Parameterized Post-Post-Newtonian Light Propagation in the Field of One Spherically-Symmetric Body*
本站小编 Free考研考试/2022-01-02
Xiao-Yan Zhu1, Bo Yang2, Chun-Hua Jiang1, Wen-Bin Lin,1,2,?School of Mathematics and Physics, University of South China, Hengyang 421001, China School of Physical Science and Technology, Southwest Jiaotong University, Chengdu 610031, China
*Supported in part by the National Natural Science Foundation of China under Grant.Nos. 11647314 *Supported in part by the National Natural Science Foundation of China under Grant.Nos.11847307
Abstract We derive a more generally parameterized post-post-Newtonian solution for the light propagation in the gravitational field of one spherically-symmetric body. Based on the solution for the light velocity, we give the formula of the light deflection when both the emitter and receiver are located in the regions far away from the body, which is the most important scenario in the real applications. Our results can be applied to more metric theories of gravitation. Keywords：PPN;spherically-symmetric metric;light propagation;light deflection
PDF (119KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article Xiao-Yan Zhu, Bo Yang, Chun-Hua Jiang, Wen-Bin Lin. Parameterized Post-Post-Newtonian Light Propagation in the Field of One Spherically-Symmetric Body*. [J], 2019, 71(12): 1455-1460 doi:10.1088/0253-6102/71/12/1455
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
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
In this section we employ an iterative method[1, 41, 42] to derive the 2PN solution for the light propagation in the spacetime characterized by the above PPN metric.
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
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 ${n}$:
$$ \frac{\text{d}^2{x}_{\rm 1PN\perp}}{\text{d}t^2} = {b}\Big[\!-\!(\alpha+\gamma-\sigma)\frac{m}{|{x}_{\rm N}|^3}-3\sigma \frac{ b^2 m }{|{x}_{\rm N}|^5}\Big], $$ $$ \frac{\text{d}{x}_{\rm 1PN\parallel}}{\text{d}t} = {n}\Big[\!-\!(\alpha+\gamma+\sigma)\frac{m}{|{x}_{\rm N}|}+\sigma\frac{ b^2m}{|{x}_{\rm N}|^3}\Big]. $$ 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
$$ {x}_{\rm 2PN\parallel} ={n}({n}\cdot{x}_{\rm 2PN}), $$ $$ {x}_{\rm 2PN\perp} = {x}_{\rm 2PN} - {n}({n}\cdot{x}_{\rm 2PN}). $$ 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.
S.AKlioner . and S. Zschocke, Class.Quantum Grav. 27(2010) 075015. [Cited within: 1]
C. M.Will , Living Rev. Relativ. 17(2014) 4. DOI:10.12942/lrr-2014-4URLPMID:28179848 The status of experimental tests of general relativity and of theoretical frameworks for analyzing them is reviewed and updated. Einstein's equivalence principle (EEP) is well supported by experiments such as the E?tv?s experiment, tests of local Lorentz invariance and clock experiments. Ongoing tests of EEP and of the inverse square law are searching for new interactions arising from unification or quantum gravity. Tests of general relativity at the post-Newtonian level have reached high precision, including the light deflection, the Shapiro time delay, the perihelion advance of Mercury, the Nordtvedt effect in lunar motion, and frame-dragging. Gravitational wave damping has been detected in an amount that agrees with general relativity to better than half a percent using the Hulse-Taylor binary pulsar, and a growing family of other binary pulsar systems is yielding new tests, especially of strong-field effects. Current and future tests of relativity will center on strong gravity and gravitational waves.
S.Zschocke , Phys. Rev. D 92 ( 2015) 063015. DOI:10.1103/PhysRevE.92.063015URLPMID:26764812 We present hydrodynamic and magnetohydrodynamic (MHD) simulations of liquid sodium flow with the PLUTO compressible MHD code to investigate influence of magnetic boundary conditions on the collimation of helicoidal motions. We use a simplified cartesian geometry to represent the flow dynamics in the vicinity of one cavity of a multiblades impeller inspired by those used in the Von-Kármán-sodium (VKS) experiment. We show that the impinging of the large-scale flow upon the impeller generates a coherent helicoidal vortex inside the blades, located at a distance from the upstream blade piloted by the incident angle of the flow. This vortex collimates any existing magnetic field lines leading to an enhancement of the radial magnetic field that is stronger for ferromagnetic than for conducting blades. The induced magnetic field modifies locally the velocity fluctuations, resulting in an enhanced helicity. This process possibly explains why dynamo action is more easily triggered in the VKS experiment when using soft iron impellers.
M. H.Soffel and W. B.Han , Phys. Lett. A 379 ( 2015) 233.
Deep space laser ranging missions like ASTROD I (Single-Spacecraft Astrodynamical Space Test of Relativity using Optical Devices) and ASTROD, together with astrometry missions like GAIA and LATOR will be able to test relativistic gravity to an unprecedented level of accuracy. More precisely, these missions will enable us to test relativistic gravity to 10-7–10-9 of the size of relativistic (post-Newtonian) effects, and will require second post-Newtonian approximation of relevant theories of gravity. The first post-Newtonian approximation is valid to 10-6 and the second post-Newtonian approximation is valid to 10-12 in terms of post-Newtonian effects in the solar system. The scalar-tensor theory is widely discussed and used in tests of relativistic gravity, especially after the interests in inflation models and in dark energy models. In the Lagrangian, intermediate-range gravity term has a similar form as cosmological term. Here we present the full second post-Newtonian approximation of the scalar-tensor theory including viable examples of intermediate-range gravity. We use Chandrasekhar’s approach to derive the metric coefficients and the equation of the hydrodynamics governing a perfect fluid in the second post-Newtonian approximation in scalar-tensor theory; all terms inclusive of O(c-4) are retained consistently in the equations of motion.
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