1 Introduction
The light propagation in the gravitational field serves key tests for the theories of gravitation. The light deflection, time delay and gravitational redshift are the fundamental predictions of general relativity (GR) and have been found in good agreement to the first post-Newtonian (PN) order with the observations.[1] With the development of technologies and the demands of the future observations, the higher PN effects on the light propagation have also been explored extensively, e.g., the light deflection,[2-15] the time delay,[16-18] and the gravitational redshift.[19, 20] At the same time, since the parametrized post-Newtonian (PPN) formalism[21, 22] can characterize the weak-field-limit metric of a broad spectrum of gravitation theories, thus it has been employed in making the theoretical predictions for the light propagation.[2-5, 7-9, 16, 23-28]
Similar to Klioner and Kopeikin's work on the motion of binary compact systems under a more generally parameterized 2PN acceleration,[29] in this work we derive the 2PN solution for the light propagation in a more generally PPN framework for the external field of the spherically-symmetric body. Compared to the conventional PPN models, this work includes more parameters, which makes the formulation of the light propagation be applicable to more gravitation theories at the 2PN order. Specifically, the combinations of these parameters may describe not only the weak-field-limit metrics of the Brans-Dicke theory (BDT) and the scalar-tensor theories discussed in Refs [25, 30, 31], but also those of the non-minimal Einstein-Yang-Mills theory and wormholes spacetime,[32-37] as well as the interaction between gravitational and electromagnetic fields beyond the Einstein-Maxwell theory.[38, 39]
Based on the solution of the light velocity, we further give the formula for the 2PN deflection of light for an interesting scenario in the astronomical applications, in which both the light emitter and receiver are located in the regions very far away from the body.
2 The Second-Order PPN Metric and Geodesic Equation for the Field of the Spherically-Symmetric Body
We consider the second-order PPN metric for the field of the spherically-symmetric body as follows,
$$ \!\!\!\! g_{00}=-1+\alpha\frac{2m}{r}-\beta\frac{ 2m^2}{r^2},\nonumber\\ \!\!\!\! g_{0i}=0,\nonumber\\ \!\!\!\! g_{ij}=\Big(1\!+\!\gamma \frac{2m}{r}+\epsilon\frac{ m^2}{r^2}\Big)\delta_{ij}\!+\!\Big(\sigma\frac{ 2 m}{r}+\varepsilon\frac{ m^2}{r^2}\Big)\frac{x^i x^j}{r^2}, $$
where the metric has signature of ($-+++$), and $m$ denotes the mass of body. The gravitational constant and the light speed in vacuum have been set as $1$. Latin indices $i$ and $j$ run from 1 to 3.$r\equiv |{x}|$ with ${x}\equiv(x^1, x^2, x^3)$ denoting the position vector of the field point $r\!\equiv\! |{x}|$ denotes the distance from the field position ${x}\!\equiv\!(x^1,x^2,x^3)$ to the body located at the coordinate origin. The PPN framework are characterized by the parameters $\alpha$, $\beta$, $\gamma$, $\sigma$, $\epsilon$ and $\varepsilon$.The parameter $\alpha$ is usually absorbed into the definition of the gravitational constant, while we keep it here for the completeness of a general second-order PPN metric, as Ref. [29] did for the general PPN acceleration. $\beta$, $\gamma$ and $\epsilon$ are the conventional PPN parameters.[16] The parameter $\varepsilon$ is introduced in Ref. [27] to include more gravitational theories. The discussions of the physical motivation for this parameter can be found in Refs. [25, 27] so we do not repeat them here. The parameter $\sigma$ is newly introduced in this work. Table 1 lists some representative PPN frameworks of the static spherically-symmetric body's field for light propagation. The relations between the parameters in these references and ours are also shown for readers' convenience.
The values of $\epsilon$ and $\varepsilon$ for some scalar-tensor theories and the Einstein-aether theory in the harmonic coordinates have been tabulated in Ref. [27].Table 2 gives the values of $\beta$, $\gamma$, $\sigma$ for GR and BDT in the harmonic coordinates. Here $\omega$ is a dimensionless constant of BDT, and when it's value goes to infinity BDT will reduce to GR.
The dynamics equation of the test particles including the photon for the metric form given by Eq. (1) can be written as
$$ \frac{\text{d}^2{x}}{\text{d}t^2}=-\frac{m}{r^3}{x}\Big\{\alpha\!-\!2(\alpha\gamma\!+\!\alpha\sigma\!+\!\beta)\frac{m}{r} \!+\!\Big[\!+\!(2\sigma\!+\!\gamma)\!+\!(4\sigma^2\!+\!6\gamma\sigma\!+\!2\gamma^2\!-\!\varepsilon\!-\! \epsilon)\frac{m}{r}\Big]\Big|\frac{\text{d}{x}}{\text{d}t}\Big|^2\nonumber\\ -\Big[3\sigma\!+\!2(\varepsilon\!-\!5\gamma\sigma\!-\!3\sigma^2)\frac{m}{r}\Big]\!\Big(\!\frac{{x}}{r}\!\cdot\!\frac{\text{d}{x}}{\text{d}t}\!\Big)^{\!\!2}\Big\}\!+\frac{2m}{r^2}\frac{\text{d}{x}}{\text{d}t}\Big[(\gamma\!+\!\alpha)\!+\!(\epsilon\!-\!2\gamma^2\!+\!2\alpha^2\!-\!2\beta)\frac{m}{r}\Big]\!\Big(\!\frac{{x}}{r}\!\cdot\!\frac{\text{d}{x}}{\text{d}t}\!\Big)\!. $$
For the photon's motion, it also satisfies the null-geodesic condition, which in the 2PN approximation reads
$ -1+\alpha\frac{2m}{r}-\beta\frac{2m^2}{r^2}+\Big(1+\gamma\frac{2m}{r}+\epsilon\frac{m^2}{r^2}\Big)\Big|\frac{\text{d}{x}}{\text{d}t}\Big|^2+\Big(\sigma\frac{2m}{r}+\varepsilon \frac{m^2}{r^2}\Big)\Big(\frac{{x}}{r}\!\cdot\!\frac{\text{d}{x}}{\text{d}t}\Big)^2=0 .$
Table 1 Comparisons of some representative PPN frameworks of the spherically-symmetric body's field for light propagation. |
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Table 2 The values of $\beta, \gamma, \sigma$ for GR and BDT with $\omega$ being a dimensionless constant in the harmonic coordinates.[40] |
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From this equation we can obtain the photon's velocity when the position and velocity direction of the photon are given. This equation can be employed to simplify the derivation of the light propagation when solving the dynamics equation.
3 The 2PN Solution for the Light Propagation
We assume there is a photon emitted at the coordinate time $t_{\rm e}$ at the position ${x}_{\rm e}$ with an initial velocity direction described by the unit vector ${n}$.
For the zeroth-order order (Minkowskian spacetime), Eqs. (2) and (3) reduce to
$$ \frac{\text{d}^2{x}}{\text{d}t^2} =0\,,\quad\Big|\frac{\text{d}{x}}{\text{d}t}\Big|= 1\,, $$
and the corresponding solutions (Newtonian solution) are
$$ \frac{d{x}_{\rm N}}{d t} ={n}\,, $$
$$ \label{x-Newton}{x}_{\rm N} = {x}_{\rm e} + {n}(t -t_{\rm e}) , $$
where ${x}_{\rm N}$ denote the Newtonian trajectory.
Before we present the PN solutions, we introduce an important parameter for the light deflection in the gravitational field --- the impact vector ${b}$, which joins the body's center and the point of the closest approach in the line of ${x}_{\rm N}$, and whose amplitude $b \equiv |{b}|$ is well-known as the impact parameter.[40, 41]
To the 1PN accuracy, Eqs. (2) and (3) reduce to
$$ \frac{\text{d}^2{x}}{\text{d}t^2}=-\Big[\alpha+(2\sigma+\gamma)\Big|\frac{\text{d}{x}}{\text{d}t}\Big|^2-3\sigma\Big(\frac{{x}}{r}\!\cdot\!\frac{\text{d}{x}}{\text{d}t}\Big)^{\!2}\Big]\frac{m{x}}{r^3}+2(\gamma+\alpha)\Big(\frac{{x}}{r}\!\cdot\!\frac{\text{d}{x}}{\text{d}t}\Big)\frac{m}{r^2}\frac{\text{d}{x}}{\text{d}t}, $$
$$ -1+\alpha\frac{2m}{r}+\Big(1+\gamma\frac{2m}{r}\Big)\Big|\frac{\text{d}{x}}{\text{d}t}\Big|^2+\sigma\frac{2m}{r}\Big(\frac{{x}}{r}\!\cdot\!\frac{\text{d}{x}}{\text{d}t}\Big)^2=0 ,\label{eq:null-1PN} $$
and the corresponding solutions can be written as the following form
$$ {x} = {x}_{\rm N} + {x}_{\rm 1PN}, $$
with ${x}_{\rm 1PN}$ being the first-order post-Newtonian term.
Substituting Eqs. (6) and (9) into Eqs. (7) and (8), and only keeping the 1PN terms, we can obtain
$$ \label{eq:geodesic-1PNOnly} \frac{\text{d}^2{x}_{\rm 1PN}}{\text{d}t^2} =\frac{m}{|{x}_{\rm N}|^3}\Big\{\!-\!\Big[(\alpha+\gamma-\sigma)+3\sigma \frac{ b^2}{|{x}_{\rm N}|^2}\Big]{x}_{\rm N}+2(\alpha+\gamma)({n}\!\cdot\!{x}_{\rm N}){n}\Big\}. $$
$$ {n}\!\cdot\!\frac{\text{d}{x}_{\rm 1PN}}{\text{d}t}=-(\alpha+\gamma+\sigma)\frac{m}{|{x}_{\rm N}|}+\sigma\frac{b^2m}{|{x}_{\rm N}|^3}. $$
We decompose ${x}_{\rm 1PN}$ into the components being parallel and perpendicular to ${n}$:
$$ {x}_{\rm 1PN\parallel} ={n}({n}\cdot{x}_{\rm 1PN}), $$
$$ {x}_{\rm 1PN\perp} = {x}_{\rm 1PN} - {n}({n}\cdot{x}_{\rm 1PN}). $$
Eqs. (10)$-$(11) then yield
$$ \frac{\text{d}^2{x}_{\rm 1PN\perp}}{\text{d}t^2} = {b}\Big[\!-\!(\alpha+\gamma-\sigma)\frac{m}{|{x}_{\rm N}|^3}-3\sigma \frac{ b^2 m }{|{x}_{\rm N}|^5}\Big], $$
$$ \frac{\text{d}{x}_{\rm 1PN\parallel}}{\text{d}t} = {n}\Big[\!-\!(\alpha+\gamma+\sigma)\frac{m}{|{x}_{\rm N}|}+\sigma\frac{ b^2m}{|{x}_{\rm N}|^3}\Big]. $$
Integrating Eq. (14), we can get
$$ \frac{\text{d}{x}_{\rm 1PN\perp}}{\text{d}t}={b}\Big[\!-\!(\alpha+\gamma+\sigma)\frac{m}{b^2}\Big(\frac{{n}\!\cdot\!{x}_{N}}{|{x}_{N}|} -\frac{{n}\!\cdot\!{x}_{\rm e}}{|{x}_{e}|}\Big)-\sigma m\Big(\frac{{n}\!\cdot\!{x}_{\rm N}}{|{x}_{\rm N}|^3} -\frac{{n}\!\cdot\!{x}_{\rm e}}{|{x}_{e}|^3}\Big)\Big]. $$
Combining Eqs. (15) and (16), we have the velocity of the particle to the 1PN accuracy
$$ \frac{\text{d}{x}_{\rm 1PN}}{\text{d}t}={n}\Big[\!-\!(\alpha+\gamma+\sigma)\frac{m}{|{x}_{\rm N}|}+\sigma \frac{ b^2m}{|{x}_{\rm N}|^3}\Big]+{b}\Big[\!-\!(\alpha+\gamma+\sigma)\frac{m}{b^2}\Big(\frac{{n}\!\cdot\!{x}_{\rm N}}{|{x}_{\rm N}|}-\frac{{n}\!\cdot\!{x}_{e}}{|{x}_{e}|}\Big)-\sigma m\Big(\frac{{n}\!\cdot\!{x}_{\rm N}}{|{x}_{\rm N}|^3} -\frac{{n}\!\cdot\!{x}_{\rm e}}{|{x}_{\rm e}|^3}\Big)\Big]. $$
Integrating Eq. (17), we can obtain the 1PN contributions to the photon's trajectory as follows,
$$ {x}_{\rm 1PN}=m{n}\Big[\!-(\alpha+\gamma+\sigma)\ln\!{\frac{|{x}_{\rm N}|+\!{n}\!\cdot\!{x}_{\rm N}}{|{x}_{\rm e}|+\!{n}\!\cdot\!{x}_{\rm e}}}+\sigma \Big(\frac{{n}\!\cdot\!{x}_{\rm N}}{|{x}_{\rm N}|}-\frac{{n}\!\cdot\!{x}_{\rm e}}{|{x}_{\rm e}|}\Big)\Big]\nonumber\\ \hskip 1cm + m\frac{{b}}{b}\Big[\!-\!(\alpha+\gamma+\sigma) \Big(1\!-\!\frac{{x}_{ \rm e}\!\cdot\!{x}_{\rm N}}{|{x}_{\rm e}||{x}_{\rm N}|}\Big)\frac{|{x}_{\rm N}|}{b}+\sigma \Big(\frac{1}{|{x}_{\rm N}|}\!-\!\frac{2}{|{x}_{\rm e}|}\!+\!\frac{{x}_{\rm e}\!\cdot\!{x}_{\rm N}}{|{x}_{\rm e}|^3}\Big)b\Big]. $$
To the 2PN accuracy, the solution of Eqs. (2) and (3) can be written as
$$ {x} ={x}_{\rm N}+ {x}_{\rm 1PN}+ {x}_{\rm 2PN}, $$
with ${x}_{\rm 2PN}$ being the 2PN correction.
Substituting Eq. (19) into Eq. (2), making use of Eqs. (6), (10), (17), (18), we can obtain
$$ \frac{\text{d}^2{x}_{\rm 2PN}}{\text{d}t^2}=\frac{{n}m^2}{|{x}_{\rm N}|^3}\Big\{(\alpha\!+\!\gamma\!+\!\sigma)\!\Big[2(\alpha\!+\!\gamma\!+\!\sigma)\!-\!3(\alpha\!+\!\gamma\!+\!5\sigma)\frac{b^2}{|{x}_{\rm N}|^2}\!+\!15\sigma\frac{ b^4}{|{x}_{\rm N}|^4}\Big]\!\ln\frac{|{x}_{\rm N}|+{n}\!\cdot\!{x}_{\rm N}}{|{x}_{\rm e}|+{n}.{x}_{\rm e}}\nonumber\\ \hskip0.5cm -(\alpha^2\!\!+\!2\alpha\sigma\!+\!2\alpha\gamma\!+\!\gamma^2\!\!+\!2\sigma\gamma\!+\!\sigma^2)\!\frac{{n}\!\cdot\!{x}_{\rm e}}{|{x}_{\rm e}|}+(\alpha^2\!\!-\!2\beta\!-\!2\alpha\gamma\!-\!3\gamma^2\!\!-\!2\alpha\sigma\!-\!4\sigma^2\!\!-\!6\sigma\gamma\!+\!\varepsilon\!+\!\epsilon)\!\frac{{n}\!\cdot\!{x}_{\rm N}}{|{x}_{\rm N}|}\nonumber\\ \hskip0.5cm -\frac{[3(\alpha\!+\!\gamma)^2\!+\!6\alpha\sigma\!+\!6\sigma\gamma\!-\!9\sigma^2]b^2({n}\!\cdot\!{x}_{\rm N})-[3(\alpha+\gamma)^2\!+\!18\alpha\sigma\!+\!18\sigma\gamma\!+\!3\sigma^2]b^2({n}\!\cdot\!{x}_{\rm e})}{|{x}_{\rm N}|^2|{x}_{\rm e}|}\nonumber\\-\frac{(\!\alpha\sigma\!-\!7\sigma\gamma\!-\!5\sigma^2\!+\!2\varepsilon)b^2({n}\!\cdot\!{x}_{\rm N})}{|{x}_{\rm N}|^3}-\frac{3\sigma(\alpha\!+\!\gamma\!+\!3\sigma)b^4({n}\!\cdot\!{x}_{\rm N})-3\sigma(\alpha\!+\!\gamma\!+\!18\sigma)b^4({n}\!\cdot\!{x}_{\rm e})}{|{x}_{\rm N}|^2|{x}_{\rm e}|^3}\nonumber\\-\frac{\sigma(\alpha\!+\!\gamma\!+\!3\sigma)b^2({n}\!\cdot\!{x}_{\rm e})}{|{x}_{\rm e}|^3}+\frac{15\sigma(\alpha\!+\!\gamma)b^4{n}\!\cdot\!({x}_{\rm N}\!-\!{x}_{\rm e})\!-\!15\sigma^2b^4({n}\!\cdot\!{x}_{\rm N})}{|{x}_{\rm N}|^4|{x}_{\rm e}|}+\frac{15\sigma^2b^6{n}\!\cdot\!({x}_{\rm N}\!-\!{x}_{\rm e})}{|{x}_{\rm N}|^4|{x}_{\rm e}|^3}\Big\}\nonumber\\+\frac{{b}m^2}{|{x}_{\rm N}|^3}\Big\{\!\!-\!\frac{(\alpha\!+\!\gamma\!+\!\sigma)^2|{x}_{\rm N}|}{b^2}\!-\!\Big[3(\alpha\!+\!\gamma)^2\!-\!3\sigma^2\!+\!15\sigma(\alpha\!+\!\gamma\!+\!\sigma)\frac{b^2}{|{x}_{\rm N}|^2}\Big]\frac{{n}\!\cdot\!{x}_{\rm N}}{|{x}_{ \rm N}|^2}\ln\!\frac{|{x}_{\rm N}|+{n}\!\cdot\!{x}_{\rm N}}{|{x}_{\rm e}|+{n}\!\cdot\!{x}_{\rm e}}\nonumber\\-\frac{(\alpha^2\!+\!2\alpha\gamma\!+\!\gamma^2\!+\!\sigma^2)}{|{x}_{\rm e}|}\!-\!\frac{(\alpha^2\!-\!2\beta\!-\!2\alpha\gamma\!-\!3\gamma^2\!-\!5\alpha\sigma\!-\!\sigma\gamma\!-\!\varepsilon\!+\!\epsilon)}{|{x}_{\rm N}|}\!-\!\frac{(\alpha\sigma\!-\!7\sigma\gamma\!-\!5\sigma^2\!+\!2\varepsilon)b^2}{|{x}_{\rm N}|^3}\nonumber\\+ \frac{{x}_{\rm N}\!\cdot\!{x}_{\rm e}}{|{x}_{\rm e}|}\Big[\frac{(\alpha\!+\!\gamma\!+\!\sigma)^2}{b^2}\!+\!\frac{3(\alpha\!+\!\gamma\!-\!\sigma)^2}{|{x}_{\rm N}|^2}\!+\!\frac{\sigma(\alpha\!+\!\gamma\!+\!\sigma)}{|{x}_{\rm e}|^2}\!+\!\frac{3\sigma(\alpha+\gamma\!-\!2\sigma)b^2}{|{x}_{\rm N}|^2|{x}_{\rm e}|^2}+\!\frac{15\sigma(\alpha\!+\!\gamma)b^2}{|{x}_{\rm N}|^4}\nonumber\\+\frac{15\sigma^2b^4}{|{x}_{\rm N}|^4|{x}_{\rm e}|^2}\Big]-\Big[\sigma(9\alpha\!+\!9\gamma\!-\!15\sigma)\!+\!\frac{15\sigma^2b^2}{|{x}_{\rm N}|^2}\Big]\frac{b^2}{|{x}_{\rm N}|^2|{x}_{\rm e}|}\!-\!\Big[2\sigma(\alpha\!+\!\gamma)\!+\!\frac{6\sigma^2b^2}{|{x}_{\rm N}|^2}\Big]\frac{b^2}{|{x}_{\rm e}|^3}\Big\}. $$
Similarly, we can decompose ${x}_{\rm 2PN}$ into the components being parallel and perpendicular to ${n}$:
$$ {x}_{\rm 2PN\parallel} ={n}({n}\cdot{x}_{\rm 2PN}), $$
$$ {x}_{\rm 2PN\perp} = {x}_{\rm 2PN} - {n}({n}\cdot{x}_{\rm 2PN}). $$
Integrating Eq. (20) for the components being perpendicular to ${n}$, we can obtain
$$ \frac{\text{d}{x}_{\rm 2PN\perp}}{\text{d}t}=\frac{m^2{b}}{b^3}\Big\{\Big[(\alpha\!+\!\gamma)^2\!-\!\sigma^2\!+\!3\sigma(\alpha\!+\!\gamma\!+\!\sigma)\frac{b^2}{|{x}_{\rm N}|^2}\Big]\frac{b^3}{|{x}_{\rm N}|^3} \ln\frac{|{x}_{\rm N}|+\!{n}\!\cdot\!{x}_{\rm N}}{|{x}_{\rm e}|\!+\!{n}\cdot{x}_{\rm e}}\nonumber\\ +\Big(2\alpha^2\!+\!2\alpha\gamma\!-\!\beta\!+\!\alpha\sigma\!-\!\frac{\sigma^2}{4}\!+\!\frac{\epsilon}{2}\!+\!\frac{\varepsilon}{4}\Big)\Big(\!\arccos\frac{{n}\!\cdot\!{x}_{\rm N}}{|{x}_{\rm N}|}\!-\!\arccos\frac{{n}\!\cdot\!{x}_{\rm e}}{|{x}_{\rm e}|}\Big)\nonumber\\ +\Big(\!\alpha^2\!\!-\!\beta\!-\!\gamma^2\!\!+\!\alpha\sigma\!+\!\frac{3\sigma^2}{4}\!+\!\frac{\epsilon}{2}\!+\!\frac{\varepsilon}{4}\Big)\!\Big(\frac{{n}\!\cdot\!{x}_{\rm e}}{|{x}_{\rm e}|^2}\!-\!\frac{{n}\!\cdot\!{x}_{\rm N}}{|{x}_{\rm N}|^2}\Big)b\!-\!\Big(\!2\sigma\gamma\!+\!\frac{3\sigma^2}{2}\!-\!\frac{\varepsilon}{2}\!-\!\frac{\sigma^2b^2}{|{x}_{\rm e}|^2}\Big)\frac{({n}\!\cdot\!{x}_{\rm e})b^3}{|{x}_{\rm e}|^4}\nonumber\\ +\Big[(\alpha\!+\!\gamma)^2\!-\!\sigma^2\!-\!3\sigma^2\frac{b^4}{|{x}_{\rm N}|^4}\!+\!2\sigma(\alpha\!+\!\gamma\!+\!\sigma)\frac{|{x}_{\rm e}|}{|{x}_{\rm N}|}\Big]\frac{({n}\!\cdot\!{x}_{\rm N})b}{|{x}_{\rm N}||{x}_{\rm e}|}\!-\!\Big(\alpha\sigma\!-\!\sigma\gamma\!-\!\frac{\sigma^2}{2}\!+\!\frac{\varepsilon}{2}\Big)\frac{({n}\!\cdot\!{x}_{\rm N})b^3}{|{x}_{\rm N}|^4}\nonumber\\ +\frac{{n}\!\cdot\!({x}_{\rm N}\!\!-\!{x}_{\rm e})b^3}{|{x}_{\rm N}||{x}_{\rm e}|^3}\!\Big[\sigma(\alpha\!+\!\gamma\!+\!\sigma)\!+\!\frac{3\sigma^2b^4}{|{x}_{\rm N}|^4}\!+\!\sigma(\alpha\!+\!\gamma)\Big(\!1\!+\!3\frac{|{x}_{\rm e}|^2}{|{x}_{\rm N}|^2}\!\Big)\!\frac{b^2}{|{x}_{\rm N}|^2}\!+\!(\alpha\!+\!\gamma)(\alpha\!+\!\gamma\!-\!\sigma)\!\frac{|{x}_{\rm e}|^2}{|{x}_{\rm N}|^2}\Big]\nonumber\\ +\Big[\!-\!(\alpha\!+\!\gamma\!+\!\sigma)^2+(\alpha\sigma\!+\!\sigma\gamma\!+\!\sigma^2)\frac{b^2}{|{x}_{\rm N}|^2}\!+\!2\sigma^2\frac{b^4}{|{x}_{\rm N}|^2|{x}_{\rm e}|^2}Big]\frac{({n}\!\cdot\!{x}_{\rm e})b}{|{x}_{\rm N}||{x}_{\rm e}|}\Big\}. $$
Substituting Eqs. (19) and (22) into Eq. (3), and making use of Eqs. (6), (17) and (18), we can obtain
$ \frac{\text{d}{x}_{\rm 2PN\parallel}}{\text{d}t}=\frac{m^2}{b^2}{n}\Big\{\!-\!(\alpha\!+\!\gamma\!+\!\sigma)^2\!-\!(\alpha\!+\!\gamma\!+\!\sigma)\Big[(\alpha\!+\!\gamma\!+\!\sigma)\!-\!\frac{3\sigma b^2}{|{x}_{\rm N}|^2}\Big]\frac{m^2({n}\!\cdot\!{x}_{\rm N})}{|{x}_{\rm N}|^3}\ln\!{\frac{|{x}_{\rm N}|+\!{n}\!\cdot\!{x}_{\rm N}}{|{x}_{\rm e}|+\!{n}\!\cdot\!{x}_{\rm e}}}\nonumber\\ \quad -(\alpha\!+\!\gamma\!+\!\sigma)^2\frac{b^2}{|{x}_{\rm N}||{x}_{\rm e}|}\!-\!\Big(\alpha^2\!\!-\!\beta\!-\!\gamma^2\!\!-\!2\sigma\gamma\!-\!\sigma^2\!\!+\!\frac{\epsilon}{2}\!+\!\frac{\varepsilon}{2}\Big)\frac{b^2}{|{x}_{\rm N}|^2}\!+\!\frac{1}{2}[(\alpha\!+\!\gamma)^2\!-\!\sigma^2]\frac{b^2}{|{x}_{\rm e}|^2}\nonumber\\ \quad +\frac{b^2({x}_{\rm N}\!\!\cdot\!{x}_{\rm e})}{|{x}_{\rm N}|^3|{x}_{\rm e}|}\Big[\!(\alpha\!+\!\gamma\!+\!\sigma)^2\!\Big(\!1\!+\!\frac{|{x}_{\rm N}|^2}{b^2}\!\Big)\!+\!\sigma(\alpha\!+\!\gamma\!+\!\sigma)\frac{|{x}_{\rm N}|^2\!\!+\!b^2}{|{x}_{\rm e}|^2}\!-\!\frac{3\sigma(\alpha\!+\!\gamma)b^2}{|{x}_{\rm N}|^2}\!+\!\frac{\sigma^2b^2}{|{x}_{\rm e}|^2}\Big(\!1\!-\!\frac{3b^2}{|{x}_{\rm N}|^2}\!\Big)\!\Big]\nonumber\\ \quad +\frac{b^4}{|{x}_{\rm N}|^4}\Big[\!\Big(\alpha\sigma\!-\!\sigma\gamma\!-\!\frac{\sigma^2}{2}\!+\!\frac{\varepsilon}{2}\Big)\!-\!4\sigma^2\frac{|{x}_{\rm N}|}{|{x}_{\rm e}|}\!-\!\sigma(\alpha\!+\!\gamma\!+\!\sigma)\frac{|{x}_{\rm N}|^3}{|{x}_{\rm e}|^3}\!+\!\frac{\sigma^2b^2}{2|{x}_{\rm N}|^2}\!+\!\sigma^2\frac{b^2|{x}_{\rm N}|}{|{x}_{\rm e}|^3}\!+\!\frac{3\sigma^2b^2}{|{x}_{\rm N}||{x}_{\rm e}|}\Big]\Big\}.$
Finally, the 2PN correction to the trajectory ${x}_{\rm 2PN}$ can be achieved via integrating Eqs. (23)$-$(24) as follows
$ {x}_{\rm 2PN}=\frac{m^2}{b}{n}\Big\{\!(\alpha\!+\!\gamma\!+\!\sigma)\Big(\!\alpha\!+\!\gamma\!+\!\sigma\!-\!\frac{\sigma b^2}{|{x}_{\rm N}|^2}\!\Big)\!\frac{b}{|{x}_{\rm N}|}\!\ln\!{\frac{|{x}_{\rm N}|\!+\!{n}\!\cdot\!{x}_{\rm N}} {|{x}_{\rm e}|\!+\!{n}\!\cdot\!{x}_{\rm e}}}\!+\!\Big(\!\alpha\sigma\!+\!\frac{\sigma^2}{4}\!+\!\frac{\varepsilon}{4}\!\Big)\!\Big(\!\frac{{n}\!\cdot\!{x}_{\rm N}}{|{x}_{\rm N}|^2}\!-\!\frac{{n}\!\cdot\!{x}_{\rm e}}{|{x}_{\rm e}|^2}\!\Big)b\nonumber\\+\Big(\!2\alpha^2\!+\!2\alpha\gamma\!-\!\beta\!+\!\alpha\sigma\!-\!\frac{\sigma^2}{4}\!+\!\frac{\epsilon}{2}\!+\!\frac{\varepsilon}{4}\Big)\Big(\!\arccos\frac{{n}\!\cdot\!{x}_{\rm N}}{|{x}_{\rm N}|}\!-\!\arccos\frac{{n}\!\cdot\!{x}_{\rm e}}{|{x}_{\rm e}|}\!\Big)\!+\![(\alpha+\gamma)^{\!2}\!-\!\sigma^2]\frac{b({n}\!\cdot\!{x}_{\rm N})}{|{x}_{\rm N}||{x}_{\rm e}|}\nonumber\\-\Big[(\alpha\!+\!\gamma)^{\!2}\!+\!2\sigma(\alpha\!+\!\gamma\!+\!\frac{\sigma}{2})\Big]\!\frac{b({n}\!\cdot\!{x}_{\rm e})}{|{x}_{\rm N}||{x}_{\rm e}|}\!-\!(\alpha\!+\!\gamma\!+\!\sigma)^2\!\frac{|{x}_{\rm N}|}{b}\!\Big(\!\frac{{n}\!\cdot\!{x}_{\rm N}}{|{x}_{\rm N}|}\!-\!\frac{{n}\!\cdot\!{x}_{\rm e}}{|{x}_{\rm e}|}\!\Big)\!-\!\sigma(\!\alpha\!-\!\sigma\!+\!\gamma)\frac{b^3({n}\!\cdot\!{x}_{\rm N})}{|{x}_{\rm N}|^3|{x}_{\rm e}|}\nonumber\\+ \frac{1}{2}[(\alpha\!+\!\gamma)^2\!-\!\sigma^2]\frac{b({n}\!\cdot\!{x}_{\rm N})}{|{x}_{\rm e}|^2}\!-\!\Big[\frac{1}{2}(\alpha\!+\!\gamma)^2\!-\!2\sigma\Big(\!\alpha\!+\!\gamma\!+\!\frac{5}{4}\sigma\!\Big)\Big]\frac{b({n}\!\cdot\!{x}_{\rm e})}{|{x}_{\rm e}|^2}\Big]\!+\!\sigma(\!\alpha\!+\!\gamma\!+\!\sigma)\frac{b^3({n}\!\cdot\!{x}_{\rm N})}{|{x}_{\rm N}||{x}_{\rm e}|^3}\nonumber\\- \sigma^2\frac{b^5[{n}\!\cdot\!({x}_{\rm N}\!-\!{x}_{\rm e})]}{|{x}_{\rm e}|^3}\!\Big(\!\frac{1}{|{x}_{\rm N}|^3}\!-\!\frac{1}{2|{x}_{\rm e}|^3}\!\Big)\!-\!\sigma(\alpha\!+\!\gamma\!+\!2\sigma)\!\frac{b^3({n}\!\cdot\!{x}_{\rm e})}{|{x}_{\rm N}||{x}_{\rm e}|^3}\!+\!\sigma(\alpha\!+\!\gamma\!+\!\sigma)\!\frac{b({n}\!\cdot\!{x}_{\rm e})}{|{x}_{\rm e}|^2}\!\Big(\!\frac{|{x}_{\rm N}|}{|{x}_{\rm e}|}\!-\!1\!\Big)\!\Big\}\nonumber\\+ \frac{m^2}{b^2}{b}\,\Big\{\Big[(\alpha\!+\!\gamma\!+\!\sigma)^2\Big(\frac{{n}\!\cdot\!{x}_{\rm N}}{|{x}_{\rm N}|}\!-\!\frac{{n}\!\cdot\!{x}_{\rm e}}{|{x}_{\rm e}|}\!\Big)\!+\!\sigma(\alpha\!+\!\gamma\!+\!\sigma)\Big(\frac{{n}\!\cdot\!{x}_{\rm N}}{|{x}_{\rm N}|^3}\!-\!\frac{{n}\!\cdot\!{x}_{\rm e}}{|{x}_{\rm e}|^3}\Big)b^2\Big]\ln\!{\frac{|{x}_{\rm N}|\!+\!{n}\!\cdot\!{x}_{\rm N}}{|{x}_{\rm e}|\!+\!{n}\!\cdot\!{x}_{\rm e}}}\nonumber\\+\Big(\!2\alpha^2\!+\!2\alpha\gamma\!-\!\beta\!+\!\alpha\sigma\!-\!\frac{\sigma^2}{4}\!+\!\frac{\epsilon}{2}\!+\!\frac{\varepsilon}{4}\Big)\!\Big(\!\!\arccos\frac{{n}\!\cdot\!{x}_{ \rm N}}{|{x}_{\rm N}|}\!-\!\arccos\frac{{n}\!\cdot\!{x}_{\rm e}}{|{x}_{\rm e}|}\!\Big)\!\frac{{n}\!\cdot\!{x}_{\rm N}}{b}\!+\!\Big(\!\alpha\sigma\!+\!\frac{\sigma^2}{4}\!+\!\frac{\varepsilon}{4}\Big)\!\frac{b^2}{|{x}_{\rm N}|^2}\nonumber\\- \Big[\sigma(\alpha\!+\!\gamma)\! +\! \sigma^2\frac{b^2}{|{x}_{\rm e}|^2}\! - \!\sigma^2\frac{b^2 |{x}_{\rm e}|}{|{x}_{\rm N}|^3}+\!\Big(\!2\sigma\gamma\!+\!\frac{3\sigma^2}{2}\!-\!\frac{\varepsilon}{2}\Big)\!\frac{|{x}_{\rm N}|^3}{|{x}_{\rm e}|^3}\Big]\frac{b^2({x}_{\rm N}\!\cdot\!{x}_{\rm e})}{|{x}_{\rm N}|^3|{x}_{\rm e}|}\!+\!\sigma(\!\alpha\!+\!\gamma\!+\!\sigma)\frac{b^2|{x}_{\rm N}|}{|{x}_{\rm e}|^3}\nonumber\\\Big(\alpha^2\!-\!\gamma^2\!-\!\beta\!+\!\alpha\sigma\!+\!\frac{3\sigma^2}{4}\!+\!\frac{\epsilon}{2}\!+\!\frac{\varepsilon}{4}\Big)\Big(\frac{{x}_{\rm N}\!\cdot\!{x}_{\rm e}}{|{x}_{\rm e}|^2}-1\Big)\!+\![(\alpha\!+\!\gamma)^2\!-\!\sigma^2]\Big(\frac{|{x}_{\rm N}|}{|{x}_{\rm e}|}\!-\!\frac{{x}_{\rm N}\!\cdot\!{x}_{\rm e}}{|{x}_{\rm N}||{x}_{\rm e}|}\Big)\nonumber\\+\sigma(\alpha\!+\!\gamma\!-\!\sigma)\frac{b^2}{|{x}_{\rm N}||{x}_{\rm e}|}\!+\!\Big(2\sigma\gamma\!-\!\alpha\sigma\!+\!\frac{5}{4}\sigma^2\!-\!\frac{3}{4}\varepsilon\Big)\frac{b^2}{|{x}_{\rm e}|^2}\!-\!\sigma^2\Big(1-\frac{|{x}_{\rm e}|^3}{|{x}_{ \rm N}|^3}\Big)\frac{b^4}{|{x}_{\rm e}|^4}\Big\}.$
The photon's trajectory in the 2PN approximation is described by the combinations of Eqs. (19) with (6), (18) and (25). The corresponding velocity is described by the summation of Eqs. (5), (17), (23) and (24).
Most integrals used in this section can be found in our previous work,[41] and the other integrals are listed in Appendix for readers' convenience.
4 The 2PN Light Deflection in the Field of a Spherically-Symmetric Body
In real applications, people are usually interested in the gravitational deflection of light when the emitter and receiver are both far away from the body. In this case, we only need the light velocity at the locations of the emitter and the receiver whose distances from the body can be approximated as infinity. In this case,
from Eqs. (5), (17), (23) and (24), we can write the light velocity at the emitter and the receiver as follows:
$$ \quad {v}_{\rm emit} \approx {n}\,, $$
$$ \quad {v}_{\rm recv} \approx {n} \Big[1-2(\alpha+\gamma+\sigma)^2\frac{m^2}{b^2}\Big]-\!\frac{{b}}{b}\Big[2(\alpha\!+\!\gamma\!+\!\sigma)\!\frac{m}{b}\!\quad \hphantom{ {v}_{\rm recv} }+ \pi\Big(\!2\alpha^2\!\!-\!\beta\!+\!2\alpha\gamma\!+\!\alpha\sigma\!-\!\frac{\sigma^2\!}{4}\!+\!\frac{\epsilon}{2}\!+\!\frac{\varepsilon}{4}\Big)\!\frac{m^2}{b^2}\Big].$$
Therefore, the parameterized second-order PN light deflection in the field of a spherically-symmetric body can be formulated as
$$ \theta_{\rm 2PN} \equiv \;\arcsin \frac{|{v}_{\rm emit} \!\times\! {v}_{\rm recv}|}{|{v}_{\rm emit}| \!|{v}_{\rm recv}|} \\ \approx \; 2(\alpha\!+\!\gamma\!+\!\sigma)\frac{m}{b}+ \pi\Big(2\alpha^2 -\beta+2\alpha\gamma + \alpha\sigma \\ \; - \frac{\sigma^2}{4}+\frac{\epsilon}{2}+\frac{\varepsilon}{4}\Big)\frac{m^2}{b^2}. $$
It can be easily checked that the 2PN light deflection for the Schwarzschild black hole in GR is $\frac{4m}{b}\!+\!\frac{15\pi}{4}\frac{m^2}{b^2}$. Our result is also consistent with the 2PN light deflection in the scalar-tensor theory given in Ref. [31].
5 Summary
In this work, we have derived the light propagation under a generally parameterized second-order post-Newtonian framework for the gravitational field of the spherically-symmetric body. Especially, we include more parameters in the PPN frame, which enables the formulations be applicable to more metric theories, as well as in different coordinates. With the development of the observational technologies, the achieved parameterized 2PN light-deflection formula may be useful in the discriminations of different gravitational theories in future.
Appendix: Lists of Integrals
Most integrals used in the above derivations have been given in Ref. [41]. Here we give some new integrals for readers' convenience.
$ \int_{t_{\rm e}}^t \!\frac{1}{|{x}_{\rm N}|^7} \text{d}t =\frac{1}{5b^2}\!\Big(\frac{{n}\!\cdot\!{x}_{\rm N}}{|{x}_{\rm N}|^5} \!-\! \frac{{n}\!\cdot\!{x}_{\rm e}}{|{x}_{\rm e}|^5}\Big) \!+\!\frac{4}{15b^4}\Big(\frac{{n}\!\cdot\!{x}_{\rm N}}{|{x}_{\rm N}|^3} \!-\! \frac{{n}\!\cdot\!{x}_{\rm e}}{|{x}_{\rm e}|^3}\Big) \!+\! \frac{8}{15b^6}\Big(\frac{{n}\!\cdot\!{x}_{\rm N}}{|{x}_{\rm N}|} \!-\! \frac{{n}\!\cdot\!{x}_{\rm e}}{|{x}_{\rm e}|}\Big), \\ \int_{t_{\rm e}}^t \!\frac{{n}\!\cdot\!{x}_{\rm N}}{|{x}_{\rm N}|^7} \text{d}t = -\frac{1}{5|{x}_{\rm N}|^5} \!+\! \frac{1}{5|{x}_{\rm e}|^5} , \\ \int_{t_{\rm e}}^t \!\frac{{n}\!\cdot\!{x}_{\rm N}}{|{x}_{\rm N}|^7} \ln{\frac{\!|{x}_{\rm N}|\!+\!{n}\!\cdot\!{x}_{\rm N}}{|{x}_{\rm e}|\!+\!{n}\!\cdot\!{x}_{\rm e}}}\text{d}t = -\frac{1}{5|{x}_{\rm N}|^5} \ln{\frac{|{x}_{\rm N}|\!+\!{n}\!\cdot\!{x}_{\rm N}}{|{x}_{\rm e}|\!+\!{n}\!\cdot\!{x}_{\rm e}}} \!-\!\frac{3}{40b^5}\Big(\!\arccos\!{\frac{\!{n}\!\cdot\!{x}_{\rm N}}{|{x}_{\rm N}|}} \!-\! \arccos{\frac{{n}\!\cdot\!{x}_{\rm e}}{|{x}_{\rm e}|}}\Big)+ \frac{3}{40b^4}\Big(\frac{{n}\!\cdot\!{x}_{\rm N}}{|{x}_{\rm N}|^2}\!-\!\frac{\!{n}\!\cdot\!{x}_{\rm e}}{|{x}_{\rm e}|^2}\Big)\nonumber\\ \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+\! \frac{1}{20b^2}\Big(\frac{{n}\!\cdot\!{x}_{\rm N}}{|{x}_{\rm N}|^4}\!-\!\frac{\!{n}\!\cdot\!{x}_{\rm e}}{|{x}_{\rm e}|^4}\Big) .$