }}^{2}}-24\frac{{m}^{2}{ \mathcal E }}{{{ \mathcal J }}^{2}}+3\,{{ \mathcal E }}^{2}\left)\right.,\end{eqnarray}$
$\begin{eqnarray}D=6m{{ \mathcal J }}^{2}\left[1-2\,{ \mathcal E }+\frac{4}{3}\frac{{m}^{2}}{{{ \mathcal J }}^{2}}\left(1-\frac{{a}^{2}}{8{m}^{2}}\right)\right],\end{eqnarray}$
$\begin{eqnarray}E=-12{m}^{2}{{ \mathcal J }}^{2}\left(1-\frac{{a}^{2}}{12{m}^{2}}\right).\end{eqnarray}$
Following the relation $\begin{eqnarray}{\dot{r}}^{2}=({r}^{4}{\dot{\phi }}^{2}){\left[\frac{{\rm{d}}(1/r)}{{\rm{d}}\phi }\right]}^{2},\end{eqnarray}$
we can obtain the radial equation as $\begin{eqnarray}\begin{array}{rcl}{\left[\frac{{\rm{d}}(1/r)}{{\rm{d}}\phi }\right]}^{2} & =& \frac{2{ \mathcal E }}{{{ \mathcal J }}^{2}}\left(1+\frac{1}{2}\,{ \mathcal E }\right)+\frac{2m}{r{{ \mathcal J }}^{2}}\\ & & -\frac{1}{{r}^{2}}\left(1-4{ \mathcal E }\frac{{a}^{2}}{{{ \mathcal J }}^{2}}\right)+\frac{2m}{{r}^{3}}\left(1+\frac{3}{2}\frac{{a}^{2}}{{{ \mathcal J }}^{2}}\right)-\frac{{a}^{2}}{{r}^{4}}.\end{array}\end{eqnarray}$
Given that the right-hand side of equation ( $\begin{eqnarray}\begin{array}{rcl}{\left[\frac{{\rm{d}}(1/r)}{{\rm{d}}\phi }\right]}^{2} & =& \left[\frac{1}{r}-\frac{1}{{a}_{r}(1+{e}_{r})}\right]\\ & & \times \left[\frac{1}{{a}_{r}(1-{e}_{r})}-\frac{1}{r}\right]\left({C}_{1}+\frac{{C}_{2}}{r}+\frac{{C}_{3}}{{r}^{2}}\right).\end{array}\end{eqnarray}$
Comparing equations ( $\begin{eqnarray}{a}_{r}=\frac{m}{-2{ \mathcal E }}\left[1+\frac{3}{2}{ \mathcal E }+\frac{1}{4}{{ \mathcal E }}^{2}+8\frac{{m}^{2}{ \mathcal E }}{{{ \mathcal J }}^{2}}\left(1+\frac{{a}^{2}}{8{m}^{2}}\right)\right],\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{e}_{r}^{2} & =& 1+\frac{2{ \mathcal E }{{ \mathcal J }}^{2}}{{m}^{2}}-{ \mathcal E }\left(8+7\frac{{ \mathcal E }{{ \mathcal J }}^{2}}{{m}^{2}}\right)\\ & & +\,{{ \mathcal E }}^{2}\left[16\frac{{ \mathcal E }{{ \mathcal J }}^{2}}{{m}^{2}}-20\left(1+\frac{2}{5}\frac{{a}^{2}}{{m}^{2}}\right)-32\frac{{m}^{2}}{{ \mathcal E }{{ \mathcal J }}^{2}}\left(1+\frac{{a}^{2}}{8{m}^{2}}\right)\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}{C}_{1}=1-\frac{4{m}^{2}}{{{ \mathcal J }}^{2}}-8\frac{{m}^{2}{ \mathcal E }}{{{ \mathcal J }}^{2}}\left(1+\frac{{a}^{2}}{4{m}^{2}}\right)-16\frac{{m}^{4}}{{{ \mathcal J }}^{4}}\left(1+\frac{{a}^{2}}{8{m}^{2}}\right),\end{eqnarray}$
$\begin{eqnarray}{C}_{2}=-2m\left(\right.1+\frac{{a}^{2}}{2{{ \mathcal J }}^{2}}\left)\right.,\end{eqnarray}$
$\begin{eqnarray}{C}_{3}={a}^{2}.\end{eqnarray}$
Equation (The solution to equation (26 ) can be expressed as 29 )-(32 ) into equation (33 ), we have 34 ), we arrive at
$\begin{eqnarray}r=\frac{{a}_{r}(1-{e}_{r}^{2})}{1+{e}_{r}\cos f},\end{eqnarray}$
where f represents the true anomaly, which satisfies $\begin{eqnarray}\left(\right.\frac{{\rm{d}}f}{{\rm{d}}\phi }{\left)\right.}^{2}={C}_{1}+\frac{{C}_{2}}{r}+\frac{{C}_{3}}{{r}^{2}}.\end{eqnarray}$
Substituting equations ( $\begin{eqnarray}\begin{array}{rcl}\frac{{\rm{d}}f}{{\rm{d}}\phi } & =& \left[\right.1-\frac{3{m}^{2}}{{{ \mathcal J }}^{2}}-\frac{13}{2}\frac{{m}^{2}{ \mathcal E }}{{{ \mathcal J }}^{2}}\left(\right.1+\frac{{a}^{2}}{13{m}^{2}}\left)\right.-\frac{67}{4}\frac{{m}^{4}}{{{ \mathcal J }}^{4}}\left(\right.1+\frac{3}{67}\frac{{a}^{2}}{{m}^{2}}\left)\right.\left]\right.\\ & & \times \left\{\right.1-\frac{{m}^{2}}{{{ \mathcal J }}^{2}}\left[\right.1+2{ \mathcal E }+10\frac{{m}^{2}}{{{ \mathcal J }}^{2}}\left(\right.1-\frac{{a}^{2}}{20{m}^{2}}\left)\right.\left]\right.\\ & & \times {e}_{r}\cos f-\frac{{m}^{4}}{4{{ \mathcal J }}^{4}}\left(\right.1-\frac{{a}^{2}}{{m}^{2}}\left)\right.{e}_{r}^{2}\cos 2f\left\}\right..\end{array}\end{eqnarray}$
By integrating equation ( $\begin{eqnarray}\begin{array}{rcl}\phi \left(\right.\frac{2\pi }{{\rm{\Phi }}}\left)\right. & =& f+\frac{{m}^{2}}{{{ \mathcal J }}^{2}}\left[\right.1+2{ \mathcal E }+10\frac{{m}^{2}}{{{ \mathcal J }}^{2}}\left(\right.1-\frac{{a}^{2}}{20{m}^{2}}\left)\right.\left]\right.\\ & & \times {e}_{r}\sin f+\frac{3}{8}\frac{{m}^{4}}{{{ \mathcal J }}^{4}}\left(\right.1-\frac{{a}^{2}}{3{m}^{2}}\left)\right.{e}_{r}^{2}\sin 2f\end{array}\end{eqnarray}$
with $\begin{eqnarray}\begin{array}{rcl}{\rm{\Phi }} & =& 2\pi \left[\right.1+3\frac{{m}^{2}}{{{ \mathcal J }}^{2}}+\frac{15}{2}\frac{{m}^{2}{ \mathcal E }}{{{ \mathcal J }}^{2}}\left(\right.1+\frac{{a}^{2}}{15{m}^{2}}\left)\right.\\ & & +\frac{105}{4}\frac{{m}^{4}}{{{ \mathcal J }}^{4}}\left(\right.1+\frac{{a}^{2}}{35{m}^{2}}\left)\right.\left]\right..\end{array}\end{eqnarray}$
To determine the time evolution of the motion, we combine equations (17 ) and (34 ), leading to
$\begin{eqnarray}\begin{array}{rcl}{r}^{2}\dot{f} & =& { \mathcal J }\left\{\right.1-{ \mathcal E }-2\frac{m}{r}-3\frac{{m}^{2}}{{{ \mathcal J }}^{2}}+{{ \mathcal E }}^{2}+2{ \mathcal E }\frac{m}{r}\\ & & -\frac{7}{2}\frac{{m}^{2}{ \mathcal E }}{{{ \mathcal J }}^{2}}\left(\right.1+\frac{{a}^{2}}{7{m}^{2}}\left)\right.+6\frac{m}{r}\frac{{m}^{2}}{{{ \mathcal J }}^{2}}-\frac{{m}^{4}}{{{ \mathcal J }}^{4}}\left(\right.\frac{67}{4}+\frac{3}{4}\frac{{a}^{2}}{{m}^{2}}\left)\right.\\ & & -\frac{{a}^{2}}{{r}^{2}}-\frac{{m}^{2}}{{{ \mathcal J }}^{2}}\left[\right.1+{ \mathcal E }-2\frac{m}{r}+\frac{{m}^{2}}{{{ \mathcal J }}^{2}}\left(\right.7-\frac{{a}^{2}}{2{m}^{2}}\left)\right.\left]\right.\\ & & \times {e}_{r}\cos f-\frac{1}{4}\frac{{m}^{4}}{{{ \mathcal J }}^{4}}\left(\right.1-\frac{{a}^{2}}{{m}^{2}}\left)\right.{e}_{r}^{2}\cos 2f\left\}\right..\end{array}\end{eqnarray}$
Using the relations 32 ) can then be rewritten in terms of u as
$\begin{eqnarray}\begin{array}{rcl}\sin f & =& \frac{{(1-{e}_{r}^{2})}^{\frac{1}{2}}\sin u}{1-{e}_{r}\cos u};\,\,\cos f=\frac{\cos u-{e}_{r}}{1-{e}_{r}\cos u};\\ & & f=2\arctan \left(\right.\sqrt{\frac{1+{e}_{r}}{1-{e}_{r}}}\tan \frac{u}{2}\left)\right.,\end{array}\end{eqnarray}$
which leads to $\begin{eqnarray}\frac{{\rm{d}}f}{{\rm{d}}t}=\frac{{(1-{e}_{r}^{2})}^{1/2}}{1-{e}_{r}\cos u}\frac{{\rm{d}}u}{{\rm{d}}t},\end{eqnarray}$
where u is the eccentric anomaly. The orbital equation in equation ( $\begin{eqnarray}r={a}_{r}(1-{e}_{r}\cos u).\end{eqnarray}$
Integrating equation (37 ) and using equations (38 )-(40 ), we derive the 2PN solution for the quasi-Keplerian motion, as given in
$\begin{eqnarray}\begin{array}{rcl}t\left(\right.\frac{2\pi }{{{\rm{T}}}_{u}}\left)\right. & =& u-{e}_{t}\sin u\\ & & +\frac{30m\,{{ \mathcal E }}^{2}}{\sqrt{-2\,{ \mathcal E }{{ \mathcal J }}^{2}}}\left(\right.1+\frac{{a}^{2}}{15{m}^{2}}\left)\right.(f-u),\end{array}\end{eqnarray}$
with $\begin{eqnarray}\begin{array}{rcl}{{\rm{T}}}_{u} & =& \frac{2\pi m}{{(-2{ \mathcal E })}^{\frac{3}{2}}}\left[\right.1-\frac{15}{4}{ \mathcal E }-\frac{105}{32}{{ \mathcal E }}^{2}\\ & & +\frac{30m{{ \mathcal E }}^{2}}{\sqrt{-2{ \mathcal E }{{ \mathcal J }}^{2}}}\left(\right.1+\frac{{a}^{2}}{15{m}^{2}}\left)\right.\left]\right.,\end{array}\end{eqnarray}$
and $\begin{eqnarray}\begin{array}{rcl}{e}_{t} & =& {e}_{r}\left[\right.1+6\,{ \mathcal E }+27\,{{ \mathcal E }}^{2}-\frac{30m{{ \mathcal E }}^{2}}{\sqrt{-2{ \mathcal E }{{ \mathcal J }}^{2}}}\\ & & \times \left(\right.1+\frac{{a}^{2}}{15{m}^{2}}\left)\right.+8\frac{{m}^{2}{ \mathcal E }}{{{ \mathcal J }}^{2}}\left(\right.1+\frac{{a}^{2}}{8{m}^{2}}\left)\right.\left]\right.,\end{array}\end{eqnarray}$
where, et represents the time eccentricity, and Tu denotes the orbital period.It is common to use υ instead of f in the quasi-Keplerian equation, where the condition is imposed that the $\sin \upsilon $ term vanishes at each PN order in $\phi (\frac{2\pi }{{\rm{\Phi }}})$ [49,57,112]. Following the method in [112], we establish 44 ) to eliminate u from equation (38 ), we have [112]
$\begin{eqnarray}\upsilon =2\arctan \left(\right.\sqrt{\frac{1+{e}_{\phi }}{1-{e}_{\phi }}}\tan \frac{u}{2}\left)\right.,\end{eqnarray}$
where $\begin{eqnarray}{e}_{\phi }=(1+{c}_{1}\epsilon +{c}_{2}{\epsilon }^{2}){e}_{r},\end{eqnarray}$
differs from er, c1 and c2 represent 1PN and 2PN corrections, respectively. Here, ε denotes the PN order. Using equation ( $\begin{eqnarray}\begin{array}{rcl}f & =& \upsilon +\epsilon \,{c}_{1}\frac{{e}_{r}}{{e}_{r}^{2}-1}\sin \upsilon +{\epsilon }^{2}\,\\ & & \times \left[\right.\left(\right.{c}_{2}-{c}_{1}^{2}\frac{{e}_{r}^{2}}{{e}_{r}^{2}-1}\left)\right.\frac{{e}_{r}}{{e}_{r}^{2}-1}\sin \upsilon +\frac{{c}_{1}^{2}}{4}\frac{{e}_{r}^{2}}{{({e}_{r}^{2}-1)}^{2}}\sin 2\upsilon \left]\right..\end{array}\end{eqnarray}$
Substituting this result into equation (35 ) and enforcing the condition that the $\sin \upsilon $ term vanishes in $\phi (\frac{2\pi }{{\rm{\Phi }}})$, we arrive at
$\begin{eqnarray}{c}_{1}=-2{ \mathcal E },\end{eqnarray}$
$\begin{eqnarray}{c}_{2}=-18{ \mathcal E }\frac{{m}^{2}}{{{ \mathcal J }}^{2}}\left(1-\frac{{a}^{2}}{18{m}^{2}}\right),\end{eqnarray}$
leading to $\begin{eqnarray}{e}_{\phi }={e}_{r}\left[1-2{ \mathcal E }-18\frac{{m}^{2}{ \mathcal E }}{{{ \mathcal J }}^{2}}\left(1-\frac{{a}^{2}}{18{m}^{2}}\right)\right],\end{eqnarray}$
$\begin{eqnarray}\phi \left(\frac{2\pi }{{\rm{\Phi }}}\right)=\upsilon +\frac{1}{8}\frac{{m}^{4}}{{{ \mathcal J }}^{4}}\left(1-\frac{{a}^{2}}{{m}^{2}}\right){e}_{r}^{2}\sin 2\upsilon .\end{eqnarray}$
The time evolution to the motion described by equation (41 ) is reformulated in terms of υ, taking the form given by
$\begin{eqnarray}\begin{array}{rcl}t\left(\right.\frac{2\pi }{{{\rm{T}}}_{u}}\left)\right. & =& u-{e}_{t}\sin u\\ & & +\frac{30m\,{{ \mathcal E }}^{2}}{\sqrt{-2\,{ \mathcal E }{{ \mathcal J }}^{2}}}\left(\right.1+\frac{{a}^{2}}{15{m}^{2}}\left)\right.(\upsilon -u).\end{array}\end{eqnarray}$
In summary, the quasi-Keplerian motion of a test particle in Simpson-Visser black-bounce spacetime can be formulated by equations (14 ), (40 ), (44 ), (50 ) and (51 ).
Specifically, the periastron advance and orbital period are crucial astronomical quantities, pivotal for testing gravitational theories [92,93,96,97,113-115]. Notably, parameter a significantly influences these quantities at the 2PN order. This effect allows us to distinguish the Simpson-Visser black hole from the Schwarzschild black hole using periastron advance and orbital period at this post-Newtonian order in our research.
$\begin{eqnarray}\begin{array}{rcl}{\rm{\Delta }}\phi \equiv {\rm{\Phi }}-2\pi & =& 6\pi \frac{{m}^{2}}{{{ \mathcal J }}^{2}}+15\frac{\pi {m}^{2}{ \mathcal E }}{{{ \mathcal J }}^{2}}\\ & & \times \left(\right.1+\frac{{a}^{2}}{15{m}^{2}}\left)\right.+\frac{105}{2}\frac{\pi {m}^{4}}{{{ \mathcal J }}^{4}}\left(\right.1+\frac{{a}^{2}}{35{m}^{2}}\left)\right.,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{{\rm{T}}}_{u} & =& \frac{2\pi m}{{(-2{ \mathcal E })}^{\frac{3}{2}}}\left[1-\frac{15}{4}{ \mathcal E }-\frac{105}{32}{{ \mathcal E }}^{2}\right.\\ & & \left.+\frac{30m{{ \mathcal E }}^{2}}{\sqrt{-2{ \mathcal E }{{ \mathcal J }}^{2}}}\left(\right.1+\frac{{a}^{2}}{15{m}^{2}}\left)\right.\right].\end{array}\end{eqnarray}$
As bounce parameter a approaches zero, equation (52 ) reduces to the 2PN periastron advance of the binary system in the extreme-mass-ratio limit, as derived in [49,116]. For practical purposes, it is convenient to express the above equation in terms of the the orbital elements, the periastron advance can then be written as follows: 54 ) provides a method to test the bounce parameter a through precessing motion.
$\begin{eqnarray}{\rm{\Delta }}\phi ={\rm{\Delta }}{\phi }_{{\rm{GR}}}+{\rm{\Delta }}{\phi }_{a},\end{eqnarray}$
where the GR contribution consists of two components: ΔΦGR=ΔΦ1PN+ΔΦ2PN [49,116], and we have $\begin{eqnarray}{\rm{\Delta }}{\phi }_{{\rm{1PN}}}=3\,\xi {(1-{e}_{t}^{2})}^{-1},\end{eqnarray}$
$\begin{eqnarray}{\rm{\Delta }}{\phi }_{{\rm{2PN}}}={\xi }^{2}{(1-{e}_{t}^{2})}^{-2}(78+51{e}_{t}^{2})/4,\end{eqnarray}$
$\begin{eqnarray}{\rm{\Delta }}{\phi }_{a}={a}^{2}\,{\xi }^{2}\,{(1-{e}_{t}^{2})}^{-2}{(G\,m)}^{-2}{c}^{4}(2+{e}_{t}^{2})/4,\end{eqnarray}$
where $\xi ={(G\,m\,n/{c}^{3})}^{2/3}$ and m=m1+m2. We reintroduce the speed of light c and the gravitational constant G in the above equations for later convenience. The result from equation (4. Application to the precession of OJ 287 and S2
Blazar OJ 287 is among the first candidates believed to host a compact system of two supermassive black holes at its center, making it a promising precision testing ground for GR and alternate gravity theories. By analyzing the accurately extracted (observed) starting epochs of ten optical outbursts of OJ 287 to track the binary orbit, the independent and dependent parameters of the binary black hole system in OJ 287 have been determined [117].
Including the leading terms associated with the bounce parameter, the precession of OJ 287 can be expressed as follows: 55 )-(57 ). Using the independent and dependent parameters of the binary black hole system in OJ 287 from table 2 of [117], including the precession rate of the major axis per period ΔΦ=38.62 ± 0.01 deg, the eccentricity e=0.657, and the present (redshifted) orbital period ${P}_{{\rm{orb}}}^{2017}=12.062$ year, etc., we obtain
$\begin{eqnarray}{\rm{\Delta }}{\phi }_{{\rm{OJ287}}}={\rm{\Delta }}{\phi }_{{\rm{1PN}}}+{\rm{\Delta }}{\phi }_{{\rm{2PN}}}+{\rm{\Delta }}{\phi }_{a},\end{eqnarray}$
where the 1PN, 2PN and bounce parameter contributions to the periastron advance are given by equations ( $\begin{eqnarray}a=(9.43\pm 0.02)\times 1{0}^{13}\,{\rm{m}},\end{eqnarray}$
or $\begin{eqnarray}{a}_{\bullet }=\frac{a}{{m}_{{\rm{OJ287}}}}=3.45\pm 0.01.\end{eqnarray}$
Based on the detection of the Schwarzschild precession in the orbit of star S2 around the supermassive black hole at the Galactic Center, Sgr A*, the authors of [97] estimated the constraint on the bounce parameter as a=(3.0 ± 2.9) × 1011m, i.e., a•=47.9 ± 45.5. While our bounds on a are less competitive with the one of [97], the bound on a• from OJ 287 is reduced by one order of magnitude than it, as shown in table 1 for a comparison. It is expected that as observational accuracy and precision continue to improve, tests involving the star S2 and in OJ 287 will provide more stringent upper bounds on the the suppression parameter a.
Table 1. Upper bounds on a. |
| a (m) | a• | data | |
|---|---|---|---|
| This work | (9.43 ± 0.02) × 1013 | 3.45 ± 0.01 | OJ 287 |
| [97] | (3.0 ± 2.9) × 1011 | 47.9 ± 45.5 | S2 |
5. Summary and discussion
Inspired by the Simpson-Visser black hole, we take this model as a description of OJ 287, whose deviation from the Schwarzschild spacetime is characterized by the bounce parameter a. Based on the Simpson-Visser black hole's metric, we obtain the 2PN orbital energy and angular momentum of a test particle. Through the application of iterative and function fitting techniques, we obtain the second post-Newtonian motion in Simpson-Visser black-bounce spacetime and find its relativistic periastron advance affected by a. The results we obtained are helpful for probing the spacetime properties of regular black holes through the observation of test particles' motion in their vicinity, and the post-Newtonian solution we derived provides a framework that can be used to constrain the bounce parameter through astronomical observations.
While it is indeed straightforward to solve the geodesic equations for a test particle in the background of a spherically symmetric Simpson-Visser black hole using the conserved energy and angular momentum, the exact motion of the test particle in this spacetime is formulated in terms of elliptic integrals, which are unable to exhibit the effects of the black hole's mass and bounce parameter on the motion of the test particle explicitly. In the post-Newtonian approximations, we can achieve the analytical relations between the orbital elements and the orbital energy and angular momentum, as well as the time dependence of the test particle's position. In the latter's case, we can clearly see how the bounce parameter a influences the orbital precession motion. This study specifically focuses on bound orbits in the Simpson-Visser black-bounce spacetime. Regarding unbound orbits, He et al [33] investigates in detail the weak-field gravitational lensing of a massive neutral particle with a relativistic initial velocity in this spacetime, providing insights into the case of unbound orbits, particularly in terms of gravitational deflection and lensing observables.
Using the data sets of OJ 287, we find the bounds on a/m=3.45 ± 0.01, which is improved by one order of magnitude than the one of the star S2, although our upper limit of a still needs further improvement. With the continuous progress in interferometric techniques, such as those provided by GRAVITY+, we could expect that the orbits of S2 and OJ 287 would be considerably improved and its bound on the bounce parameter would be tightened further. It is important to note that the constraints we obtained are strictly preliminary. This is because we compared the theoretical prediction for the additional precession with the measured OJ 287 precession, which was obtained without modeling the exotic effects. A genuine constraint could only be derived by explicitly modeling the exotic effect of interest and estimating the corresponding parameters. Thus, our results are just a guess, nonetheless an important one, about what could be possible to obtain in real, dedicated data analysis.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant Nos. 12303079, 12481540180 and 12475057). We thank for the support of the postdoctoral program of purple Mountain Observatory, Chinese Academy of Sciences.