The path of light rays traversing a binary system

How is the orbit of a photon affected by the field of a binary system? This is a restricted three-body problem with gravitational interactions. Adding one body to the famous two-body problem of a photon in a gravitational field brings a remarkable complexity to the problem. It is however believed that in a complete treatment of such motions and with today's highly accurate astronomical observations in hand, these effects must be taken into account. This subject has been a field of interest since the astrometric space mission like GAIA planned to position the stars and celestial objects with one microarcsecond level of accuracy [1]. At this level, many subtle effects are potentially observable. One of them would probably be the deflection of light by the field of a binary system. When a photon moves in the weak gravity field of an isolated spherical body, the leading order corrections to the photon (Newtonian path) are scaled by $\frac{{Gm}}{{c}^{2}R},$ where R is the closest distance to the gravitating body the photon can reach and m is the mass of such a body. Considering the scales characterizing our problem, it is tempting to ask whether there are any relativistic effects produced by the second body in the binary that we should be concerned about or if the traces left by the companying body still remain a challenge for future missions?

In this work, we assume that photons move on the null paths and are influenced by a binary system in a circular motion. We also assume that the radius of the binary orbit is large, compared to the distance between the photon and the member of the binary which we call the main body. In adequately weak gravity regimes, effects like the deviation of the photon trajectory from a straight line may be well described by the post-Newtonian approximation. The latter assumption permits an approximate solution to our problem by reducing the three-body problem to the motion of a photon in an effective field of the orbiting bodies. The body of the paper is divided in two parts. In the first part we determine the path of light rays traversing nonspinning binary system with arbitrary masses and explore the tidally perturbed photon path in Newtonian order; in the second part we allow the binary members to be slowly rotating and calculate the light deviation angle. The assumption of slow rotation allows us to capture the effects with the dragging of inertial frames and to neglect the rotational deformation of the main body.

To better discuss the dynamics of these systems, we introduce three lengthscales: one scale is given by the radius of the circular orbit, denoted by r_o, and the other two are given by the masses in the system expressed in geometrized units, $G=c=1.$ Let ρ be the separation between the photon and the main body. We assume that (i) ρ is comparable to the mass of the main body, m₁, and (ii) the second body in the binary is remote, $\rho \ll {r}_{{\rm{o}}}.$ To be precise, we focus on photons that traverse in a region bounded by $\rho \lt {\rho }_{\mathrm{max}}\ll {r}_{{\rm{o}}}.$ There is also a post-Newtonian zone which is bounded by $\rho \gt {\rho }_{\mathrm{min}}\gg {m}_{1}$ and an overlap region in which the effects of the second body on the geometry of the spacetime can be described by tidal potentials (see figure 1). These are given by a linear perturbation about the Schwarzschild solution, so that the perturbed metric satisfies the vacuum field equations, perturbatively.

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It is straightforward to apply tidal effects to a system comprising a photon and a gravitating body, but solving the equations corresponding to the path of the photon in this perturbed spacetime would be tricky. A suitable framework for solving the path based on osculating orbits has already been suggested in [2]. In this framework, the motion of the photon is at all times described by a family of null geodesics, called osculating trajectories, characterized by a set of constants of motion. A nice reformulation of the various aspects of the original problem can be achieved by the evolution of these constants if we know the interacting forces. The evolution of this set of constants is constrained by a set of first-order differential equations, known as the osculating conditions. In the first part of this paper we calculate the tidal forces acting on the photon by employing a metric calculated in a paper by Taylor and Poisson [3]. In their work, the main gravitating body was a black hole; they used this metric to find the dynamics of a black hole, i.e., those produced by an influx of gravitational energy across the horizon (like tidal heating or an increase in mass, angular momentum, and surface area [4]) of the black hole. Our interest in this paper is mostly in the consequences of the tidal interactions on photon orbital dynamics. The metric given by Taylor and Poisson is not necessarily restricted to the weak field approximation and can be equally used in small hole approximation, in which the gravitating body is assumed to be much smaller than the second body. In our problem, we need to know the tidal fields and their effects on the structure of spacetime around the gravitating body, all within the weak field approximation to general relativity.

Assuming that the photon path would not be too different from the Schwarzschild light paths, we will include the force components in the osculating equations given in [2] to obtain the observable variations at any point of the path. Our results can be evaluated in any coordinate system centered on the main body; this guarantees that those are observable quantities. For example, we find that the leading order total deflection angle depends on the second body position and varies from a minimum equal to $4\frac{{m}_{1}}{{R}_{*}}\left[1-\pi \left(\frac{{m}_{2}}{{m}_{1}}\right){\left(\frac{{R}_{*}}{{r}_{{\rm{o}}}}\right)}^{3}\right]$ to a maximum $4\frac{{m}_{1}}{{R}_{*}}\left[1+2\pi \left(\frac{{m}_{2}}{{m}_{1}}\right){\left(\frac{{R}_{*}}{{r}_{{\rm{o}}}}\right)}^{3}\right],$ where ${R}_{*}$ is the closest distance to the main body that the photon could reach if it were an isolated body.

The structure of this paper is as follows: in section 2, we start with the metric describing a tidal environment around a nonrotating gravitating body. We find the tidally perturbed metric in Schwarzchid coordinates by a coordinate transformation (from harmonic coordinates). The tidal perturbations are considered static fields on a flat background. In section 3, we calculate different components of the perturbing force and the equations governing the photon path in Newtonian order. In section 4, the evolution of constants of motion is discussed. Other corrections to the photon deflection angle may be sourced by rotational forces or the coupling of rotational and tidal fields. We follow this discussion in section 5 through 1.5PN order by incorporating previously neglected terms in the black hole metric. The discussion will be concluded with the summary and conclusions of the results in section 6.

The general description of the spacetime around a spherical gravitating body in an arbitrary tidal environment is given by a metric, which has the following form in Schwarzschild coordinates, $\left(t,\rho ,{\theta }^{A}=\left(\theta ,\phi \right)\right)$:

$$\begin{eqnarray}&&{g}_{{tt}}=-\left(1-2\displaystyle \frac{{m}_{1}}{\rho }\right)-{\rho }^{2}{\left(1-2\displaystyle \frac{{m}_{1}}{\rho }\right)}^{2}{{\mathcal{E}}}^{q}+O\left(\displaystyle \frac{{\rho }^{3}}{{{\mathcal{R}}}^{2}{\mathcal{L}}}\right),\end{eqnarray} \tag{ 2.1a }$$

$$\begin{eqnarray}&&{g}_{\rho t}=0,\end{eqnarray} \tag{ 2.1b }$$

$$\begin{eqnarray}&&{g}_{{At}}=\displaystyle \frac{2}{3}{\rho }^{3}{{\mathcal{B}}}_{a}^{q}{{\rm{\Omega }}}_{A}^{a}+O\left(\displaystyle \frac{v}{c}\displaystyle \frac{{\rho }^{3}}{{{\mathcal{R}}}^{2}{\mathcal{L}}}\right),\end{eqnarray} \tag{ 2.1c }$$

$$\begin{eqnarray}&&{g}_{\rho A}=-\left[{m}_{1}{\rho }^{3}+\displaystyle \frac{1}{3}{m}_{1}^{3}{\left(1-\displaystyle \frac{{m}_{1}}{\rho }\right)}^{-2}\right]{{\mathcal{E}}}_{a}^{q}{{\rm{\Omega }}}_{A}^{a}+O\left(\displaystyle \frac{{\rho }^{3}}{{{\mathcal{R}}}^{2}{\mathcal{L}}}\right),\end{eqnarray} \tag{ 2.1d }$$

$$\begin{eqnarray}&&{g}_{\rho \rho }={\left(1-2\displaystyle \frac{{m}_{1}}{\rho }\right)}^{-1}-{\rho }^{2}{{\mathcal{E}}}^{q}+O\left(\displaystyle \frac{v}{c}\displaystyle \frac{{\rho }^{3}}{{{\mathcal{R}}}^{2}{\mathcal{L}}}\right),\end{eqnarray} \tag{ 2.1e }$$

$$\begin{eqnarray}{g}_{{AB}} & = & {\rho }^{2}\left[{{\rm{\Omega }}}_{{AB}}-{\rho }^{2}{\left(1-\frac{{m}_{1}}{\rho }\right)}^{-1}{\left(1-2\frac{{m}_{1}}{\rho }\right)}^{2}{{\rm{\Omega }}}_{{AB}}{{\mathcal{E}}}^{q}\right.\\ & & +\left.\left(\frac{1}{3}\frac{{m}_{1}^{3}}{\rho }{\left(1-\frac{{m}_{1}}{\rho }\right)}^{-1}-{m}_{1}\rho \left(1-\frac{{m}_{1}}{\rho }\right)\right)\left(2{{\mathcal{E}}}_{{ab}}{{\rm{\Omega }}}_{A}^{a}{{\rm{\Omega }}}_{B}^{b}+{{\mathcal{E}}}^{q}{{\rm{\Omega }}}_{{AB}}\right)\right]\\ & & +O\left(\frac{{\rho }^{3}}{{{\mathcal{R}}}^{2}{\mathcal{L}}}\right){g}_{{ab}}.\end{eqnarray} \tag{ 2.1f }$$

Here m₁ is gravitational radius of our body, ${{\rm{\Omega }}}_{A}^{a}=\frac{\partial {{\rm{\Omega }}}^{a}}{\partial {\theta }^{A}}$, which satisfies ${{\rm{\Omega }}}^{{AB}}{{\rm{\Omega }}}_{A}^{a}{{\rm{\Omega }}}_{B}^{b}={\gamma }^{{ab}}$ and ${{\rm{\Omega }}}_{a}^{A}{{\rm{\Omega }}}^{a}=0.$ Furthermore, ${{\rm{\Omega }}}_{a}^{A}={{\rm{\Omega }}}^{{AB}}{{\rm{\Omega }}}_{{aB}},$ ${{\rm{\Omega }}}^{a}:= \frac{{x}^{a}}{r}=\left(\mathrm{sin}\theta \mathrm{cos}\phi ,\mathrm{sin}\theta \mathrm{sin}\phi ,\mathrm{cos}\theta \right)$ is a unit radial vector, ${{\rm{\Omega }}}_{{AB}}$ is the metric on the unit two-sphere, x^as are coordinates of a Cartesian system, and ${\gamma }_{{ab}}={\delta }_{{ab}}-{{\rm{\Omega }}}_{a}{{\rm{\Omega }}}_{b}$ is a projection tensor on the direction orthogonal to ${{\rm{\Omega }}}^{a}.$ The domain of validity of the metric is the spherical body neighborhood. To find this metric, we used the results of [3], and performed a coordinate transformation from the harmonic coordinates $\left({x}^{0},{x}^{a}\right)$ to $\left(t,\rho ,\theta ,\phi \right),$ given by ${x}^{0}={ct},{x}^{a}=r{{\rm{\Omega }}}^{a}\left({\theta }^{A}\right),$ $r=\rho -{m}_{1}.$ In [3], Taylor and Poisson examined the motion and the tidal dynamics of a black hole placed within a post-Newtonian external spacetime. The tidal environment is presented as a functional of arbitrary external potentials $\left\{{{\mathcal{E}}}^{q},{{\mathcal{E}}}_{a}^{q},{{\mathcal{E}}}_{{ab}}^{q},{{\mathcal{B}}}_{a}^{q}\right\}$ built from the lowest-order tidal moments, ${{\mathcal{E}}}_{{ab}}$ and ${{\mathcal{B}}}_{{ab}}.$ Our work is a continuation of their effort, and this coordinate transformation brings the metric of the distorted black hole to the new form given in (2.1). This metric applies to the spacetime outside any spherical body immersed in a tidal environment. It is linearized about the Schwarzschild solution and satisfies (approximate) vacuum field equations. The metric is accurate to all orders in $\frac{{m}_{1}}{\rho }$, and the neglected terms involve the fourth order in $\frac{\rho }{{r}_{{\rm{o}}}}$ that comes from higher-order tidal moments or the time derivative of quadrapole moments. In these terms, ${\mathcal{L}}$ and ${\mathcal{R}}$ are respectively the scales of spatial inhomogeneity and the radius of curvature in the external spacetime in which the black hole moves.

In studying the path of a photon through a curved geometry, it is a well-known practice to consider the metric perturbations as fields defined on a flat background spacetime. In this way the tidal environment is described by the weak field limit of the metric. Following the same practice, we take $\frac{{m}_{1}}{\rho }$ as well as the potential terms produced by the second body, i.e. the linear perturbation terms to flat Minkowski space, to be small. Although the potentials that characterize the tidal perturbations are completely general, in this work we require that the gravitating body be a member of a binary system, the radius of the binary be large and the terms beyond the quadrapole tidal moments be negligible³ . For this procedure to make sense, we should invoke the slow motion limit as well. We introduce a small dimensionless parameter and denote it by :

$$\begin{eqnarray}&&\epsilon := \displaystyle \frac{{m}_{1}}{\rho }\approx {\left(\displaystyle \frac{\rho }{{r}_{{\rm{o}}}}\right)}^{4}.\end{eqnarray} \tag{ 2.2 }$$

In this limit, the nonzero components of the metric are thus written as

$$\begin{eqnarray}&&{g}_{{tt}}=-\left(1-2\displaystyle \frac{{m}_{1}}{\rho }+{\rho }^{2}{{\mathcal{E}}}^{q}\right)+O\left({\epsilon }^{2}\right),\end{eqnarray} \tag{ 2.3a }$$

$$\begin{eqnarray}&&{g}_{\rho \rho }=1+2\displaystyle \frac{{m}_{1}}{\rho }-{\rho }^{2}{{\mathcal{E}}}^{q}+O\left({\epsilon }^{2}\right),\end{eqnarray} \tag{ 2.3b }$$

$$\begin{eqnarray}&&{g}_{{AB}}={\rho }^{2}{{\rm{\Omega }}}_{{AB}}\left[1-{\rho }^{2}{{\mathcal{E}}}^{q}+O\left({\epsilon }^{2}\right)\right],\end{eqnarray} \tag{ 2.3c }$$

where ${{\mathcal{E}}}^{q}=\frac{1}{{c}^{2}}{{\mathcal{E}}}_{{ab}}{{\rm{\Omega }}}^{a}{{\rm{\Omega }}}^{b}$ and ${{\mathcal{E}}}_{{ab}}$ are a symmetric-trace free tensor, which its components for circular orbit motion are calculated by matching the local metric with the global metric that includes the gravitating body and the external spacetime [3] and [12]. These nonvanishing components of the tidal moments are given by

$$\begin{eqnarray}&&{{\mathcal{E}}}_{11}=-\displaystyle \frac{{m}_{2}}{2{r}_{{\rm{o}}}^{3}}\left[1+3\mathrm{cos}2\psi +O{\left(\displaystyle \frac{u}{c}\right)}^{2}\right],\end{eqnarray} \tag{ 2.4a }$$

$$\begin{eqnarray}&&{{\mathcal{E}}}_{12}={{\mathcal{E}}}_{21}=-\displaystyle \frac{{m}_{2}}{2{r}_{{\rm{o}}}^{3}}\left[1+3\mathrm{sin}2\psi +O{\left(\displaystyle \frac{u}{c}\right)}^{2}\right],\end{eqnarray} \tag{ 2.4b }$$

$$\begin{eqnarray}&&{{\mathcal{E}}}_{22}=-\displaystyle \frac{{m}_{2}}{2{r}_{{\rm{o}}}^{3}}\left[1-3\mathrm{cos}2\psi +O{\left(\displaystyle \frac{u}{c}\right)}^{2}\right],\end{eqnarray} \tag{ 2.4c }$$

$$\begin{eqnarray}&&{{\mathcal{E}}}_{33}=-\left({{\mathcal{E}}}_{11}+{{\mathcal{E}}}_{22}\right)=-\displaystyle \frac{{m}_{2}}{{r}_{{\rm{o}}}^{3}}+O{\left(\displaystyle \frac{u}{c}\right)}^{2}.\end{eqnarray} \tag{ 2.4d }$$

Here u is the relative orbital velocity and ψ is the phase related to the angular velocity, ω, of the tidal moments with respect to the frame moving with the black hole, $\psi =\omega t=\sqrt{\frac{\left({m}_{1}+{m}_{2})\right)}{{r}_{{\rm{o}}}^{3}}}t\left(1+O{\left(\frac{u}{c}\right)}^{2}\right).$ It can also be defined in terms of the orbital velocity of the second body, $\bar{\omega },$ and the precessional angular frequency of the black hole frame relative to the barycentric frame, Ω, by $\psi =\bar{\omega }\bar{t}-{\rm{\Omega }}t,$ in which $\bar{t}$ is the global time coordinate. The scalar potential, ${{\mathcal{E}}}^{q},$ is given by

$$\begin{eqnarray}&&{{\mathcal{E}}}^{q}=A\left[1-3{\mathrm{cos}}^{2}\theta +3{\mathrm{sin}}^{2}\theta \mathrm{cos}2\left(\phi -\psi \right)+O\left(\epsilon \right)\right],\end{eqnarray} \tag{ 2.5 }$$

where $A:= -\frac{{m}_{2}}{2{r}_{{\rm{o}}}^{3}}$ is a dimensionful parameter.

Before exploring the behavior of the photon in this spacetime, there are some points that we would like to add:

• (1)  
  Although the gravitomagnetic tidal (vector) potentials exist in a relativistic description of the geometry around a gravitating body, they do not appear in weak field limit and slow motion context ($v/c\ll 1,$ where v is the tidal environment velocity scale). This is also true for gravito-electric vector and tensor potentials.
• (2)  
  The spherical symmetry of the Schwarzschild geometry is perturbationally broken, so the Birkhoff theorem allows a non-static vacuum spacetime. On the other hand, the spacetime will not admit $r=3{m}_{1}$ or any other hypersurface as a photon sphere because it is not invariant under so(3) group of symmetry.
• (3)  
  The tidal geometry is described by a scalar potential and each scalar potential transforms as such under parity. This ensures that the deformed metric be conserved under a parity transformation.
• (4)  
  If we expand the tidal potential given in (2.5), in terms of spherical harmonics, we will have

  $$\begin{eqnarray}&&{{\mathcal{E}}}^{q}=\sqrt{\displaystyle \frac{\pi }{5}}A\left[-{Y}_{2}^{0}+\sqrt{24}\left({Y}_{2}^{2}+{Y}_{2}^{-2}\right)+O\left(\epsilon \right)\right].\end{eqnarray} \tag{ 2.6 }$$

  It is observed that the coefficients of ${Y}_{2}^{\pm 1}$ are zero, and the metric components are invariant under $\theta \to \pi -\theta .$ This fact demonstrates an additional symmetry with respect to the plane $\theta =\frac{\pi }{2}.$ The presence of the second body has greatly reduced the isometry (Lie) group of Schwarzschild spacetime to (discrete) subgroups, namely parity and mirror symmetry.

The first step before writing the null geodesic equations is to calculate the Christoffel symbols for this metric. The perturbed geodesic equations will then be written as

$$\begin{eqnarray}&&\displaystyle \frac{{{\rm{d}}}^{2}{x}^{\mu }}{{\rm{d}}{\lambda }^{2}}+{\bar{{\rm{\Gamma }}}}_{\nu \sigma }^{\mu }\displaystyle \frac{{\rm{d}}{x}^{\nu }}{{\rm{d}}\lambda }\displaystyle \frac{{\rm{d}}{x}^{\sigma }}{{\rm{d}}\lambda }={F}^{\mu },\end{eqnarray} \tag{ 3.1 }$$

where

$$\begin{eqnarray}&&{F}^{\mu }=-\delta {{\rm{\Gamma }}}_{\nu \sigma }^{\mu }\displaystyle \frac{{\rm{d}}{x}^{\nu }}{{\rm{d}}\lambda }\displaystyle \frac{{\rm{d}}{x}^{\sigma }}{{\rm{d}}\lambda },\quad {{\rm{\Gamma }}}_{\nu \sigma }^{\mu }={\bar{{\rm{\Gamma }}}}_{\nu \sigma }^{\mu }+\delta {{\rm{\Gamma }}}_{\nu \sigma }^{\mu }.\end{eqnarray} \tag{ 3.2 }$$

Here $\bar{{\rm{\Gamma }}}{\rm{s}}$ and $\delta {\rm{\Gamma }}{\rm{s}}$ are components of the Schwarzschild–Christoffel symbols and their perturbations, respectively. The appendix gives a full list of these symbols in the weak field limit. The right-hand side of (3.1) represents the perturbing force components.

There does not seem to be much hope for solving the set of equation (3.1) in a general case. Fortunately, there are some special cases in which the remaining symmetries simplify the task. It is easy to see that the plane $\theta =\pi /2$ satisfies the equation (3.1) for⁴ $\mu =\theta$. Therefore, if we specialize the case in which photons move in the equatorial plane of the binary system, i.e., if the inclination parameter, i, is exactly $\pi /2$ and the binary orbit is edge-on, the direction of angular momentum will be conserved and the photon will not leave the plane because ${F}^{\theta }=0.$ The hypersurface $\theta =\pi /2$ is a photon surface in the sense that every affine null geodesic on it is an affine null geodesic of spacetime. This is a direct result of theorem II in [5] which states that if the trace-free part of the extrinsic curvature on a hypersurface is zero, it will be a photon surface. Let y^a be the coordinates on the hypersurface $\theta =\pi /2$ and let ${n}_{a}={\delta }_{a}^{\theta }$ and ${K}_{{ab}}=-{{\rm{\Gamma }}}_{{ab}}^{\theta }$ be the unit normal vector and the extrinsic curvature tensor field on it, respectively. A glance at the appendix reveals that ${K}_{{ab}}=0$ on this hypersurface, so its trace-free part would be zero.

Other components of the tidal force applied to the photon are

$$\begin{eqnarray}&&{F}^{t}=\displaystyle \frac{1}{2}{\rho }^{2}\left({\rho }^{2}{\dot{\phi }}^{2}+{\dot{\rho }}^{2}-{\dot{t}}^{2}\right){\partial }_{t}{{\mathcal{E}}}^{q}-2\rho \dot{t}\dot{\rho }{{\mathcal{E}}}^{q}-{\rho }^{2}\dot{t}\dot{\phi }{\partial }_{\phi }{{\mathcal{E}}}^{q}+O\left({\epsilon }^{2}\right),\end{eqnarray} \tag{ 3.3a }$$

$$\begin{eqnarray}&&{F}^{\rho }=\rho \left({\dot{\rho }}^{2}-{\dot{t}}^{2}-{\rho }^{2}{\dot{\phi }}^{2}\right){{\mathcal{E}}}^{q}+{\rho }^{2}\left(\dot{t}{\partial }_{t}{{\mathcal{E}}}^{q}+\dot{\rho }{\partial }_{\phi }{{\mathcal{E}}}^{q}\right)+O\left({\epsilon }^{2}\right),\end{eqnarray} \tag{ 3.3b }$$

$$\begin{eqnarray}&&{F}^{\phi }=\displaystyle \frac{1}{2}\left({\rho }^{2}{\dot{\phi }}^{2}-{\dot{\rho }}^{2}-{\dot{t}}^{2}\right){\partial }_{\phi }{{\mathcal{E}}}^{q}+{\rho }^{2}{\partial }_{t}{{\mathcal{E}}}^{q}+2\rho \dot{\rho }\dot{\phi }{{\mathcal{E}}}^{q}+O\left({\epsilon }^{2}\right).\end{eqnarray} \tag{ 3.3c }$$

Even with preceding simplifications, integrating the equations of motion and determining the photon path will not be simple. Although the perturbing forces are complicated, we believe that these forces keep the form of the photon path. Therefore, the leading correction to the Schwarzschild null curves can be determined by the method of variation of parameters (VoPs). As described in [2], in this method one employs a known solution of a fiducial system of differential equations as an ansatz for solving the system under study. The known solution, ${x}_{G}^{\mu },$ plays the role of instantaneously tangential (or osculating) trajectories to the true path, ${x}^{\mu }\left(\lambda \right).$ Each point on the true path with parameter λ coincides geometrically and dynamically with an osculating trajectory with one specified set of constants, ${I}^{\alpha };$ the number of these constants is equal to the number of phase space independent variables. The constants are furnished with λ dependence and evolve along a flow line on a cross section of phase space that is consistent with the null condition. Each and every flow line is constrained by the osculating conditions

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\dot{x}}_{G}^{\mu }}{\partial {I}^{\alpha }}{\dot{I}}^{\alpha }={F}^{\mu },\ \ \ \displaystyle \frac{\partial {x}_{G}^{\mu }}{\partial {I}^{\alpha }}{\dot{I}}^{\alpha }=0,\end{eqnarray} \tag{ 3.4 }$$

but different flow lines may correspond to very different interactions. This reformulation of the problem provides an insight into the effects of perturbing forces on various aspects of the motion. In the framework of VoP, the perturbing forces are not assumed to be small and the only restriction is that the forces must preserve the shape of the path. More on this method may be found in [6].

Following [2], these constants are called osculating elements, and the Schwarzschild null geodesics family, with parameter λ, is chosen as ${x}_{G}^{\mu }.$ We recall that in weak field approximations, the relations describing forced Schwarzschild null geodesics are

$$\begin{eqnarray}&&{\rho }^{-1}\left(\phi \right)=\displaystyle \frac{\mathrm{sin}\left(\phi -{\rm{\Delta }}\left(\phi \right)\right)}{R\left(\phi \right)}+\displaystyle \frac{3{m}_{1}}{2{R}^{2}\left(\phi \right)}\left[1+\displaystyle \frac{1}{3}\mathrm{cos}2\left(\phi -{\rm{\Delta }}\left(\phi \right)\right)\right],\end{eqnarray} \tag{ 3.5a }$$

$$\begin{eqnarray}&&\phi \left(\lambda \right)={\rm{\Delta }}\left(\lambda \right)+{\displaystyle \int }_{0}^{\lambda }L\left(\tilde{\lambda }\right){\rho }^{-2}\left(\tilde{\lambda }\right){\rm{d}}\tilde{\lambda },\end{eqnarray} \tag{ 3.5b }$$

$$\begin{eqnarray}&&t\left(\lambda \right)=T\left(\lambda \right)+{\displaystyle \int }_{0}^{\lambda }\displaystyle \frac{E\left(\tilde{\lambda }\right)}{1-2{m}_{1}{\rho }^{-1}\left(\tilde{\lambda }\right)}{\rm{d}}\tilde{\lambda },\end{eqnarray} \tag{ 3.5c }$$

$$\begin{eqnarray}&&{\dot{\rho }}^{2}={E}^{2}-\displaystyle \frac{{L}^{2}}{{\rho }^{2}}\left(1-\displaystyle \frac{2{m}_{1}}{\rho }\right).\end{eqnarray} \tag{ 3.5d }$$

Here the dot denotes the derivative with respect to λ and the osculating elements on these trajectories are $\left\{E,L,R,{\rm{\Delta }},T\right\}.$ Some osculating elements, like R, T, and Δ, that characterize the initial position on the fiducial trajectory are usually referred to as the positional elements; these elements typically change whenever the geometry of spacetime is affected with a new interaction. Other elements, like L and E, which are called principal elements, determine on which trajectory the photon is moving. We can ignore T and evolve t explicitly by using

$$\begin{eqnarray}&&\overset{\prime}{t}=\displaystyle \frac{E}{L}\displaystyle \frac{{\rho }^{2}}{1-2\frac{{m}_{1}}{\rho }},\end{eqnarray} \tag{ 3.6 }$$

which is equivalent to the evolving of t − T directly. The flow lines that satisfy (3.4) are given by

$$\begin{eqnarray}&&\overset{\prime}{E}=\displaystyle \frac{{R}^{2}}{L}\displaystyle \frac{1}{{\mathrm{sin}}^{2}\left(\phi -{\rm{\Delta }}\right)}{F}^{t}+O\left({\epsilon }^{2}\right),\end{eqnarray} \tag{ 3.7a }$$

$$\begin{eqnarray}&&\overset{\prime}{{\rm{\Delta }}}=-\displaystyle \frac{{R}^{3}}{{L}^{2}}\displaystyle \frac{1}{\mathrm{sin}\left(\phi -{\rm{\Delta }}\right)}{F}^{\rho }+O\left({\epsilon }^{2}\right),\end{eqnarray} \tag{ 3.7b }$$

$$\begin{eqnarray}&&\overset{\prime}{L}=\displaystyle \frac{{R}^{4}}{L}\displaystyle \frac{1}{{\mathrm{sin}}^{4}\left(\phi -{\rm{\Delta }}\right)}{F}^{\phi }+O\left({\epsilon }^{2}\right),\end{eqnarray} \tag{ 3.7c }$$

$$\begin{eqnarray}&&\overset{\prime}{R}=\displaystyle \frac{{R}^{4}}{{L}^{2}}\displaystyle \frac{\mathrm{cos}\left(\phi -{\rm{\Delta }}\right)}{{\mathrm{sin}}^{2}\left(\phi -{\rm{\Delta }}\right)}{F}^{\rho }+O\left({\epsilon }^{2}\right),\end{eqnarray} \tag{ 3.7d }$$

$$\begin{eqnarray}&&\overset{\prime}{t}=\displaystyle \frac{E}{{R}^{2}L}\mathrm{sin}\left(\phi -{\rm{\Delta }}\right)\left[\mathrm{sin}\left(\phi -{\rm{\Delta }}\right)+4{m}_{1}\right]+O\left({\epsilon }^{2}\right).\end{eqnarray} \tag{ 3.7e }$$

In these equations, ϕ has been used instead of λ (or t), as an independent variable and the prime sign denotes a derivative with respect to ϕ. The equations governing the evolution of E and L could have been calculated by the covariant formulation of osculating conditions [8].

The variations in the osculating elements are assumed to be small; therefore, one can achieve a good approximation of these elements by substituting the leading order expressions

$$\begin{eqnarray}&&\rho =\displaystyle \frac{{R}_{*}}{\mathrm{sin}\left(\phi -{{\rm{\Delta }}}_{*}\right)}\left(1+O\left(\epsilon \right)\right),\end{eqnarray} \tag{ 3.8a }$$

$$\begin{eqnarray}&&\dot{t}={E}_{*}\left(1+O\left(\epsilon \right)\right),\end{eqnarray} \tag{ 3.8b }$$

$$\begin{eqnarray}&&{\dot{\rho }}^{2}={E}_{*}^{2}-{L}_{*}^{2}\left(\displaystyle \frac{{\mathrm{sin}}^{2}\left(\phi -{{\rm{\Delta }}}_{*}\right)}{{R}_{*}^{2}}+O\left(\epsilon \right)\right),\end{eqnarray} \tag{ 3.8c }$$

$$\begin{eqnarray}&&\dot{\phi }=\displaystyle \frac{{L}_{*}}{{R}_{*}^{2}}{\mathrm{sin}}^{2}\left(\phi -{{\rm{\Delta }}}_{*}\right)+O\left(\epsilon \right),\end{eqnarray} \tag{ 3.8d }$$

into the right-hand side of equation (3.7). Here the set of constants that the photon trajectory could have assumed, if the main body was in a complete isolation, are denoted by $\left\{{E}_{*},{L}_{*},{R}_{*},{{\rm{\Delta }}}_{*}\right\}.$ Using (2.5), the different components of the perturbing force in the plane $\theta =\pi /2$ are given by

$$\begin{eqnarray}&&{F}^{t}=2{E}_{*}{L}_{*}A\left[\displaystyle \frac{1}{k}\mathrm{cot}\left(\phi -{{\rm{\Delta }}}_{*}\right)\left[1+3\mathrm{cos}2\left(\phi -\psi \right)\right]+3\mathrm{sin}2\left(\phi -\psi \right)\right]+O\left({\epsilon }^{2}\right),\end{eqnarray} \tag{ 3.9a }$$

$$\begin{eqnarray}{F}^{\rho } & = & -2{E}_{*}{L}_{*}A\left[k\mathrm{sin}\left(\phi -{{\rm{\Delta }}}_{*}\right)\left[1+3\mathrm{cos}2\left(\phi -\psi \right)\right]\right.\\ & & +\left.3\mathrm{cos}\left(\phi -{{\rm{\Delta }}}_{*}\right)\mathrm{sin}2\left(\phi -\psi \right)\left[\displaystyle \frac{{E}_{*}{R}_{*}^{2}\omega }{{\mathrm{sin}}^{2}\left(\phi -{{\rm{\Delta }}}_{*}\right)}-{L}_{*}\right]\right]+O\left({\epsilon }^{2}\right),\end{eqnarray} \tag{ 3.9b }$$

$$\begin{eqnarray}&&{F}^{\phi }=2{E}_{*}^{2}A\left[-k\left[1+3\mathrm{cos}2\left(\phi -\psi \right)\right]\mathrm{sin}2\left(\phi -{{\rm{\Delta }}}_{*}\right)\right.\\ &&\qquad +\left.3\mathrm{sin}2\left(\phi -\psi \right)\left[1+\displaystyle \frac{{L}_{*}}{{E}_{*}}\omega -{k}^{2}{\mathrm{sin}}^{2}\left(\phi -{{\rm{\Delta }}}_{*}\right)\right]\right]+O\left({\epsilon }^{2}\right).\end{eqnarray} \tag{ 3.9c }$$

Here $k:= {E}_{*}{R}_{*}/{L}_{*}$ is a dimensionless parameter, which its value can be found by substituting $\rho ={R}_{*}$ in (3.5d), $k=1+O\left(\epsilon \right).$ In the derivation of the force components, we have used the square root of (3.8c) with a minus sign, $\dot{\rho }=-{E}_{*}\mathrm{cos}\left(\phi -{{\rm{\Delta }}}_{*}\right)+O\left(\epsilon \right).$

We recall that in the preceding equations, ψ is a function of t, so, formally, its explicit dependence on ϕ must be considered. However, these equations will turn out to be simpler by recognizing that in the problem under study, osculating elements undergo two types of change. The first is an oscillation with a period equal to the period of tidal moments (or a multiple of it); the second is a steady drift in ϕ. In other words, we have two timescales: t and ${\omega }^{-1}.$ On a short timescale, the photon moves on a geodesy of the Schwarzschild spacetime with a static tidal field, characterized by $\left\{E,L,R,{\rm{\Delta }}\right\}.$ Over a longer timescale, ${\omega }^{-1}\approx \sqrt{\frac{{m}_{1}+{m}_{2}}{{r}_{{\rm{o}}}^{3}}}\approx \frac{{m}_{1}+{m}_{2}}{{\epsilon }^{\displaystyle \frac{15}{8}}},$ the displacement of the second body changes ψ and causes the path to evolve. It is clearly desirable to compute an accurate trajectory for the whole domain of the validity of ϕ (or t). However, the linear perturbation method is limited to producing a model for the trajectory over a rapid timescale (or a snapshot of the trajectory) in which the trajectory is described by the deviations from the pivot elements, $\left\{{E}_{*},{L}_{*},{R}_{*},{{\rm{\Delta }}}_{*}\right\},$ and the new configurations (of the second body) are neglected. Such approximations fall out of phase with the true path after the longer timescale $\approx {\epsilon }^{\displaystyle \frac{-15}{8}}{m}_{1}.$ In fact a rigorous prescription for the path can be provided by a two-timescale analysis, but considering the order of accuracy by which the short term secular changes are captured, at the cost of discarding long term effects, it is sensible to assume that $\dot{\psi }\ll \dot{\phi }$ or

$$\begin{eqnarray}&&\displaystyle \frac{\omega \dot{t}}{\dot{\phi }}\approx \displaystyle \frac{{R}_{*}^{2}{E}_{*}\omega }{{L}_{*}}\displaystyle \frac{1}{{\mathrm{sin}}^{2}\left(\phi -{{\rm{\Delta }}}_{*}\right)}\ll 1.\end{eqnarray} \tag{ 3.10 }$$

After invoking this approximation upon (3.9), the components of the perturbing force can be written as

$$\begin{eqnarray}&&{F}^{t}=2{E}_{*}{L}_{*}A\mathrm{cot}\left(\phi -{{\rm{\Delta }}}_{*}\right)\left[3\displaystyle \frac{\mathrm{cos}\left(\phi -2\psi +{{\rm{\Delta }}}_{*}\right)}{\mathrm{cos}\left(\phi -{{\rm{\Delta }}}_{*}\right)}+1\right],\end{eqnarray} \tag{ 3.11a }$$

$$\begin{eqnarray}&&{F}^{\rho }=2{E}_{*}{L}_{*}A\mathrm{sin}\left(\phi -{{\rm{\Delta }}}_{*}\right)\left[3\displaystyle \frac{\mathrm{sin}\left(\phi -2\psi +{{\rm{\Delta }}}_{*}\right)}{\mathrm{sin}\left(\phi -{{\rm{\Delta }}}_{*}\right)}-1\right],\end{eqnarray} \tag{ 3.11b }$$

$$\begin{eqnarray}&&{F}^{\phi }=2{E}_{*}^{2}A\mathrm{cos}\left(\phi -{{\rm{\Delta }}}_{*}\right)\left[3\displaystyle \frac{\mathrm{sin}\left(\phi -2\psi +{{\rm{\Delta }}}_{*}\right)}{\mathrm{sin}\left(\phi -{{\rm{\Delta }}}_{*}\right)}-1\right].\end{eqnarray} \tag{ 3.11c }$$

The evolution of the conserved quantities will then be reproduced by substituting the preceding equations into (3.7) as

$$\begin{eqnarray}&&\overset{\prime}{{\rm{\Delta }}}=-2{{AR}}_{*}^{2}\left[3\displaystyle \frac{\mathrm{sin}\left(\phi -2\psi +{{\rm{\Delta }}}_{*}\right)}{\mathrm{sin}\left(\phi -{{\rm{\Delta }}}_{*}\right)}-1\right],\end{eqnarray} \tag{ 3.12a }$$

$$\begin{eqnarray}&&\displaystyle \frac{\overset{\prime}{R}}{{R}_{*}}=2{{AR}}_{*}^{2}\mathrm{cot}\left(\phi -{{\rm{\Delta }}}_{*}\right)\left[3\displaystyle \frac{\mathrm{sin}\left(\phi -2\psi +{{\rm{\Delta }}}_{*}\right)}{\mathrm{sin}\left(\phi -{{\rm{\Delta }}}_{*}\right)}-1\right],\end{eqnarray} \tag{ 3.12b }$$

$$\begin{eqnarray}&&\displaystyle \frac{\overset{\prime}{E}}{{E}_{*}}=2{{AR}}_{*}^{2}\displaystyle \frac{\mathrm{cos}\left(\phi -{{\rm{\Delta }}}_{*}\right)}{{\mathrm{sin}}^{3}\left(\phi -{{\rm{\Delta }}}_{*}\right)}\left[3\displaystyle \frac{\mathrm{cos}\left(\phi -2\psi +{{\rm{\Delta }}}_{*}\right)}{\mathrm{cos}\left(\phi -{{\rm{\Delta }}}_{*}\right)}+1\right],\end{eqnarray} \tag{ 3.12c }$$

$$\begin{eqnarray}&&\displaystyle \frac{\overset{\prime}{L}}{{L}_{*}}=2{{AR}}_{*}^{2}\displaystyle \frac{\mathrm{cos}\left(\phi -{{\rm{\Delta }}}_{*}\right)}{{\mathrm{sin}}^{3}\left(\phi -{{\rm{\Delta }}}_{*}\right)}\left[3\displaystyle \frac{\mathrm{sin}\left(\phi -2\psi +{{\rm{\Delta }}}_{*}\right)}{\mathrm{sin}\left(\phi -{{\rm{\Delta }}}_{*}\right)}-1\right].\end{eqnarray} \tag{ 3.12d }$$

It is easily seen that the corrections imposed on the Schwarzschild null paths by the second body are of the order of ${{AR}}_{*}^{2},$ that is $O\left({\epsilon }^{\displaystyle \frac{7}{4}}\right).$ These corrections are so small that, even for photons with large impact parameters, we can neglect higher-order terms in (2.3) with confidence. These equations appear to be singular at $\phi ={{\rm{\Delta }}}_{*},\pi +{{\rm{\Delta }}}_{*}.$ The singularities correspond to the fact that Schwarzschild geodesics lose their geometric meaning for asymptotic values of ϕ.

In the following section, we will give expressions for osculating elements; once these functions are known, the path is obtained from equation (3.5). Considering the long separation of the two timescales, we assumed that the geometry around the main body has time translation symmetry. Before concluding this section, we are interested in using this symmetry to calculate energy of the perturbed system, $\bar{E},$ as a constant on the path. The Killing vector whose conserved quantity is energy is $K=\frac{\partial }{\partial t}.$ The conserved energy is then given by

$$\begin{eqnarray}&&\bar{E}=-{p}_{\mu }{K}^{\mu }=\dot{t}\left(1-\displaystyle \frac{2{m}_{1}}{\rho }+{\rho }^{2}{{\mathcal{E}}}^{q}+O\left({\epsilon }^{2}\right)\right).\end{eqnarray} \tag{ 3.13 }$$

This is a perturbationally constant quantity even for the paths outside of the equatorial plane. Moreover, for the trajectories constrained to the equatorial plane (photon surface), the direction of the angular momentum is also conserved.

The expression for the evolution of Δ in (3.12a) can now be used to obtain the explicit dependence on ψ at each ϕ. This gives

$$\begin{eqnarray}{\rm{\Delta }}\left(\phi \right) & = & {{\rm{\Delta }}}_{*}+2{{AR}}_{*}^{2}\left[\phi \left(1-3\mathrm{cos}2\left(\psi -{{\rm{\Delta }}}_{*}\right)\right)\right.\\ & & +\left.3\mathrm{sin}2\left(\psi -{{\rm{\Delta }}}_{*}\right)\mathrm{ln}\left(\mathrm{sin}\left|{{\rm{\Delta }}}_{*}-\phi \right|\right)\right].\end{eqnarray} \tag{ 4.1 }$$

Figure 2 shows the variations of Δ at different points of the path for different ψ directions at the initial condition ${{\rm{\Delta }}}_{*}=0.$ The contribution of the second body in the total deflection angle of a light ray passing near a binary system is

$$\begin{eqnarray}&&{\displaystyle \int }_{{{\rm{\Delta }}}_{*}}^{{{\rm{\Delta }}}_{*}+\pi }\overset{\prime}{{\rm{\Delta }}}{\rm{d}}\phi =2\pi {{AR}}_{*}^{2}\left[1-3\mathrm{cos}2\left(\psi -{{\rm{\Delta }}}_{*}\right)\right].\end{eqnarray} \tag{ 4.2 }$$

This implies that for edge-on binaries, the second body gives a maximum increase in the deflection angle when $\psi -{{\rm{\Delta }}}_{*}=\left\{\frac{\pi }{2},\frac{3\pi }{2}\right\}$ and a maximum decrease is obtained when $\psi -{{\rm{\Delta }}}_{*}=\left\{0,\pi \right\}.$ In [9], it is shown that among orbits with arbitrary inclination, the maximal possible light deflection is attained for edge-on binaries ($i=\pi /2$). Our calculations complement the preceding criteria, in the sense that we determine the changes in light deflection as well as other positional elements of these binaries⁵ as functions of the relative angle, $\psi -{{\rm{\Delta }}}_{*}.$ The various gravitational effects in the light propagation are additive in weak field limit, so for the total deflection angle, δ, we will find that

$$\begin{eqnarray}&&4\displaystyle \frac{{m}_{1}}{{R}_{*}}-4\pi {{AR}}_{*}^{2}\leqslant \delta \leqslant 4\displaystyle \frac{{m}_{1}}{{R}_{*}}+8\pi {{AR}}_{*}^{2}.\end{eqnarray} \tag{ 4.3 }$$

By integrating (3.12b), one can obtain

$$\begin{eqnarray}\displaystyle \frac{R}{{R}_{*}} & = & 1+2{{AR}}_{*}^{2}\left[6\mathrm{sin}2\left({{\rm{\Delta }}}_{*}-\psi \right)\left(1-\mathrm{cot}\left(\phi -{{\rm{\Delta }}}_{*}\right)\right)\right.\\ & & -\left.\left(1-3\mathrm{cos}2\left({{\rm{\Delta }}}_{*}-\psi \right)\right)\mathrm{ln}\;{\mathrm{sin}}^{2}\left(\phi -{{\rm{\Delta }}}_{*}\right)\right].\end{eqnarray} \tag{ 4.4 }$$

This amounts to a total change in R equal to

$$\begin{eqnarray}&&12\pi {{AR}}_{*}^{3}\mathrm{sin}2\left({{\rm{\Delta }}}_{*}-\psi \right).\end{eqnarray} \tag{ 4.5 }$$

The photon gets a maximum value of total R deflection when $\psi -{{\rm{\Delta }}}_{*}=\left\{\frac{\pi }{4},\frac{3\pi }{4},\frac{5\pi }{4},\frac{7\pi }{4}\right\};$ no change in total R is expected when $\psi -{{\rm{\Delta }}}_{*}=\left\{\frac{\pi }{2},\pi ,\frac{3\pi }{2}\right\}.$

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Obviously, one can find a different parametrization of the path by first integrating (3.7e) and keeping the leading order terms to produce

$$\begin{eqnarray}&&t\left(\phi \right)=\displaystyle \frac{{E}_{*}}{{L}_{*}{R}_{*}^{2}}\left[\displaystyle \frac{1}{2}\phi -\displaystyle \frac{1}{4}\mathrm{sin}2\left(\phi -{{\rm{\Delta }}}_{*}\right)+4{m}_{1}\mathrm{cos}\left(\phi -{{\rm{\Delta }}}_{*}\right)\right],\end{eqnarray} \tag{ 4.6 }$$

and then omitting ϕ in favor of t numerically. The expressions describing the evolution of other osculating elements, up to the leading order in , are

$$\begin{eqnarray}&&\displaystyle \frac{E}{{E}_{*}}=1+{{AR}}_{*}^{2}\left[6\mathrm{sin}2\left({{\rm{\Delta }}}_{*}-\psi \right)\mathrm{cot}\left(\phi -{{\rm{\Delta }}}_{*}\right)-\displaystyle \frac{1+3\mathrm{cos}2\left({{\rm{\Delta }}}_{*}-\psi \right)}{{\mathrm{sin}}^{2}\left(\phi -{{\rm{\Delta }}}_{*}\right)}\right],\end{eqnarray} \tag{ 4.7a }$$

$$\begin{eqnarray}&&\quad \displaystyle \frac{L}{{L}_{*}}=1+{{AR}}_{*}^{2}\left[-6\mathrm{sin}2\left({{\rm{\Delta }}}_{*}-\psi \right){\mathrm{cot}}^{3}\left(\phi -{{\rm{\Delta }}}_{*}\right)+\displaystyle \frac{1-3\mathrm{cos}2\left({{\rm{\Delta }}}_{*}-\psi \right)}{{\mathrm{sin}}^{2}\left(\phi -{{\rm{\Delta }}}_{*}\right)}\right].\end{eqnarray} \tag{ 4.7b }$$

In figures 2–5, we show the evolution of the path under the influence of the second body and initial condition ${{\rm{\Delta }}}_{*}=0.$ To avoid the singularities at asymptotes, $\phi =0,\pi ,$ the ϕ variations do not cover the whole domain, also we have chosen ${{AR}}_{*}^{2}=1.$ The four figures show the two positional elements $\left({\rm{\Delta }},R\right)$ and the two principal elements $\left(E,L\right)$ as functions of ϕ and ψ. The comparison of the behavior of these elements at asymptotes shows that there is a secular change in $\left({\rm{\Delta }},R\right)$, but there is no change in the values of the principal elements in these limits. For particles moving on timelike bound orbits, it is believed that the changes in principal elements occur when perturbing forces include a dissipative part [10]. Likewise, equations (4.7b) and (4.7a) imply that $L\left(0\right)=L\left(\pi \right)$ and $E\left(0\right)=E\left(\pi \right)$ for massless particles, even though the total angular momentum is not conserved in our problem. Although this result is obtained for a tidal interaction and edge-on binaries, it seems to be also valid for general inclinations.

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The geometry of spacetime around a slowly rotating star could be again deformed by a companion body on a circular orbit in the star's equatorial plane. The photon's path in this gravitational field is quite complicated. However, the method of VoP allows us to calculate the light deflection angle without solving the null geodesic equation. We interpret the effects of these tidal/dragging fields on Schwarzschild null geodesics as a force; its transverse component would be responsible for the path deviation.

5.1. Description of the geometry

Let us introduce the various scales that enter the description of this system. The mass of the main (companion) body is m₁(m₂). We take their spin vectors ${S}_{1,2}^{a}$ to be related to dimensionless quantities ${\chi }_{1,2}^{a}$ by ${S}_{\mathrm{1,2}}={\chi }_{\mathrm{1,2}}{m}_{1,2}^{2}$ which either aligned or anti-aligned with the orbital angular momentum vector. We also demand that ${\chi }_{1}\ll 1$ so that the main body is rotating slowly and the unperturbed metric can be obtained from the Kerr solution by neglecting all terms beyond linear order in ${\chi }_{1}.$ We focus on photons that traverse in empty annulus described in the introduction. In this region the metric is a vacuum solution of Einstein equations which was determined by Poisson in [11]. Nonvanishing components of the perturbed metric up to the first and a half post-Newtonian approximation (1.5PN) are given by

$$\begin{eqnarray}&&{g}_{{tt}}=-\left(1-2\displaystyle \frac{{m}_{1}}{\rho }\right)-{\rho }^{2}\left(1-4\displaystyle \frac{{m}_{1}}{\rho }\right){{\mathcal{E}}}^{q}+2{m}_{1}\rho {{\mathcal{K}}}^{d},\end{eqnarray} \tag{ 5.1a }$$

$$\begin{eqnarray}&&{g}_{{At}}=\displaystyle \frac{2{m}_{1}^{2}}{\rho }{\chi }_{A}^{d}+\displaystyle \frac{2}{3}{\rho }^{3}{{\mathcal{B}}}_{A}^{q}-{\rho }^{2}{m}_{1}{\gamma }^{d}{F}_{A}^{d},\end{eqnarray} \tag{ 5.1b }$$

$$\begin{eqnarray}&&{g}_{\rho \rho }=\left(1+2\displaystyle \frac{{m}_{1}}{\rho }+4{\left(\displaystyle \frac{{m}_{1}}{\rho }\right)}^{2}\right)-{\rho }^{2}{{\mathcal{E}}}^{q},\end{eqnarray} \tag{ 5.1c }$$

$$\begin{eqnarray}&&{g}_{{AB}}={\rho }^{2}{{\rm{\Omega }}}_{{AB}}\left[1-{\rho }^{2}{{\mathcal{E}}}^{q}\right],\end{eqnarray} \tag{ 5.1d }$$

in which the standard Boyer–Lindquist coordinates and the Regge–Wheeler gauge are adopted for the background and the perturbation, respectively. ${\gamma }^{d}$ is a gauge constant associated with a uniform rotation around the main body's spinning axis [11]. The perturbation terms are constructed from irreducible potentials $\left\{{{\mathcal{E}}}^{q},{{\mathcal{B}}}_{A}^{q},{\chi }_{A}^{d}\right\}$ that are linear in ${\chi }_{1},{{\mathcal{E}}}_{{ab}},{{\mathcal{B}}}_{{ab}},$ as well as bilinear potentials $\left\{{{\mathcal{F}}}_{A}^{d},{{\mathcal{K}}}^{d}\right\}$ that couple ${\chi }_{1}$ to ${{\mathcal{E}}}_{{ab}}$ or ${{\mathcal{B}}}_{{ab}}.$ The angular dependence of these potentials are shown to be

$$\begin{eqnarray}&&{{\mathcal{E}}}^{q}={{\mathcal{E}}}_{0}\left(1-3{\mathrm{cos}}^{2}\theta \right)+{{\mathcal{E}}}_{2}{\mathrm{sin}}^{2}\theta \mathrm{cos}2\left(\psi -\phi \right),\end{eqnarray} \tag{ 5.2a }$$

$$\begin{eqnarray}&&{{\mathcal{B}}}_{\theta }^{q}=\displaystyle \frac{1}{2}{{\mathcal{B}}}_{1}\mathrm{cos}\theta \mathrm{sin}\left(\phi -\psi \right),\end{eqnarray} \tag{ 5.2b }$$

$$\begin{eqnarray}&&{{\mathcal{B}}}_{\phi }^{q}=\displaystyle \frac{1}{2}{{\mathcal{B}}}_{1}\mathrm{sin}\theta \left(2{\mathrm{cos}}^{2}\theta -1\right)\mathrm{cos}\left(\phi -\psi \right),\end{eqnarray} \tag{ 5.2c }$$

$$\begin{eqnarray}&&{{\mathcal{F}}}_{A}^{d}=2{\chi }_{1}{{\mathcal{E}}}_{0}{\delta }_{A}^{\phi }{\mathrm{sin}}^{2}\theta ,\end{eqnarray} \tag{ 5.2d }$$

$$\begin{eqnarray}&&{\chi }_{A}^{d}=-{\chi }_{1}{\delta }_{A}^{\phi }{\mathrm{sin}}^{2}\theta ,\end{eqnarray} \tag{ 5.2e }$$

$$\begin{eqnarray}&&{{\mathcal{K}}}^{d}={{\mathcal{B}}}_{1}{\chi }_{1}\mathrm{sin}\theta \mathrm{cos}\left(\psi -\phi \right),\end{eqnarray} \tag{ 5.2f }$$

And the tidal moments are given by

$$\begin{eqnarray}&&{{\mathcal{E}}}_{0}=A\left[1+\displaystyle \frac{{m}_{1}}{2m}{u}^{2}-6\displaystyle \frac{{m}_{2}}{m}{\chi }_{2}{u}^{3}+O\left({u}^{4}\right)\right],\end{eqnarray} \tag{ 5.3a }$$

$$\begin{eqnarray}&&{{\mathcal{E}}}_{2}=3A\left[1-\displaystyle \frac{3{m}_{1}+4{m}_{2}}{2m}{u}^{2}-2\displaystyle \frac{{m}_{2}}{m}{\chi }_{2}{u}^{3}+O\left({u}^{4}\right)\right],\end{eqnarray} \tag{ 5.3b }$$

$$\begin{eqnarray}&&{{\mathcal{B}}}_{1}=3{Au}\left[1-\displaystyle \frac{{m}_{2}}{m}{\chi }_{2}u+O\left({u}^{2}\right)\right],\end{eqnarray} \tag{ 5.3c }$$

When the couplings of the star's spin vector and the tidal moments are applied to a circular binary, the tidal moments will incorporate terms that involve powers of u, the orbital velocity. It is related to r_o by the post-Newtonian relation

$$\begin{eqnarray}&&{u}^{2}=\displaystyle \frac{{Gm}}{{r}_{{\rm{o}}}}\left[1-\left(3-\eta \right)\displaystyle \frac{{Gm}}{{c}^{2}{r}_{{\rm{o}}}}+O{\left(\displaystyle \frac{{Gm}}{{c}^{2}{r}_{{\rm{o}}}}\right)}^{3/2}\right].\end{eqnarray} \tag{ 5.4 }$$

Here another mass combination $\eta := {m}_{1}{m}_{2}/{m}^{2}$ is introduced. It is assumed that ${u}^{2}\approx m/{r}_{{\rm{o}}}\ll {\chi }_{1}\ll 1;$ this ensures that the coupling terms between ${\chi }_{1}$ and the tidal quadrapole moments dominate over the time derivative of the tidal quadrupole moments as well as the tidal octupole moments.

We next follow the procedure described in previous sections to calculate the effect of rotation and/or companion body on the deviation of light. The nonvanishing components of Christoffel symbols, ${{\rm{\Gamma }}}_{\mu \nu }^{\rho }$ and ${{\rm{\Gamma }}}_{\mu \nu }^{\theta },$ built from metric (5.1) are listed in the appendix. In the calculation of Christoffel symbols the time derivatives of tidal potentials are neglected in favor of the short term changes in the photon path.

Under the stationary perturbation approximation the energy of the system, $\bar{E}=-\dot{t}{g}_{{tt}},$ is perturbationally constant on the photon path. Before moving on, it is worthwhile to examine the hypersurface $\theta =\pi /2.$ A glance at the appendix reveals that ${{\rm{\Gamma }}}_{\mu \nu }^{\theta }=0$ on this hypersurface, so the reasonings similar to those of section 3 show that ${F}^{\theta }=0$ on it, and the hypersurface $\theta =\pi /2$ is a photon surface. It seems that this occurs whenever rotations about the star's rotation axis are the source of corrections to the Schwarzschild path.

5.2. Transverse force and deviation angle

Using (5.2) and (5.3), the ρ-component of the perturbing force, (3.2), is given by

$$\begin{eqnarray}{F}^{\rho } & = & -\delta {{\rm{\Gamma }}}_{{tt}}^{\rho }{\dot{t}}^{2}-2\delta {{\rm{\Gamma }}}_{t\phi }^{\rho }\dot{t}\dot{\phi }-\delta {{\rm{\Gamma }}}_{\phi \phi }^{\rho }{\dot{\phi }}^{2}-\delta {{\rm{\Gamma }}}_{\rho \rho }^{\rho }{\dot{\rho }}^{2}-2\delta {{\rm{\Gamma }}}_{\rho \phi }^{\rho }\dot{\rho }\dot{\phi }\\ & = & 2A\displaystyle \frac{{L}_{*}^{2}}{{R}_{*}}\left\{\displaystyle \frac{{m}_{1}}{{R}_{*}}\left[1+3\mathrm{cos}\left(\phi -\psi \right)\right]\right.\\ & & +3\displaystyle \frac{{m}_{1}}{{R}_{*}}\sqrt{\displaystyle \frac{{m}_{1}}{{r}_{0}}}{\chi }_{1}\mathrm{cos}\left(\phi -\psi \right)-\displaystyle \frac{B}{A}\displaystyle \frac{{m}_{1}}{{R}_{*}}{\chi }_{1}{\mathrm{sin}}^{4}\left({{\rm{\Delta }}}_{*}-\phi \right)\\ & & -3\sqrt{\displaystyle \frac{m}{{r}_{0}}}\left(1-\displaystyle \frac{{m}_{2}}{m}\sqrt{\displaystyle \frac{m}{{r}_{0}}}{\chi }_{2}-\displaystyle \frac{1}{2}\left(3-\eta \right)\displaystyle \frac{m}{{r}_{0}}-\displaystyle \frac{{m}_{1}}{{R}_{*}}\mathrm{sin}\left(\phi -{{\rm{\Delta }}}_{*}\right)\right)\mathrm{cos}\left(\phi -\psi \right)\\ & & -2\displaystyle \frac{{m}_{1}}{{R}_{*}}{\gamma }^{d}{\chi }_{1}\mathrm{sin}\left(\phi -{{\rm{\Delta }}}_{*}\right)-\left[1+3\mathrm{cos}2\left(\phi -\psi \right)\right]\mathrm{sin}\left(\phi -{{\rm{\Delta }}}_{*}\right)\\ & & -\displaystyle \frac{{m}_{1}}{2{r}_{0}}\left(1-3\displaystyle \frac{3{m}_{1}+4{m}_{2}}{{m}_{1}}\mathrm{cos}2\left(\phi -\psi \right)\right)\mathrm{sin}\left(\phi -{{\rm{\Delta }}}_{*}\right)\\ & & -6\displaystyle \frac{{m}_{2}}{{r}_{0}}\sqrt{\displaystyle \frac{m}{{r}_{0}}}{\chi }_{2}\left[1+\mathrm{cos}2\left(\phi -\psi \right)\right]\mathrm{sin}\left(\phi -{{\rm{\Delta }}}_{*}\right)\\ & & +\displaystyle \frac{{m}_{1}}{2{R}_{*}}\left[3+\mathrm{cos}2\left(\phi -{{\rm{\Delta }}}_{*}\right)+{\mathrm{sin}}^{2}\left(\phi -{{\rm{\Delta }}}_{*}\right)\right]\left[1+3\mathrm{cos}2\left(\phi -\psi \right)\right]\\ & & -3\left[1-\displaystyle \frac{3{m}_{1}}{2{R}_{*}}\displaystyle \frac{\mathrm{sin}\left(\phi -{{\rm{\Delta }}}_{*}\right)\left(1+\frac{1}{3}\mathrm{cos}2\left(\phi -{{\rm{\Delta }}}_{*}\right)\right)}{{\mathrm{cos}}^{2}\left(\phi -{{\rm{\Delta }}}_{*}\right)}\right.\\ & & -\left.\displaystyle \frac{3{m}_{1}}{{R}_{*}}\mathrm{sin}\left(\phi -{{\rm{\Delta }}}_{*}\right)-\displaystyle \frac{3{m}_{1}+4{m}_{2}}{2{r}_{0}}-\displaystyle \frac{2{m}_{2}}{{r}_{0}}\sqrt{\displaystyle \frac{m}{{r}_{0}}}{\chi }_{2}\right]\\ & & \left.\times \mathrm{cos}\left(\phi -{{\rm{\Delta }}}_{*}\right)\mathrm{sin}2\left(\psi -\phi \right)\right\}.\end{eqnarray} \tag{ 5.5 }$$

Here $B:= -\frac{{m}_{1}}{2{r}_{{\rm{o}}}^{3}}$ and the relation ${k}^{2}:= {\left(\frac{{E}_{*}{R}_{*}}{{L}_{*}}\right)}^{2}=1-2{m}_{1}\mathrm{sin}\left(\phi -{{\rm{\Delta }}}_{*}\right){/R}_{*}$ have been used. There are some points about this result: (i) this force component has complicated angular dependences to the relative angle between the photon and companion body, $\phi -\psi ,$ as well as to the relative angle to the incoming asymptote, $\phi -{{\rm{\Delta }}}_{*}.$ (ii) Equation (5.5) reduces to the expression given in (3.11b), as ${m}_{\mathrm{1,2}}$ goes to zero but $A$ is kept fixed. (iii) The term proportional to B can be traced back to the the odd-parity rotational vector potential ${\chi }_{A}^{d}$ defined in (5.2e) and can be considered as a pure dragging force component. (iv) Equation (5.5) loses its validity in asymptotic values of ϕ.

The evolution of Δ can then be reproduced by substituting the preceding expression into (3.7b) and the explicit dependence on ψ at each ϕ is as follows

$$\begin{eqnarray}{\rm{\Delta }}\left(\phi \right) & = & {{AR}}_{*}^{2}\left\{15{\alpha }_{1}\mathrm{cos}\left(\phi -2\psi +{{\rm{\Delta }}}_{*}\right)\right.\\ & & -\left(\displaystyle \frac{{\alpha }_{1}}{2}+\displaystyle \frac{27}{2}{\alpha }_{1}\mathrm{cos}2\left({{\rm{\Delta }}}_{*}-\psi \right)+{\alpha }_{2}\right)\mathrm{cos}\left(\phi -{{\rm{\Delta }}}_{*}\right)\\ & & +\left(7{\alpha }_{1}\mathrm{cos}2\left({{\rm{\Delta }}}_{*}-\psi \right)+\displaystyle \frac{{\alpha }_{2}}{3}\right){\mathrm{cos}}^{3}\left(\phi -{{\rm{\Delta }}}_{*}\right)\\ & & +\left[3{\alpha }_{1}-{\alpha }_{3}\mathrm{sin}\left(\phi -{{\rm{\Delta }}}_{*}\right)+7{\alpha }_{1}{\mathrm{sin}}^{2}\left(\phi -{{\rm{\Delta }}}_{*}\right)\right]\mathrm{sin}\left(\phi -{{\rm{\Delta }}}_{*}\right)\mathrm{sin}2\left({{\rm{\Delta }}}_{*}-\psi \right)\\ & & \times 3{\alpha }_{1}\mathrm{sin}\left(\phi -\psi \right)+\displaystyle \frac{1}{2}{\alpha }_{3}\mathrm{sin}2\left(\phi -{{\rm{\Delta }}}_{*}\right)\mathrm{cos}2\left({{\rm{\Delta }}}_{*}-\psi \right)+{\alpha }_{4}\phi \\ & & +\left(\phi -{{\rm{\Delta }}}_{*}\right)\left[-{\alpha }_{5}\mathrm{sin}\left({{\rm{\Delta }}}_{*}-\psi \right)+{\alpha }_{3}\mathrm{cos}2\left({{\rm{\Delta }}}_{*}-\psi \right)\right]-\displaystyle \frac{1}{2}{\alpha }_{3}\mathrm{sin}2\left(\phi -\psi \right)\\ & & +\left[{\alpha }_{5}\mathrm{cos}\left({{\rm{\Delta }}}_{*}-\psi \right)+{\alpha }_{3}\mathrm{sin}2\left({{\rm{\Delta }}}_{*}-\psi \right)\right]\mathrm{ln}\left|\mathrm{sin}\left(\phi -{{\rm{\Delta }}}_{*}\right)\right|\\ & & +{\alpha }_{1}\left(1+3\mathrm{cos}2\left({{\rm{\Delta }}}_{*}-\psi \right)\right)\mathrm{ln}\left|\displaystyle \frac{1+\mathrm{cos}\left(\phi -{{\rm{\Delta }}}_{*}\right)}{\mathrm{sin}\left(\phi -{{\rm{\Delta }}}_{*}\right)}\right|\\ & & +\left.3{\alpha }_{1}\mathrm{sin}2\left({{\rm{\Delta }}}_{*}-\psi \right)\mathrm{ln}\left|\displaystyle \frac{1+\mathrm{sin}\left(\phi -{{\rm{\Delta }}}_{*}\right)}{\mathrm{cos}\left(\phi -{{\rm{\Delta }}}_{*}\right)}\right|\right\}.\end{eqnarray} \tag{ 5.6 }$$

The coefficients in this relation are defined as

$$\begin{eqnarray}{\alpha }_{1} & := & -2\displaystyle \frac{{m}_{1}}{{R}_{*}},\\ {\alpha }_{2} & := & 4\displaystyle \frac{{m}_{1}^{2}}{{{AR}}_{*}^{4}}{\chi }_{1},\\ {\alpha }_{3} & := & -6\left(1-\displaystyle \frac{3{m}_{1}+4{m}_{2}}{2{r}_{{\rm{o}}}}-\displaystyle \frac{2{m}_{2}}{{r}_{{\rm{o}}}}{\chi }_{2}\sqrt{\displaystyle \frac{m}{{r}_{{\rm{o}}}}}\right),\\ {\alpha }_{4} & := & 2+\displaystyle \frac{{m}_{1}}{{r}_{0}}-\displaystyle \frac{12{m}_{2}}{{r}_{0}}{\chi }_{2}\sqrt{\displaystyle \frac{m}{{r}_{{\rm{o}}}}}+4\displaystyle \frac{{m}_{1}}{{R}_{*}}{\gamma }^{d}{\chi }_{1},\\ {\alpha }_{5} & := & \sqrt{\displaystyle \frac{m}{{r}_{{\rm{o}}}}}\left(1+\displaystyle \frac{{m}_{2}}{m}{\chi }_{2}\sqrt{\displaystyle \frac{m}{{r}_{{\rm{o}}}}}+\displaystyle \frac{1}{2}\left(3-\eta \right)\displaystyle \frac{m}{{r}_{0}}-\displaystyle \frac{{m}_{1}}{{R}_{*}}{\chi }_{1}\right).\end{eqnarray} \tag{ 5.7 }$$

The total deflection due to tidal and dragging forces are given by

$$\begin{eqnarray}\delta & = & {{AR}}_{*}^{2}\left\{\right.{\alpha }_{1}+\displaystyle \frac{4}{3}{\alpha }_{2}+\pi {\alpha }_{4}-\mathrm{cos}2\left({{\rm{\Delta }}}_{*}-\psi \right)\left(17{\alpha }_{1}-\pi {\alpha }_{3}\right)\\ & & -\mathrm{sin}\left({{\rm{\Delta }}}_{*}-\psi \right)\left(6{\alpha }_{1}+\pi {\alpha }_{5}\right)\left.\right\}.\end{eqnarray} \tag{ 5.8 }$$

This result also reduces to the expression given in equation (4.2), as ${m}_{\mathrm{1,2}}$ goes to zero but $A$ is kept fixed. Extremum values of δ occur when $\mathrm{sin}\left({{\rm{\Delta }}}_{*}-\psi \right)=0$ or $\mathrm{sin}\left({{\rm{\Delta }}}_{*}-\psi \right)=\left(-1/4\right)\left(6{\alpha }_{1}+\pi {\alpha }_{5}\right)/\left(17{\alpha }_{1}-\pi {\alpha }_{3}\right).$ Equation (5.8) is the final outcome of the work carried out in the second part of this paper. The method exploited here can be extended to provide higher-order correction terms to δ due to tidal deformation of the main body.

In this paper, we considered a photon in the gravitational field of a binary system and found the form of trajectories invoking the weak field and slow motion approximations. In the first part, we found the magnitude of secular changes to the path of a light ray propagating in the field of a spherical gravitating body that arises in the presence of a companion body. The binary system of stars is allowed to be slowly rotating and a generalized component of tidal/dragging force which affects the deflection of the photon path is calculated in the second part of the article. This work illustrates the application of the osculating elements formalism to the computation of photon path evolutions in the Schwarzschild spacetime, as well as the tidal geometry around a black hole in reducing a photon-binary problem to an effective two-body problem. This work shows how a companion star and/or slow rotation deviates a light ray from the path dictated by the main lens.

It is known that edge-on binaries deflect the light more than others [9]; this criterion is further developed in this paper. Provided that the companion body is at a right distance and at a right orbital phase, in principle, it can be traced with a sensitive detector. The right distance, r_o, and phase, $\psi -{{\rm{\Delta }}}_{*},$ are determined by the geometry of spacetime around the main lens. Equation (3.12) indicates that edge-on binaries with ${r}_{{\rm{o}}}=O\left({\epsilon }^{-\displaystyle \frac{5}{4}}{m}_{2}\right)$ and $\psi -{{\rm{\Delta }}}_{*}=\left\{\frac{\pi }{2},\frac{3\pi }{2}\right\}$ can produce deflections of the order of $O\left({\epsilon }^{\displaystyle \frac{7}{4}}\right).$ Therefore, these corrections have potentially detectable effects on GAIA (or any other μas $\approx {10}^{-8}$ astrometric mission) observables if $\epsilon \geqslant {10}^{-5}.$ This condition holds for known typical binary systems. In this estimation, it is assumed that the photon passes by the edge of the star.

The importance of determining such corrections to Schwarzschild null geodesics can be understood when astrometric information is read off the light paths at a high precision level. The astrometric observations of a GAIA-like satellite has been modeled within the PPN formalism to post-Newtonian gravity. The sensitivity of such astronomy to some PPN parameters [13] as well as the light bending due to the quadrapole moment of an axisymmetric planet (like Jupiter) [14] has been established. In such feasibility studies, one must consider the light deflection in the vicinity of a star which is (i) a member of a binary system and (ii) crosses one of the astrometric fields during GAIA's mission. A comparison between the details of the background field corresponding to the time when the star is at the center of the field with that when it is not will help to extract the contribution of the binary system. The difference between the two patterns may then be fitted to the deflection model to determine whether the effect is detectable or not. In spite of the common belief that 50% of stars are a member of a binary (or multiple) system [15] and the order of correction terms is in GAIA's accuracy level, finding a favorable field and binary system with known ephemerids is a big challenge that is out of the scope of this paper. On the other hand, the trace of wide binary lenses studied in this paper may be found in microlensing light curves. If repeating microlensing peaks [16] appear in the microlensing light curves during GAIA's mission period, it will be a good signal for a binary candidate with ${m}_{1}\ll {r}_{{\rm{o}}}.$ Analytical results derived in this paper can be used in models fitted to the microlensing light curves. Such models provide a convenient framework for mutual verification of the results in an astrometric field common in different sky missions.

The nonvanishing Christoffel symbols built from metric (2.3) are

${{\rm{\Gamma }}}_{{tt}}^{t}=-{{\rm{\Gamma }}}_{\rho \rho }^{t}=\frac{1}{2}{\rho }^{2}\frac{\partial }{\partial t}{{\mathcal{E}}}^{q},$	${{\rm{\Gamma }}}_{{tt}}^{\rho }=-{{\rm{\Gamma }}}_{\rho \rho }^{\rho }=\frac{{m}_{1}}{{\rho }^{2}}+\rho {{\mathcal{E}}}^{q},$	${{\rm{\Gamma }}}_{{tt}}^{\theta }={{\rm{\Gamma }}}_{\rho \rho }^{\theta }=\frac{1}{2}\frac{\partial }{\partial \theta }{{\mathcal{E}}}^{q},$	$\begin{array}{l}{{\rm{\Gamma }}}_{{tt}}^{\phi }=-{{\rm{\Gamma }}}_{\rho \rho }^{\phi }=\frac{1}{2{\mathrm{sin}}^{2}\theta }\\ \times \frac{\partial }{\partial \phi }{{\mathcal{E}}}^{q},\end{array}$
${{\rm{\Gamma }}}_{t\rho }^{t}=\frac{{m}_{1}}{{\rho }^{2}}+\rho {{\mathcal{E}}}^{q},$	${{\rm{\Gamma }}}_{\rho t}^{\rho }=\frac{1}{2}{\rho }^{2}\frac{\partial }{\partial t}{{\mathcal{E}}}^{q},$	${{\rm{\Gamma }}}_{\theta t}^{\theta }=-\frac{1}{2}{\rho }^{2}\frac{\partial }{\partial t}{{\mathcal{E}}}^{q},$	${{\rm{\Gamma }}}_{\phi t}^{\phi }=-\frac{1}{2}{\rho }^{2}\frac{\partial }{\partial t}{{\mathcal{E}}}^{q},$
${{\rm{\Gamma }}}_{{tA}}^{t}=\frac{1}{2}{\rho }^{2}\frac{\partial }{\partial {\theta }^{A}}{{\mathcal{E}}}^{q},$	${{\rm{\Gamma }}}_{\rho A}^{\rho }=-\frac{1}{2}{\rho }^{2}\frac{\partial }{\partial {\theta }^{A}}{{\mathcal{E}}}^{q},$	${{\rm{\Gamma }}}_{\theta \rho }^{\theta }=\frac{1}{\rho }-\rho {{\mathcal{E}}}^{q},$	${{\rm{\Gamma }}}_{\phi \rho }^{\phi }=\frac{1}{\rho }-\rho {{\mathcal{E}}}^{q},$
${{\rm{\Gamma }}}_{{AB}}^{t}=-\frac{1}{2}{\rho }^{4}{{\rm{\Omega }}}_{{AB}}\frac{\partial }{\partial t}{{\mathcal{E}}}^{q},$	${{\rm{\Gamma }}}_{{AB}}^{\rho }={\rho }^{3}{{\rm{\Omega }}}_{{AB}}{{\mathcal{E}}}^{q},$	${{\rm{\Gamma }}}_{\theta A}^{\theta }=-\frac{1}{2}{\rho }^{2}\frac{\partial }{\partial {\theta }^{A}}{{\mathcal{E}}}^{q},$	$\begin{array}{l}{{\rm{\Gamma }}}_{\theta \theta }^{\phi }=\frac{1}{2}{\rho }^{2}{\mathrm{sin}}^{2}\theta \\ \times \frac{\partial }{\partial \phi }{{\mathcal{E}}}^{q},\end{array}$
	${{\rm{\Gamma }}}_{\phi \phi }^{\theta }=-\mathrm{sin}\theta \mathrm{cos}\theta$	${{\rm{\Gamma }}}_{\phi \phi }^{\phi }=-\frac{1}{2}{\rho }^{2}\frac{\partial }{\partial \phi }{{\mathcal{E}}}^{q}.$
		$+\frac{1}{2}{\rho }^{2}{\mathrm{sin}}^{2}\theta \frac{\partial }{\partial \theta }{{\mathcal{E}}}^{q},$

The nonvanishing Christoffel symbols built from metric (5.1) are

${{\rm{\Gamma }}}_{{tt}}^{\rho }=f\left[\frac{{m}_{1}}{{\rho }^{2}}+f\rho {{\mathcal{E}}}^{q}\left(1+\frac{{m}_{1}}{\rho }\right)-{m}_{1}{{\mathcal{K}}}^{d}\right],$	${{\rm{\Gamma }}}_{{tt}}^{\theta }=\frac{1}{2}\left(1-\frac{4{m}_{1}}{\rho }\right)\frac{\partial }{\partial \theta }{{\mathcal{E}}}^{q}-\frac{{m}_{1}}{\rho }\frac{\partial }{\partial \theta }{{\mathcal{K}}}^{d},$
${{\rm{\Gamma }}}_{{tA}}^{\rho }=f\left[\frac{{m}_{1}^{2}}{{\rho }^{2}}{\chi }_{A}^{d}-{\rho }^{2}{{\mathcal{B}}}_{A}^{q}+\rho {m}_{1}{\gamma }^{d}{{\mathcal{F}}}_{A}^{d}\right],$	${{\rm{\Gamma }}}_{\theta t}^{\theta }=-\frac{1}{2}{\rho }^{2}\frac{\partial }{\partial t}{{\mathcal{E}}}^{q},$
${{\rm{\Gamma }}}_{\rho \rho }^{\rho }=-\frac{{m}_{1}}{{\rho }^{2}}{f}^{-1}-\rho {{\mathcal{E}}}^{q}\left(1-\frac{{m}_{1}}{\rho }\right),$	${{\rm{\Gamma }}}_{t\rho }^{\theta }=-\frac{2{m}_{1}^{2}}{{\rho }^{4}}\left[1+\frac{{m}_{1}}{\rho }{f}^{-1}\right]{\chi }_{\theta }^{d}+\left[1-\frac{2{m}_{1}}{3\rho }{f}^{-1}\right]{{\mathcal{B}}}_{\theta }^{q}$
	$+\frac{{m}_{1}}{\rho }\left[1-\frac{{m}_{1}}{\rho }{f}^{-1}\right]{\gamma }^{d}{{\mathcal{F}}}_{\theta }^{d},$
${{\rm{\Gamma }}}_{\rho A}^{\rho }=-\frac{1}{2}f{\rho }^{2}{\partial }_{A}{{\mathcal{E}}}^{q},$	${{\rm{\Gamma }}}_{t\phi }^{\theta }=\frac{{m}_{1}^{2}}{{\rho }^{3}}\left(\frac{\partial }{\partial \phi }{\chi }_{\theta }^{d}-\frac{\partial }{\partial \theta }{\chi }_{\phi }^{d}\right)+\frac{1}{3}\rho \left(\frac{\partial }{\partial \phi }{{\mathcal{B}}}_{\theta }^{q}-\frac{\partial }{\partial \theta }{{\mathcal{B}}}_{\phi }^{q}\right)$
	$-\frac{1}{2}{m}_{1}{\gamma }^{d}\left(\frac{\partial }{\partial \phi }{{\mathcal{F}}}_{\theta }^{d}-\frac{\partial }{\partial \theta }{{\mathcal{F}}}_{\phi }^{d}\right),$
${{\rm{\Gamma }}}_{{AB}}^{\rho }=f{\rho }^{3}{{\rm{\Omega }}}_{{AB}}\left(2-f\right){{\mathcal{E}}}^{q},$	${{\rm{\Gamma }}}_{\rho \rho }^{\theta }=\frac{1}{2}\frac{\partial }{\partial \theta }{{\mathcal{E}}}^{q},$
	${{\rm{\Gamma }}}_{\theta \rho }^{\theta }=\frac{1}{\rho }-\rho {{\mathcal{E}}}^{q},$
	${{\rm{\Gamma }}}_{\theta A}^{\theta }=-\frac{1}{2}{\rho }^{2}\frac{\partial }{\partial {\theta }^{A}}{{\mathcal{E}}}^{q},$
	${{\rm{\Gamma }}}_{\phi \phi }^{\theta }=-\mathrm{sin}\theta \mathrm{cos}\theta +\frac{{\rho }^{2}}{2}{\mathrm{sin}}^{2}\theta \frac{\partial }{\partial \theta }{{\mathcal{E}}}^{q}.$

Here $f:= 1-\frac{2{m}_{1}}{\rho }$. In this list the time derivatives of the potentials have been neglected.