# Parameterized second post-Newtonian framework with conservation Laws

Yu-Mei Wu<EMAIL_ADDRESS>CAS Key Laboratory of Theoretical Physics,
Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190,
China  School of Physical Sciences, University of Chinese Academy of Sciences,
No. 19A Yuquan Road, Beijing 100049, China  Qing-Guo Huang  Corresponding
author<EMAIL_ADDRESS>CAS Key Laboratory of Theoretical Physics, Institute
of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
School of Physical Sciences, University of Chinese Academy of Sciences, No.
19A Yuquan Road, Beijing 100049, China  School of Fundamental Physics and
Mathematical Sciences Hangzhou Institute for Advanced Study, UCAS, Hangzhou
310024, China  Center for Gravitation and Cosmology, College of Physical
Science and Technology, Yangzhou University, Yangzhou 225009, China

###### Abstract

We parameterize the second post-Newtonian (2PN) metric for a gravitating
system of fluids in the generalized harmonic gauge, and find that there are
only three independent 2PN parameters (i.e. $\omega$, $\delta$ and
$\delta_{2}$) for satisfying some conservation laws including the
conservations of energy, momentum, angular-momentum and the uniform motion of
the center-of-mass.

##  I Introduction

Was Einstein right? Put general relativity (GR) to the test! Will (1995) The
experimental tests prevail in every gravitational observation predicted by GR
in different scales. For instance, the solar system provides a terrific weak-
field regime Will (2014, 2018) where the tests like light deflection and
perihelion advance of Mercury are conducted, and the binary-pulsar as a
strong-gravity astrophysical system Stairs (2003) gives access to the
comparison between the decrease in its orbital period and gravitational-wave
energy loss. In recent years, the thrilling detection of gravitational waves
originated from binary black holes Abbott  _et al._ (2016) and the image of
black holes Akiyama  _et al._ (2019a, b), through possible direct reflection
of the strong field of black holes, mark the advent of a new era for
gravitational tests.

GR has passed all the gravitational tests so far with flying colors, but due
to the non-renormalizability of the theory itself and interpretation puzzle
for the cosmological constant problem, the efforts to seek its alternatives
have never ceased. However, from an observational point of view, in order to
incorporate the possible deviation from GR systematically rather than focus on
a certain modified gravitational theory, parameterizing gravitational theory
from different aspects according to the application range of experiments is an
effective method. For instance, the parameterized post-Newtonian (PPN)
framework, parameterized post-Keplerian (PPK) framework Damour and Taylor
(1992) and parameterized post-Einsteinian (PPE) framework Yunes and Pretorius
(2009) have been constructed. Among them, the PPN formalism is a powerful tool
governing the realm of weak-field tests. The idea of PPN framework was
originated from Eddington. He parameterized the first post-Newtonian (1PN)
limit111In order to avoid ambiguity, we use “1PN (order)” to refer to “the
first post-Newtonian (order)”, and “PN (theory)” to refer to “post-Newtonian
(theory) of all orders”. However, the traditional notation “PPN (metric)”
specially refers to parameterized post-Newtonian (metric) at the first order.
of a Schwarzschild metric with two arbitrary parameters (namely, $\gamma$ and
$\beta$) in 1922 Eddington (1923), where $\gamma$ and $\beta$ can be regarded
to separately measure the spatial curvature and nonlinearity produced by
gravity. Will and Nordtvedt extended the approach during 1968–1972 by
introducing another eight arbitrary parameters in addition to $\gamma$ and
$\beta$ in front of the independent 1PN potentials to develop the modern
version of PPN framework Nordtvedt (1968); Thorne and Will (1971); Will
(1971a, b); Will and Nordtvedt (1972), where the new parameters are classified
according to whether there exist preferred location/frame effects and
conserved total momentum. The PPN framework encompasses a large amount of
alternative metric theories Will (2014); Chen  _et al._ (2015); Hohmann
(2015); Mohseni Sadjadi (2017) and provides a broad range of testable
phenomena. Up to now, the experimental tests performed in the solar system put
relatively strong constraints on the PPN parameters and therefore on the
modified gravitational theories Will (2014).

After the success of the PPN framework, the attempts to generalize the
parameterization to the second post-Newtonian (2PN) order were made from
different directions. For example, the Schwarzschild metric is directly
expanded to 2PN order and then parameterize the spatial-spatial component
$g_{ij}$ to account for a future more accurate measurement of the light
deflection Epstein and Shapiro (1980); Deng (2015). A more ambitious effort
from Nordtvedt and Benacquista tried to include any possible metric theory at
this order and has obtained the independent parameters on the requirement of
Lorentz invariance of Lagrangian Benacquista and Nordtvedt (1988); Benacquista
(1992); Nordtvedt (1993). But Demour pointed out that these parameters are not
indeed constants and introduced a field-theory-based framework to find out
that only two independent 2PN parameters are needed in the “tensor-
multiscalar” theories Damour and Esposito-Farese (1996). Thereafter, several
specific modified theories have also been calculated to their 2PN
approximation Xie  _et al._ (2009); Xie and Huang (2008), but a parameterized
second post-Newtonian (P2PN) theory parallel to PPN theory has not been
actually constructed.

On one hand, the traditional weak field tests is expected to obtain
unprecedented precision with present and future astrometric missions, such as
Gaia Perryman  _et al._ (2001) and Laser Astrometric Test of Relativity
(LATOR) Turyshev  _et al._ (2004), to ${\cal O}(10^{-9}\sim 10^{-7})$ of the
size of relativistic effects, which is beyond the level of 1PN effects
$(GM_{\odot}/(R_{\odot}c^{2})\sim{\cal O}(10^{-6}))$ and requires taking the
2PN effects into account. On the other hand, the information of a 2PN
parameter also resides in the strong field in a parameterized Kerr metric
Johannsen and Psaltis (2011), and can be extracted with present and future
gravitational wave test Carson and Yagi (2020) and the horizon-scale images of
black hole Psaltis  _et al._ (2020). The reason is the n-th post-Newtonian
approximation acts as the asymptotic form of a black hole metric. Actually,
the parameterized Kerr metrics in Johannsen and Psaltis (2011); Konoplya  _et
al._ (2016) match with the PPN metric in the weak-field region Johannsen and
Psaltis (2011); Konoplya  _et al._ (2016). If we require a higher order
accuracy, in order to keep the compatibility of parameterization in the weak-
field and strong-field regions, a reasonable P2PN formalism is also necessary.

In this paper, we will mainly follow Will’s approach to construct a reasonable
P2PN framework. However, it is rather difficult to seek out all the possible
2PN potentials, especially the ones absent in the 2PN limit of GR. Through a
brief analysis on the PPN framework, we find a compromise to simplify the P2PN
structure and concentrate on searching the independent P2PN parameters under
fairly stringent constraints on the metric theory; that is, the theory should
contain no preferred location/frame effects and satisfy the conservation laws.
Our paper is organized as follows. We give a brief review on the PPN framework
in Section II. In Section III, we construct a primitive P2PN metric and
establish the gravitational stress-energy pseudotensor with the help of some
transformation tricks and obtain the parameter constraints. The gauge that the
metric with the constraints satisfies is also discussed. Finally, we give a
summary and a brief discussion in Section IV. Some details are added in
Appendix A and B.

We adopt the following conventions and notations in the paper. The signature
of metric takes $(-,+,+,+)$. Greek indices take the values from 0 to 3, while
Latin indices take the values from 1 to 3. Einstein’s summation rule over
repeated indices is used, even when we only have spatial indices and are not
careful about raising and lowering the indices. Parentheses around the indices
indicate the indices being symmetrized and square brackets indicate anti-
symmetrization; for example, $A_{(i}B_{j)}=\frac{1}{2}(A_{i}B_{j}+A_{j}B_{i})$
and $A_{[i}B_{j]}=\frac{1}{2}(A_{i}B_{j}-A_{j}B_{i})$. Bold letters
$\mathbf{v}=v^{i}$ denote spatial vectors. Since the post-Newtonian
expressions of GR are essentially the retarded solutions to wave equations
expanded in the near-zone field and involve Poisson-like potentials and their
generalizations, here we take the definitions of the Poisson potential,
superpotential and superduperpotential for a soure $f$ following Will’s
notation in Pati and Will (2000),

$\displaystyle\mathrm{P}(f)$ $\displaystyle\equiv$
$\displaystyle\frac{1}{4\pi}\int\frac{f(t,\mathbf{x}^{\prime})}{|\mathbf{x}-\mathbf{x^{\prime}}|}d^{3}x^{\prime},\qquad\qquad\qquad\quad\nabla^{2}\mathrm{P}(f)=-f,$
$\displaystyle\mathrm{S}(f)$ $\displaystyle\equiv$
$\displaystyle\frac{1}{4\pi}\int
f(t,\mathbf{x}^{\prime})|\mathbf{x}-\mathbf{x^{\prime}}|d^{3}x^{\prime},\qquad\qquad\,\nabla^{2}\mathrm{S}(f)=2\mathrm{P}(f),$
$\displaystyle\mathrm{SD}(f)$ $\displaystyle\equiv$
$\displaystyle\frac{1}{4\pi}\int
f(t,\mathbf{x}^{\prime})|\mathbf{x}-\mathbf{x^{\prime}}|^{3}d^{3}x^{\prime},\\!\qquad\qquad\nabla^{2}\mathrm{SD}(f)=12\mathrm{S}(f).$
(1)

We also set $G=1$ throughout the paper, where $G$ the gravitational constant.

##  II  A brief review of the parameterized Post-Newtonian framework

Before constructing the P2PN formalism, we give a brief review on the PPN
framework and delineate the method for parameterization of metric.

For a self-gravitating fluid system with the energy-momentum tensor expressed
by the proper mass density $\rho$, the internal energy per unit mass $\Pi$,
the pressure $P$ and the four-velocity field $u^{\alpha}$ as follows

$\displaystyle
T^{\alpha\beta}=\left(\rho(1+\Pi/c^{2})+P/c^{2}\right)u^{\alpha}u^{\beta}+Pg^{\alpha\beta},$
(2)

the PPN metric in the generalized standard gauge is

$\displaystyle g_{00}$ $\displaystyle=$
$\displaystyle-1+\frac{2}{c^{2}}U+\frac{1}{c^{4}}(-2\beta
U^{2}+3\beta_{1}\Phi_{1}-2\beta_{2}\Phi_{2}+2\beta_{3}\Phi_{3}+6\beta_{4}\Phi_{4})+\frac{1}{c^{4}}(\beta_{6}\Phi_{6}+\xi\Phi_{W}+\Phi^{\rm{PF}})+\mathcal{O}(c^{-6}),$
$\displaystyle g_{0j}$ $\displaystyle=$ $\displaystyle-\frac{4}{c^{3}}\Delta
V_{j}-\frac{1}{2c^{3}}\Delta_{1}\partial_{tj}X+\frac{1}{c^{3}}\Phi^{\rm{PF}}_{j}+\mathcal{O}(c^{-5}),$
$\displaystyle g_{jk}$ $\displaystyle=$ $\displaystyle(1+\frac{2}{c^{2}}\gamma
U)\delta_{jk}+\mathcal{O}(c^{-4}).$ (3)

Here $U$ is the Newtonian gravitational potential given by

$\displaystyle
U=U(\mathbf{x},t)\equiv\int\frac{\rho^{*}(t,\mathbf{x}^{\prime})}{|\mathbf{x}-\mathbf{x^{\prime}}|}d^{3}x^{\prime}=\int\frac{\rho^{*}{{}^{\prime}}}{|\mathbf{x}-\mathbf{x^{\prime}}|}d^{3}x^{\prime}=\mathrm{P}(4\pi\rho^{*}),$
(4)

where $\rho^{*}$ is the rescaled mass density related to the proper mass
density $\rho$ by $\rho^{*}=\sqrt{-g}\rho u^{0}/c$. The rescaled mass density
is also called the conserved mass density because it satisfies the continuity
equation, Poisson and Will (2016),

$\displaystyle\partial_{t}\rho^{*}+\partial_{j}(\rho^{*}v^{j})=0,$ (5)

where $v^{j}$ is a three-velocity field in $u^{\alpha}$, namely
$u^{\alpha}=(u^{0}/c)(c,\mathbf{v})$. Some other potentials in Eq. (3) are

$\displaystyle\Phi_{1}$ $\displaystyle\equiv$
$\displaystyle\mathrm{P}(4\pi\rho^{*}v^{2}),\quad\qquad\Phi_{2}\equiv\mathrm{P}(4\pi\rho^{*}U),\quad\qquad\Phi_{3}\equiv\mathrm{P}(4\pi\rho^{*}\Pi),\quad\qquad\Phi_{4}\equiv\mathrm{P}(4\pi
P),$ $\displaystyle V^{j}$ $\displaystyle\equiv$
$\displaystyle\mathrm{P}(4\pi\rho^{*}v^{j}),\quad\qquad
X\equiv\int\rho^{*}{{}^{\prime}}|\mathbf{x}-\mathbf{x^{\prime}}|d^{3}x^{\prime}=\mathrm{S}(4\pi\rho^{*}),$
$\displaystyle\Phi_{6}$ $\displaystyle\equiv$
$\displaystyle\int\frac{\rho^{*}{{}^{\prime}}v_{j}^{\prime}v_{k}^{\prime}(x-x^{\prime})^{j}(x-x^{\prime})^{k}}{|\mathbf{x}-\mathbf{x^{\prime}}|^{3}}d^{3}x^{\prime},\qquad\Phi_{W}\equiv\int\frac{\rho^{*}{{}^{\prime}}\rho^{*}{{}^{\prime\prime}}(x-x^{\prime})_{j}}{|\mathbf{x}-\mathbf{x^{\prime}}|^{3}}\left[\frac{(x^{\prime}-x^{\prime\prime})^{j}}{|\mathbf{x}-\mathbf{x^{\prime\prime}}|}-\frac{(x-x^{\prime\prime})^{j}}{|\mathbf{x^{\prime}}-\mathbf{x^{\prime\prime}}|}\right]d^{3}x^{\prime}d^{3}x^{\prime\prime};$
(6)

and the rest are preferred-frame potentials,

$\displaystyle\Phi^{\rm{PF}}=\alpha_{1}w^{2}U+\alpha_{2}w^{j}w^{k}\partial_{jk}X+\alpha_{3}w^{j}V_{j},\qquad\qquad\Phi^{\rm{PF}}_{j}=\alpha_{4}w_{j}U+\alpha_{5}w^{k}\partial_{jk}X,$
(7)

where $w^{j}$ is the velocity of the PPN coordinate system relative to the
universal preferred frame. We make some clarifications in regard to the metric
in Eq. (3):
(1) When
$\beta=\beta_{1}=\beta_{2}=\beta_{3}=\beta_{4}=\Delta=\Delta_{1}=\gamma=1$ and
all other parameters vanish, the metric reduces to the 1PN metric in GR.
(2) The additional potentials absent in GR, i.e. $\Phi_{6}$, $\Phi_{W}$,
$\Phi^{\rm{PF}}$ and $\Phi^{\rm{PF}}_{j}$, are related to preferred-
location/frame effects. $\Phi_{6}$ and $\Phi_{W}$ appear in theories with
preferred-location effects caused by, for example, a galaxy-induced anisotropy
Will (2014), while $\Phi^{\rm{PF}}$ and $\Phi^{\rm{PF}}_{j}$ are involved in
some well-motivated vector-tensor Foster and Jacobson (2006) and tensor-
vector-scalar Sagi (2009) theories that include a dynamical timelike vector
field, such as Einstein-Æther theory.
(3) A metric theory based on an invariant action principle automatically
admits a conservation law for total momentum and is called “semi-
conservative”. Furthermore, if the theory also admits a conserved angular
momentum and a center-of-mass with a uniform motion (which means the existence
of a preferred frame is impossible due to the break of local Lorentz
invariance), it is called “fully-conservative”. For a fully-conservative
metric theory without preferred-location effects, only two independent
parameters $\beta,\gamma$ retain, and the other non-vanishing parameters
including $\beta_{1},\beta_{2},\beta_{3},\beta_{4},\Delta,\Delta_{1}$ are all
determined by $\beta,\gamma$. Scalar-tensor theories Damour and Esposito-
Farese (1992), such as Brans-Dicke theory, fall into this category.
So we conclude that in a fully-conservative PPN theory without preferred
effects
($\beta_{6}=\xi=\alpha_{1}=\alpha_{2}=\alpha_{3}=\alpha_{4}=\alpha_{5}=0$),
the potentials are just the same as those in GR but with the coefficients in
front of them replacing by the 1PN parameters; furthermore, we can extract the
independent ones $\gamma$ and $\beta$ from all these 1PN parameters.

##  III The Parameterized Second Post-Newtonian Formalism

In this section, we extend the former parameterization to 2PN order, or
equivalently we need to expand the metric to the following order, Pati and
Will (2000),

$\displaystyle g_{00}\text{\quad to\quad}\mathcal{O}(c^{-6}),\qquad
g_{0j}\text{\quad to\quad}\mathcal{O}(c^{-5}),\qquad g_{jk}\text{\quad
to\quad}\mathcal{O}(c^{-4}).$ (8)

In principle, the general P2PN framework can be constructed parallely to that
of PPN in the original paper Will (1971a) through listing all the possible 2PN
potentials and parametrizing them with arbitrary parameters, and then choosing
a gauge to get rid of the redundant degrees of freedom (non-independent
potentials). However, the difficulties also reside in two sides
correspondingly: one is to find potentials absent in GR, similar to $\Phi_{W}$
in the PPN (see Eq. (6)), that need elaborate construction; the other is to
choose a suitable gauge that is compatible from the 1PN order to the 2PN order
since the gauge-fixing process involves both orders.

Therefore, instead of trying to build the general P2PN framework once and for
all, we focus on seeking the independent P2PN parameters in a fully-
conservative theory without preferred-location/frame effects first. From the
discussion in Sec. II, this can be achieved by first parameterizing the 2PN
metric in GR in a specific gauge, and then figuring out the constraints on
these parameters by considering some conservation laws. Such a method of
parametrization will certainly break the gauge that stands in GR, but we may
find the corresponding gauge for the parameterized metic that meets the
parameter constraints hopefully.

###  III.1 The P2PN metric

We need to note that the choice of generalized standard gauge of the PPN
metric in Eq. (3) is only a reflection of historical development based on
Chandrasekhar’s work Chandrasekhar (1965) on the classic approach to post-
Newtonian theory. By contrast, the harmonic gauge prevails in the modern
approach to PN theory, and hence we choose to parameterize the 2PN metric in
the harmonic gauge, Pati and Will (2000, 2002)222A systematic method to
calculate the post-Newtonian approximation to 3.5 order for unspecified matter
fields is given in Pati and Will (2000), and the matter fields are only
consisting of baryons in Pati and Will (2002). Here, we need to generalize the
calculation to perfect fluid to 2PN order in the Appendix A.,

$\displaystyle g_{00}$ $\displaystyle=$
$\displaystyle-1+\frac{2}{c^{2}}U+\frac{1}{c^{4}}\left(-2\beta
U^{2}+\Psi_{1}\right)+\frac{1}{c^{6}}(\frac{4}{3}\delta
U^{3}+U\Psi_{2}+8\lambda
V^{2}+\mathcal{E}+\mathcal{X}+\mathcal{G}+\mathcal{H}),$ $\displaystyle
g_{0j}$ $\displaystyle=$ $\displaystyle-\frac{4}{c^{3}}\Delta
V^{j}+\frac{1}{c^{5}}(\chi UV^{j}+\mathcal{V}^{j}),$ $\displaystyle g_{jk}$
$\displaystyle=$ $\displaystyle\left[1+\frac{2}{c^{2}}\gamma
U+\frac{1}{c^{4}}(2\omega
U^{2}+\Psi_{3})\right]\delta_{jk}+\frac{1}{c^{4}}\mathcal{M}_{jk},$ (9)

in which,

$\displaystyle\Psi_{1}\equiv
3\beta_{1}\Phi_{1}-2\beta_{2}\Phi_{2}+2\beta_{3}\Phi_{3}+6\beta_{4}\Phi_{4}+\beta_{0}\ddot{X},$
$\displaystyle\Psi_{2}\equiv-6\text{$\delta_{1}$}\text{$\Phi_{1}$}+4\text{$\delta_{2}$}\text{$\Phi_{2}$}-4\text{$\delta_{3}$}\text{$\Phi_{3}$}-12\text{$\delta_{4}$}\text{$\Phi_{4}$}-2\delta_{0}\ddot{X},$
$\displaystyle\mathcal{E}\equiv
3\kappa\Sigma(U^{2})-3\kappa_{1}\Sigma(\Phi_{1})-2\kappa_{2}\Sigma(\Phi_{2})-2\kappa_{3}\Sigma(\Phi_{3})+2\kappa_{4}\Sigma(\Phi_{4})-\kappa_{0}\Sigma(\ddot{X})$
$\displaystyle\qquad+\frac{7}{4}\sigma_{1}\Sigma(v^{4})+9\sigma_{2}\Sigma(v^{2}U)-8\sigma_{3}\Sigma(v_{j}V^{j})+3\sigma_{4}\Omega(v^{2})+4\sigma_{5}\mathrm{T}(v^{2})-2\sigma_{6}\Omega(U)+12\sigma_{7}\mathrm{T}(U)$
$\displaystyle\mathcal{X}\equiv\frac{3}{2}\mu_{1}\ddot{X_{1}}-\mu_{2}\ddot{X_{2}}+\mu_{3}\ddot{X_{3}}+3\mu_{4}\ddot{X_{4}}+\frac{1}{12}\mu_{0}\mathop{\mathrm{Y}}\limits^{(4)}$
$\displaystyle\mathcal{G}\equiv-12\xi_{1}G_{1}-8\xi_{2}G_{2}+16\xi_{3}G_{3}+16\xi_{4}G_{4}+\xi_{5}G_{5},$
$\displaystyle\mathcal{H}\equiv-8\eta_{1}H_{1}-8\eta_{2}H_{2},$
$\displaystyle\mathcal{V}^{j}\equiv-2\chi_{1}V_{1}^{j}-4\chi_{2}V_{2}^{j}-4\chi_{3}V_{3}^{j}-4\chi_{4}V_{4}^{j}+8\chi_{5}\phi_{2}^{j}-2\chi_{0}\ddot{X}^{j}-16\iota_{1}K_{1}^{j}-12\iota_{2}K_{2}^{j},$
$\displaystyle\Psi_{3}\equiv-\omega_{1}\Phi_{1}-2\omega_{2}\Phi_{2}+2\omega_{3}\Phi_{3}-2\omega_{4}\Phi_{4}+\omega_{0}\ddot{X},$
$\displaystyle\mathcal{M}^{jk}\equiv
4\theta_{1}\Phi_{1}^{jk}+4\theta_{2}P_{2}^{jk},$ (10)

where the overdot denotes the derivative with respect to time coordinate and
$\mathop{\mathrm{Y}}\limits^{(4)}=\partial_{t}^{(4)}Y$. Here most of the
potentials in Eqs. (9) and (10) can be conveniently classified into several
particular classes given in Eq. (1), namely

$\displaystyle\Sigma(f)$ $\displaystyle\equiv$
$\displaystyle\int\frac{\rho^{*}{{}^{\prime}}f^{\prime}}{|\mathbf{x}-\mathbf{x^{\prime}}|}d^{3}x^{\prime}=\mathrm{P}(4\pi\rho^{*}f),\qquad\qquad\quad\Sigma^{j}(f)\equiv\int\frac{\rho^{*}{{}^{\prime}}v^{\prime
j}f^{\prime}}{|\mathbf{x}-\mathbf{x^{\prime}}|}d^{3}x^{\prime}=\mathrm{P}(4\pi\rho^{*}v^{j}f),$
$\displaystyle\Sigma^{ij}(f)$ $\displaystyle\equiv$
$\displaystyle\int\frac{\rho^{*}{{}^{\prime}}v^{\prime i}v^{\prime
j}f^{\prime}}{|\mathbf{x}-\mathbf{x^{\prime}}|}d^{3}x^{\prime}=\mathrm{P}(4\pi\rho^{*}v^{\prime
i}v^{j}f),$ $\displaystyle\mathrm{X}(f)$ $\displaystyle\equiv$
$\displaystyle\int\rho^{*}{{}^{\prime}}f^{\prime}|\mathbf{x}-\mathbf{x^{\prime}}|d^{3}x^{\prime}=\mathrm{S}(4\pi\rho^{*}f),\qquad\quad\,\mathrm{X}^{j}(f)\equiv\int\rho^{*}{{}^{\prime}}v^{\prime
j}f^{\prime}|\mathbf{x}-\mathbf{x^{\prime}}|d^{3}x^{\prime}=\mathrm{S}(4\pi\rho^{*}v^{\prime
j}f),$ $\displaystyle\mathrm{Y}(f)$ $\displaystyle\equiv$
$\displaystyle\int\rho^{*}{{}^{\prime}}f^{\prime}|\mathbf{x}-\mathbf{x^{\prime}}|^{3}d^{3}x^{\prime}=\mathrm{SD}(4\pi\rho^{*}f),$
(11)

and then the unspecified potentials can be defined by

$\displaystyle
V_{1}^{j}\equiv\Sigma^{j}(v^{2}),\quad\,\,V_{2}^{j}\equiv\Sigma^{j}(U),\quad\,\,V_{3}^{j}\equiv\Sigma^{j}(\Pi),\quad\,\,V_{4}^{j}\equiv\Sigma^{j}(P/\rho^{*}),$
$\displaystyle\phi_{2}^{j}\equiv\Sigma(V^{j}),\quad\,\,{X}^{j}\equiv\mathrm{X}^{j}(1),\quad\,\,K_{1}^{j}\equiv\mathrm{P}(U^{,k}V^{k,j}),\quad\,\,K_{2}^{j}\equiv\mathrm{P}(U^{,j}\dot{U}),$
$\displaystyle\Phi_{1}^{ij}\equiv\Sigma^{ij}(1),\quad\,\,P_{2}^{ij}\equiv\mathrm{P}(U^{,i}U^{,j}),\quad
P_{2}\equiv P_{2}^{ii}=\Phi_{2}-\frac{1}{2}U^{2},$
$\displaystyle\Omega(v^{2})\equiv\Sigma(\Pi
v^{2}),\quad\,\,\mathrm{T}(v^{2})\equiv\Sigma(P/\rho^{*}v^{2}),\quad\,\,\Omega(U)\equiv\Sigma(\Pi
U),\quad\,\,\mathrm{T}(U)\equiv\Sigma(P/\rho^{*}U),$ $\displaystyle
X_{1}\equiv\mathrm{X}(v^{2}),\quad\,\,X_{2}\equiv\mathrm{X}(U),\quad\,\,X_{3}\equiv\mathrm{X}(\Pi),\quad\,\,X_{4}\equiv\mathrm{X}(P/\rho^{*}),\quad\,\,Y\equiv\mathrm{Y}(1),$
$\displaystyle
G_{1}\equiv\mathrm{P}(\dot{U}^{2}),\quad\,\,G_{2}\equiv\mathrm{P}(U\ddot{U}),\quad\,\,G_{3}\equiv-\mathrm{P}(\dot{U}^{,k}V^{k}),\quad\,\,G_{4}\equiv\mathrm{P}(V^{i,j}V^{j,i}),\quad\,\,G_{5}\equiv-\mathrm{P}(\dot{V}^{k}U^{,k})$
$\displaystyle
H_{1}\equiv\mathrm{P}(\Phi_{1}^{ij}U^{,ij}),\quad\,\,H_{2}\equiv\mathrm{P}(P_{2}^{ij}U^{,ij}),$
(12)

where the commas in the definition denote the partial derivatives with respect
to the chosen coordinates. Note that terms of half-odd-integer PN order, i.e.
1.5 PN order exactly in $g_{00}$, representing the dissipative radiation-
reaction, are not taken into account. Eq. (9) recovers the 2PN metric in GR if
$\chi=\xi_{5}=0$ and the other parameters in Eqs. (9) and (10) are all equal
to 1.

###  III.2 The conservation laws and parameter constraints

Even though a large number of parameters are introduced in the general metric
Eq. (9), many of them are not actually independent once the metric is required
to satisfy some conservation laws.

It is well known, for example in Landau and Lifschits (1975), that the usual
version of energy-momentum conservation expressed by the vanishing of
convariant divergence of $T^{\alpha\beta}$,

$\displaystyle\nabla_{\beta}T^{\alpha\beta}=0,$ (13)

cannot afford the global conserved quantities since it is a direct consequence
of local conservation of energy-momentum ($T^{\alpha\beta}$) and the
contribution from gravitational fields is implicit. On the other hand, another
version compatible with Eq. (13) expressed by the vanishing of ordinary
divergence of a pseudotensor $\Theta^{\alpha\beta}$, namely

$\displaystyle\partial_{\beta}\Theta^{\alpha\beta}=0,$ (14)

can be exploited to determine the conserved quantities. Here an appropriate
$\Theta^{\alpha\beta}$ should follow two rules: (1) it contains contributions
from both the matter fields and the gravitational fields; and (2) it reduces
to $T^{\alpha\beta}$ in the flat spacetime. For example, in GR, the exact
version of $\Theta^{\alpha\beta}$ takes
$\Theta^{\alpha\beta}=(-\tilde{g})(T^{\alpha\beta}+t_{LL}^{\alpha\beta})$,
where $\tilde{g}$ is the determinant of the metric, and
$t_{LL}^{\alpha\beta}$, the famous Landau-Lifshitz pseudotensor, acts as the
contribution from the gravitational fields Landau and Lifschits (1975). Then,
the total momentum over a region $\mathcal{V}$ can be formally defined by the
three-dimensional integral, Poisson and Will (2016),

$\displaystyle P^{\alpha}:=\frac{1}{c}\int_{\mathcal{V}}\Theta^{\alpha
0}d^{3}x.$ (15)

When we take the limit of $\mathcal{V}$ to include all of the three-
dimensional space of an asymptotically-flat spacetime, $P^{\alpha}$ can be
proven to be invariant over time by use of Eq. (14). Similarly, the total
angular momentum can be formally defined by

$\displaystyle
J^{\alpha\beta}:=\frac{2}{c}\int_{\mathcal{V}}x^{[\alpha}\Theta^{\beta]0}d^{3}x$
(16)

which is conserved over all of the three-dimensional space only when
$\Theta^{\alpha\beta}$ is also symmetric. More specifically, the limit of
$P^{\alpha}$ and $J^{\alpha\beta}$ of an asymptotically-flat spacetime are
identified : $P^{0}$ is the total energy, $P^{i}$ the total three-momentum,
$J^{ij}$ the total angular-momentum three-tensor and $J^{0j}$ determines the
motion of the center-of-mass.

From now on, the metric in Eq. (9) is considered to be fully conservative, or
equivalently we need to find the corresponding symmetric
$\Theta^{\alpha\beta}$ satisfying Eq. (14). In the PPN formalism, such a
$\Theta^{\alpha\beta}$ assumed to be
$(1+c^{-2}AU)(T^{\alpha\beta}+t^{\alpha\beta})$ has been found by determining
the constant $A$ and the symmetric gravitational stress-energy pseudotensor
$t^{\alpha\beta}$ with the use of Eqs. (13) and (14) in Will (1971b). When we
extend to the P2PN case and follow the rules for an appropriate
$\Theta^{\alpha\beta}$, without loss of generality, we assume

$\displaystyle\Theta^{\alpha\beta}=\left(1+\frac{1}{c^{2}}AU+\frac{1}{c^{4}}(BU^{2}+\Psi_{4})\right)(T^{\alpha\beta}+t^{\alpha\beta}),$
(17)

with

$\displaystyle\Psi_{4}=B_{1}\Phi_{1}+B_{2}\Phi_{2}+B_{3}\Phi_{3}+B_{4}\Phi_{4}+B_{0}\ddot{X},$
(18)

where $A$, $B$ and $B_{k}$ ($k=0,1,...,4$) are constants. We need to point out
that $t^{\alpha\beta}$ acting as the contribution from gravitational fields
should be constructed from various gravitational potentials (e.g., $U$,
$V^{j}$ and potentials contained in Eq. (10)) and their derivatives, but not
contain any of the fluid variables ($\rho^{*}$, $v^{j}$, $\Pi$ and $P$)
explicitly.

Substituting Eq. (17) into Eq. (14), and utilizing Eq. (13) which can be
rewritten by

$\displaystyle
0=\nabla_{\beta}T^{\alpha\beta}=\partial_{\beta}T^{\alpha\beta}+\Gamma^{\alpha}_{\mu\beta}T^{\mu\beta}+\Gamma^{\beta}_{\mu\beta}T^{\alpha\mu},$
(19)

we convert Eq. (14) for $\Theta^{\alpha\beta}$ to the equation for
$t^{\alpha\beta}$,

$\displaystyle\partial_{\beta}\left[\left(1+\frac{1}{c^{2}}AU+\frac{1}{c^{4}}(BU^{2}+\Psi_{4})\right)t^{\alpha\beta}\right]$
$\displaystyle=$
$\displaystyle\left(1+\frac{1}{c^{2}}AU+\frac{1}{c^{4}}(BU^{2}+\Psi_{4})\right)(\Gamma^{\alpha}_{\mu\beta}T^{\mu\beta}+\Gamma^{\beta}_{\mu\beta}T^{\alpha\mu})-\left(\frac{1}{c^{2}}A\partial_{\beta}U+\frac{1}{c^{4}}\partial_{\beta}(BU^{2}+\Psi_{4})\right)T^{\alpha\beta},$
(20)

where the energy-momentum tensor to the required order are given by

$\displaystyle c^{-2}T^{00}=$
$\displaystyle\rho^{*}\left\\{1+\frac{1}{c^{2}}\left[\frac{1}{2}v^{2}+(2-3\gamma)U+\Pi\right]+\frac{1}{c^{4}}\left[\frac{3}{8}v^{4}+(2-\frac{1}{2}\gamma)Uv^{2}+\frac{1}{2}\Pi
v^{2}+P/\rho^{*}v^{2}-4\Delta v_{j}V^{j}\right.\right.$
$\displaystyle\left.\left.+(4-2\beta-6\gamma+\frac{15}{2}\gamma^{2}-3\omega)U^{2}+(2-3\gamma)U\Pi+\Psi_{1}-\frac{3}{2}\Psi_{3}-\frac{1}{2}\mathcal{M}\right]\right\\}$
$\displaystyle c^{-1}T^{0j}=$
$\displaystyle\rho^{*}v^{j}\left\\{1+\frac{1}{c^{2}}\left[\frac{1}{2}v^{2}+(2-3\gamma)U+\Pi+P/\rho^{*}\right]+\frac{1}{c^{4}}\left[\frac{3}{8}v^{4}+(2-\frac{1}{2}\gamma)Uv^{2}+\frac{1}{2}\Pi
v^{2}+P/\rho^{*}v^{2}-4\Delta v_{j}V^{j}\right.\right.$
$\displaystyle\left.\left.+(4-2\beta-6\gamma+\frac{15}{2}\gamma^{2}-3\omega)U^{2}+(2-3\gamma)U\Pi+2UP/\rho^{*}+\Psi_{1}-\frac{3}{2}\Psi_{3}-\frac{1}{2}\mathcal{M}\right]\right\\}$
$\displaystyle-\frac{4}{c^{4}}\Delta PV^{j}$ $\displaystyle T^{jk}=$
$\displaystyle\rho^{*}v^{j}v^{k}\left\\{1+\frac{1}{c^{2}}\left[\frac{1}{2}v^{2}+(2-3\gamma)U+\Pi+P/\rho^{*}\right]+\frac{1}{c^{4}}\left[\frac{3}{8}v^{4}+(2-\frac{1}{2}\gamma)Uv^{2}+\frac{1}{2}\Pi
v^{2}+P/\rho^{*}v^{2}-4\Delta v_{j}V^{j}\right.\right.$ (21)
$\displaystyle\left.\left.+(4-2\beta-6\gamma+\frac{15}{2}\gamma^{2}-3\omega)U^{2}+(2-3\gamma)U\Pi+2UP/\rho^{*}+\Psi_{1}-\frac{3}{2}\Psi_{3}-\frac{1}{2}\mathcal{M}\right]\right\\}$
$\displaystyle+P\left\\{1-\frac{2}{c^{2}}\gamma
U+\frac{1}{c^{4}}\left[(4\gamma^{2}-2\omega)U^{2}-\Psi_{3}\right]\right\\}\delta^{jk}-\frac{1}{c^{4}}P\mathcal{M}^{jk},$

with $\mathcal{M}\equiv\mathcal{M}^{ll}$, and the Christoffel symbols are
given by

$\displaystyle\Gamma^{0}_{00}=$
$\displaystyle-\frac{1}{c^{3}}\partial_{t}U+\frac{1}{c^{5}}\left[(-2+2\beta)U\partial_{t}U-\frac{1}{2}\partial_{t}\Psi_{1}+4\Delta
V^{l}\partial_{l}U\right],$ $\displaystyle\Gamma^{0}_{0j}=$
$\displaystyle-\frac{1}{c^{2}}\partial_{j}U+\frac{1}{c^{4}}\left[(-2+2\beta)U\partial_{j}U-\frac{1}{2}\partial_{j}\Psi_{1}\right],$
$\displaystyle\Gamma^{0}_{jk}=$
$\displaystyle\frac{4}{c^{3}}\Delta\partial_{(j}V_{k)}+\frac{1}{c^{5}}\left[\frac{1}{2}\partial_{t}\mathcal{M}_{jk}+(-8\gamma\Delta-\chi)V_{(j}\partial_{k)}U+(8\Delta-\chi)U\partial_{(j}V_{k)}-\partial_{(j}\mathcal{V}_{k)}\right]$
$\displaystyle+\left\\{\frac{1}{c^{3}}\gamma\partial_{t}U+\frac{1}{c^{5}}\left[(2\gamma+2\omega)U\partial_{t}U+\frac{1}{2}\partial_{t}\Psi_{3}+4\gamma\Delta
V^{l}\partial_{l}U\right]\right\\}\delta_{jk}$ $\displaystyle\Gamma^{j}_{00}=$
$\displaystyle-\frac{1}{c^{2}}\partial_{j}U+\frac{1}{c^{4}}\left[-4\Delta\partial_{t}V_{j}+(2\beta+2\gamma)U\partial_{j}U-\frac{1}{2}\partial_{j}\Psi_{1}\right]$
$\displaystyle+\frac{1}{c^{6}}\left[(-4\Delta+\chi)V^{j}\partial_{t}U+(8\gamma\Delta+\chi)U\partial_{t}V^{j}+\partial_{t}\mathcal{V}^{j}-8\lambda
V^{k}\partial_{j}V^{k}+(-4\beta\gamma-4\gamma^{2}-2\delta+2\omega)U^{2}\partial_{j}U\right.$
$\displaystyle\left.+(-\frac{1}{2}\Psi_{2}+\Psi_{3})\partial_{j}U+\mathcal{M}^{jk}\partial_{k}U+U\partial_{j}(\gamma\Psi_{1}-\frac{1}{2}\Psi_{2})-\frac{1}{2}\partial_{j}(\mathcal{E}+\mathcal{X}+\mathcal{G}+\mathcal{H})\right]$
$\displaystyle\Gamma^{j}_{0k}=$
$\displaystyle\frac{4}{c^{3}}\Delta\partial_{[j}V_{k]}+\frac{1}{c^{5}}\left[\frac{1}{2}\partial_{t}\mathcal{M}_{jk}-4\Delta
V^{j}\partial_{k}U+\chi
V_{[j}\partial_{k]}U+(-8\gamma\Delta-\chi)U\partial_{[j}V_{k]}-\partial_{[j}\mathcal{V}_{k]}\right]$
$\displaystyle+\left\\{\frac{1}{c^{3}}\gamma\partial_{t}U+\frac{1}{c^{5}}\left[(-2\gamma^{2}+2\omega)U\partial_{t}U+\frac{1}{2}\partial_{t}\Psi_{3}\right]\right\\}\delta_{jk}$
$\displaystyle\Gamma^{j}_{kn}=$
$\displaystyle\frac{1}{c^{2}}\gamma(\delta_{jn}\partial_{k}U+\delta_{jk}\partial_{n}U-\delta_{kn}\partial_{j}U)+\frac{1}{c^{4}}\left\\{\frac{1}{2}\partial_{k}\mathcal{M}_{jn}+\frac{1}{2}\partial_{n}\mathcal{M}_{jk}-\frac{1}{2}\partial_{j}\mathcal{M}_{kn}\right.$
(22)
$\displaystyle\left.+\delta_{jn}\partial_{k}\left[(-\gamma^{2}+\omega)U^{2}+\frac{1}{2}\Psi_{3}\right]+\delta_{jk}\partial_{n}\left[(-\gamma^{2}+\omega)U^{2}+\frac{1}{2}\Psi_{3}\right]-\delta_{kn}\partial_{j}\left[(-\gamma^{2}+\omega)U^{2}+\frac{1}{2}\Psi_{3}\right]\right\\}.$

In principle, the form of a symmetric $t^{\alpha\beta}$ and the constants $A$,
$B$ and $B_{k}$ ($k=0,1,...,4$) can be obtained by solving Eq. (20). In
following part of this subsection, we only work out the constraints on the
parameters by taking into account the conservation laws, but not the explicit
form of $t^{\alpha\beta}$.

We notice that Eq. (20) is to turn the quantities involving the fluid
variables on the right-hand side into a combination of gradients and time
derivatives of gravitational fields on the left-hand side. In order to solve
Eq. (20), several transformation tricks would be used. For example, we take
advantage of Eq. (1) and also Eq. (11) in a reverse direction to convert the
various source into field quantities,

$\displaystyle-f=\nabla^{2}\mathrm{P}(f)=\partial_{k}\partial_{k}\mathrm{P}(f),\quad-4\pi\rho^{*}f=\partial_{k}\partial_{k}\Sigma(f),\quad-4\pi\rho^{*}v^{j}f=\partial_{k}\partial_{k}\Sigma^{j}(f),\quad-4\pi\rho^{*}v^{i}v^{j}f=\partial_{k}\partial_{k}\Sigma^{ij}(f),$
(23)

and use the following transformation identity valid for any potential $\Phi$,

$\displaystyle-f\partial_{\alpha}\Phi$ $\displaystyle=$
$\displaystyle-\partial_{\alpha}[\partial_{k}\Phi\partial_{k}\mathrm{P}(f)]+\partial_{k}[\partial_{\alpha}\Phi\partial_{k}\mathrm{P}(f)]+\partial_{k}\Phi\partial_{k}\partial_{\alpha}\mathrm{P}(f),$
$\displaystyle\text{or}\quad-f\partial_{\alpha}\Phi$ $\displaystyle=$
$\displaystyle-\partial_{\alpha}[\partial_{k}\Phi\partial_{k}\mathrm{P}(f)]+2\partial_{k}[\partial_{(\alpha}\Phi\partial_{k)}\mathrm{P}(f)]-\partial_{\alpha}\mathrm{P}(f)\nabla^{2}\Phi.$
(24)

Some identities between potentials,

$\displaystyle\partial_{t}U+\partial_{j}V^{j}=0,$ (25)
$\displaystyle\partial_{t}U^{2}\\!+\partial_{j}(UV^{j}\\!+V_{2}^{j}-\phi_{2}^{j}\\!\\!+2K_{1}^{j}\\!+2K_{2}^{j})=0,$
(26)
$\displaystyle\partial_{t}(7U^{2}\\!+2\Phi_{1}\\!-\\!2\Phi_{2}\\!+4\Phi_{3}\\!+2\ddot{X})\\!+\partial_{j}(8UV^{j}\\!+2V_{1}^{j}\\!+4V_{2}^{j}\\!+4V_{3}^{j}\\!+4V_{4}^{j}\\!-\\!8\phi_{2}^{j}\\!+2\ddot{X}^{j}\\!+16K_{1}^{j}\\!+12K_{2}^{j})=\mathcal{O}(c^{-2}),$
(27)
$\displaystyle\partial_{t}V^{j}+\partial_{i}\left[4\Phi_{1}^{ij}+4P_{2}^{ij}-\delta^{ij}(U^{2}-2\Phi_{2}-4\Phi_{4})\right]=\mathcal{O}(c^{-2}),$
(28)

are also useful. The identities in Eqs. (25) and (26) are actually the results
of the continuity equation Eq. (5), because the continuity equation Eq. (5)
leads to $\partial_{t}F=\int\rho^{*}{{}^{\prime}}(\partial_{t}h+v^{\prime
k}\partial_{k^{\prime}}h)d^{3}x^{\prime}$ for any function
$F(t,x)=\int\rho^{*}{{}^{\prime}}h(t,\mathbf{x},\mathbf{x}^{\prime})d^{3}x^{\prime}$
defined by $\rho^{*}$. The third identity in Eq. (27) and the fourth in Eq.
(28) are equivalent to the conservation equation of Newtonian total energy
(the sum of kinetic, gravitational, and internal energies) Will (1971b) and
the Euler’s equation, respectively,

$\displaystyle\rho^{*}\partial_{t}\left(\frac{1}{2}v^{2}+\Pi\right)+\rho^{*}v^{j}\partial_{j}\left(\frac{1}{2}v^{2}+\Pi\right)+\partial_{j}(Pv^{j})-\rho^{*}v^{j}\partial_{j}U=\mathcal{O}(c^{-2}),$
$\displaystyle\rho^{*}\frac{dv^{j}}{dt}=\rho^{*}\partial_{j}U-\partial_{j}P+\mathcal{O}(c^{-2}),$
(29)

which can be directly derived from Eq. (19) for $\alpha=0$ and $\alpha=j$ to
corresponding order after inserting the continuity equation Eq. (5). Equipped
with the transformation relations in Eqs. (23-28), we are ready to solve Eq.
(20) and obtain the corresponding parameter constraints.

Substituting Eqs. (21) and (22) into Eq. (20) and taking advantage of Eqs.
(23–28), for $\alpha=0$, we have

$\displaystyle\partial_{\beta}\left[\left(1+\frac{1}{c^{2}}AU+\frac{1}{c^{4}}(BU^{2}+\Psi_{4})\right)t^{0\beta}\right]$
$\displaystyle=$
$\displaystyle\frac{1}{c}\frac{1}{4\pi}\partial_{t}\left\\{\frac{1}{2}(6\gamma-2A-5)\partial_{j}U\partial_{j}U\right\\}+\frac{1}{c}\frac{1}{4\pi}\partial_{j}\left\\{-(3\gamma-A-2)\partial_{j}U\partial_{t}U+2(3\gamma-A-3)\partial_{k}U\partial_{[k}V_{j]}\right\\}$
(30)
$\displaystyle+\frac{1}{c^{3}}\frac{1}{4\pi}\partial_{t}\left\\{(-\frac{23}{2}-5A-2B+5\beta+\frac{1}{2}\beta_{2}+15\gamma+6A\gamma-15\gamma^{2}+6\omega)U\partial_{j}U\partial_{j}U+\partial_{j}\left[(-\Psi_{1}+\frac{3}{2}\Psi_{3}-\Psi_{4}+\frac{1}{2}\mathcal{M})\right.\right.$
$\displaystyle\qquad\qquad\quad\left.\left.-\frac{1}{2}(3+A-3\gamma)\Phi_{1}+\frac{1}{2}(A+3-3\gamma)\Phi_{2}-(3+A-3\gamma)\Phi_{3}-\frac{1}{2}(3+A-3\gamma)\ddot{X}\right]\partial_{j}U\right.$
$\displaystyle\qquad\qquad\quad\left.-(3+A-3\gamma)\left(\partial_{j}U\partial_{t}V^{j}-\frac{1}{2}\partial_{j}V^{k}\partial_{k}V^{j}\right)+2\Delta(\partial_{k}V^{j}\partial_{k}V^{j})-\frac{1}{2}(3+A-3\gamma-\beta_{0})(\partial_{t}U)^{2}\right\\}$
$\displaystyle+\frac{1}{c^{3}}\frac{1}{4\pi}\partial_{j}\left\\{-\frac{1}{2}(3+A-3\gamma)\partial_{[k}\left[8((1-\tau)V_{{j}{]}}U)+2V_{1{j]}}+4(1-2\tau)V_{2{j]}}+4V_{3{j]}}+4V_{4{j]}}-8(1-\tau)\phi_{2{j]}}+2\ddot{X}_{j]}\right.\right.$
$\displaystyle\qquad\qquad\quad\left.\left.+16(1-\tau)K_{1{j]}}+12(1-\frac{4}{3}\tau)K_{2{j]}}\right]\partial_{k}U+2\partial_{[k}V_{{j}{]}}\partial_{k}\left(-\frac{3}{2}\Psi_{1}+\frac{3}{2}\Psi_{3}-\Psi_{4}+\frac{1}{2}\mathcal{M}\right)\\!\right.$
$\displaystyle\qquad\qquad\quad\left.-\\!\Delta\partial_{k}V^{j}\partial_{k}(-U^{2}+2\Phi_{2}-4\Phi_{4})\\!+\\!8\Delta\partial_{k}V^{l}\partial_{[k}(\Phi_{1{{j}{]}{l}}}+P_{2{{j}{]}{l}}})-4\Delta\left(V^{k}\partial_{j}U\partial_{k}U-\frac{1}{2}V^{j}\partial_{k}U\partial_{k}U\right)\right.$
$\displaystyle\qquad\qquad\quad\left.+2(-3-2B+6\beta+6\gamma-15\gamma^{2}-2A+6A\gamma+6\omega-4(3+A-3\gamma)\tau)U\partial_{k}U\partial_{[k}V_{j]}\right.$
$\displaystyle\qquad\qquad\quad\left.-(3+A-3\gamma)\partial_{k}V^{j}\partial_{t}V^{k}+\left(\frac{19}{2}+2B-4\beta-\beta_{2}-\frac{27}{2}\gamma+15\gamma^{2}+\frac{9}{2}A-6\gamma
A+2\Delta-6\omega\right)U\partial_{t}U\partial_{j}U\right.$
$\displaystyle\qquad\qquad\quad\left.-\partial_{t}\left(-\Psi_{1}+\frac{3}{2}\Psi_{3}-\Psi_{4}+\frac{1}{2}\mathcal{M}\right)\partial_{j}U+\partial_{j}\left[\frac{1}{2}(3+A-3\gamma-3\beta_{1})\Phi_{1}+\frac{1}{2}(3\gamma-4\Delta+2\beta_{2}-A-3)\Phi_{2}\right.\right.$
$\displaystyle\qquad\qquad\quad\left.\left.+(3+A-3\gamma-\beta_{3})\Phi_{3}+(4\Delta-3\beta_{4})\Phi_{4}+\frac{1}{2}(3+A-3\gamma-\beta_{0})\ddot{X}\right]\partial_{t}U\right\\}+\frac{1}{c^{3}}Q_{1},$

where $\tau$ is a new free parameter introduced during transformation for the
full use of the identities in Eq. (26) and Eq. (27) , and

$\displaystyle
Q_{1}=\partial_{t}U\left[\frac{1}{2}(-3\beta_{1}+2\gamma+1)\rho^{*}v^{2}-(8\pi)^{-1}(-1+2\beta-\beta_{2})\partial_{k}U\partial_{k}U+(1-\beta_{3})\rho^{*}\Pi+(-3\beta_{4}+3\gamma)P\right].$
(31)

Note that apart from the terms in $Q_{1}$, the solution has been arranged into
a combination of time derivatives and gradients of gravitational fields with
the use of transformation tricks (the representative examples are given in
Appendix B; in particular, $\tau$ is introduced in Eq. (67)). Therefore, in
order to keep the result compatible with the expression
$\partial_{\beta}\left[\left(1+c^{-2}AU+c^{-4}(BU^{2}+\Psi_{4})\right)t^{0\beta}\right]$,
$Q_{1}$ must vanish, or equivalently, all the coefficients in $Q_{1}$ must
vanish, namely

$\displaystyle\beta_{1}=\frac{1}{3}(2\gamma+1),$
$\displaystyle\quad\beta_{2}=2\beta-1,\quad\beta_{3}=1,\quad\beta_{4}=\gamma.$
(32)

For $\alpha=j$ in Eq. (20), we first solve the equation to the 1PN order,
i.e., to $\mathcal{O}(c^{-2})$,

$\displaystyle\partial_{\beta}\left[\left(1+\frac{1}{c^{2}}AU+\frac{1}{c^{4}}(BU^{2}+\Psi_{4})\right)t^{j\beta}\right]$
$\displaystyle=$
$\displaystyle\frac{1}{4\pi}\partial_{k}\left\\{\partial_{j}U\partial_{k}U-\delta_{jk}\left(\frac{1}{2}\partial_{l}U\partial_{l}U\right)\right\\}$
(33)
$\displaystyle+\frac{1}{c^{2}}\frac{1}{4\pi}\partial_{t}\left\\{(8\Delta+2(A-5\gamma+1))\partial_{k}U\partial_{[j}V_{k]}+(4\Delta-\beta_{0}+A-5\gamma+1)(\partial_{t}U\partial_{j}U)\right\\}$
$\displaystyle+\frac{1}{c^{2}}\frac{1}{4\pi}\partial_{k}\left\\{4\Delta\left[\partial_{k}V^{l}\partial_{l}V^{j}\\!+\\!\partial_{j}V^{l}\partial_{l}V^{k}\\!-\\!\partial_{j}V^{l}\partial_{k}V^{l}\\!-\\!\partial_{l}V^{j}\partial_{l}V^{k}\\!+\\!\partial_{t}V^{j}\partial_{k}U\\!+\\!\partial_{t}V^{k}\partial_{j}U\\!+\\!\delta_{jk}\partial_{l}V_{m}\partial_{[l}V_{m]}\\!-\\!\delta_{jk}\partial_{t}V^{l}\partial_{l}U\right]\right.$
$\displaystyle\qquad\qquad\quad-\frac{1}{2}\left(4\Delta\\!-\beta_{0}\\!+A\\!-5\gamma\\!+1\right)\delta_{jk}(\partial_{t}U\partial_{t}U)\\!+\frac{1}{4}\partial_{(j}U\partial_{k)}\Psi_{1}\\!-\frac{1}{2}\delta_{jk}\partial_{l}U\partial_{l}\Psi_{1}$
$\displaystyle\qquad\qquad\quad\left.+(2\beta-A-2+5\gamma-\beta_{2})\left[-(U\partial_{k}U\partial_{j}U)+\frac{1}{2}\delta_{jk}(U\partial_{l}U\partial_{l}U)\right]\right.$
$\displaystyle\qquad\qquad\quad+(A-5\gamma+1)\left[\partial_{j}V^{l}\partial_{l}V^{k}\\!+\partial_{t}V^{k}\partial_{j}U\\!+\partial_{k}V^{j}\partial_{t}U-\delta_{jk}\left(\frac{1}{2}\partial_{l}V^{m}\partial_{m}V^{l}\\!+\partial_{t}V^{l}\partial_{l}U\right)\right]$
$\displaystyle\qquad\qquad\quad+(A\\!-5\gamma\\!+1)\left[\frac{1}{4}\partial_{j}U\partial_{k}(-2\Phi_{2}+U^{2}+4\Phi_{4})-\frac{1}{4}\delta_{jk}\partial_{l}U\partial_{l}(-\Phi_{2}+U^{2}+2\Phi_{4})\right]$
$\displaystyle\qquad\qquad\quad\left.+(A\\!-5\gamma\\!+1)\left[\partial_{k}(\Phi_{1}^{jl}+P_{2}^{jl})\partial_{l}U-\partial_{l}(\Phi_{1}^{jk}+P_{2}^{jk})\partial_{l}U\right]\right\\}+\frac{1}{c^{2}}Q_{2}^{j}+\mathcal{O}(c^{-4}),$

where

$\displaystyle
Q_{2}^{j}=\partial_{j}U\left[\left(-\frac{1}{2}-\gamma+\frac{3}{2}\beta_{1}\right)\rho^{*}v^{2}+(8\pi)^{-1}(2\beta-\beta_{2}-1)\partial_{k}U\partial_{k}U+(-1+\beta_{3})\rho^{*}\Pi+(-3\gamma+3\beta_{4})P\right].$
(34)

Similarly, requiring that all the coefficients in $Q_{2}$ vanish leads to the
same constraints on parameters as those in Eq. (32). However, the symmetry of
$t^{jk}$ has not been fully considered in the above transformation. For
example, the symmetric counterpart of the term
$\partial_{k}(\Phi_{1}^{jl}+P_{2}^{jl})\partial_{l}U$, namely
$\partial_{j}(\Phi_{1}^{kl}+P_{2}^{kl})\partial_{l}U$, is not contained in Eq.
(33). Here $t^{jk}$ is required to be symmetric, and hence the coefficients of
such non-symmetric terms should vanish, which leads to a constraint on the
constant $A$ introduced in Eq. (17), i.e.

$\displaystyle A=5\gamma-1.$ (35)

In addition, $t^{j0}$ should also be symmetric. Comparing the time derivative
part of Eq. (33) and the gradient part of Eq. (30) at the 1PN order, we get
another two constraints,

$\displaystyle\Delta=\frac{1}{2}(\gamma+1),\qquad\beta_{0}=1.$ (36)

As a matter of fact, the solution for $\alpha=j$ to the 1PN order and the
consequent parameter constraints have been obtained in the fundamental work
for the PPN formalism by Will in Will (1971b), despite with some different
arguments.

For $\alpha=j$, at the 2PN order, substituting Eqs. (21) and (22) and
inserting the PPN parameter constraints in Eqs. (32), (35) and (36) into Eq.
(20), we obtain

$\displaystyle\partial_{\beta}\left[\left(1+\frac{1}{c^{2}}AU+\frac{1}{c^{4}}(BU^{2}+\Psi_{4})\right)t^{j\beta}\right]$
$\displaystyle\supset\,$
$\displaystyle\frac{1}{c^{4}}\\!\left\\{(\\!-\\!1\\!-\\!2B\\!+\\!2\beta\\!-\\!10\gamma\\!+\\!15\gamma^{2}\\!+\\!10\omega)\rho^{*}v^{j}v^{k}U\partial_{k}U\\!+(\\!-8\Delta\\!+\\!\chi)\rho^{*}v^{k}V^{j}\partial_{k}U\\!+\\!\rho^{*}v^{l}v^{k}\partial_{k}\mathcal{M}^{jl}\\!+\\!P\partial_{k}\mathcal{M}^{jk}\\!+\\!\rho^{*}\\!\mathcal{M}^{jk}\partial_{k}U\right.$
(37)
$\displaystyle\qquad+\rho^{*}v^{j}v^{k}\partial_{k}\left(-\frac{1}{2}\Psi_{1}+\frac{5}{2}\Psi_{3}-\Psi_{4}+\frac{1}{2}\mathcal{M}\right)+\rho^{*}v^{k}\left[\\!-2\Delta
v^{2}\\!-\\!(4\Delta-\chi)U\\!-\\!4\Delta\Pi\\!-\\!4\Delta
P\right]\partial_{k}V^{j}\\!+\rho^{*}v^{k}\partial_{k}\mathcal{V}^{j}$
$\displaystyle\qquad-\frac{1}{2}\rho^{*}\partial_{j}(\mathcal{G}\\!+\mathcal{H}\\!+\mathcal{X}\\!+\mathcal{E})\\!+\\!(-1\\!-2B\\!+2\beta\\!-7\gamma\\!+12\gamma^{2}\\!+4\omega)PU\partial_{j}U\\!+\\!(\\!-1\\!+2\beta)\rho^{*}\Pi
U\partial_{j}U$
$\displaystyle\qquad+\left(-2-B+4\beta-5\gamma+\frac{15}{2}\gamma^{2}-2\delta+5\omega\right)\rho^{*}U^{2}\partial_{j}U+\rho^{*}\left(-\Psi_{1}-\frac{1}{2}\Psi_{2}+\frac{5}{2}\Psi_{3}-\Psi_{4}+\frac{1}{2}\mathcal{M}\right)\partial_{j}U$
$\displaystyle\qquad+\rho^{*}v^{2}\left[\left(\\!-\frac{3}{8}\\!-\frac{\gamma}{2}\right)v^{2}\\!+\\!\left(-\frac{3}{2}\\!+\beta\\!-2\gamma\\!-2\omega\right)U\\!+\\!\left(\\!-\frac{1}{2}\\!-\gamma\right)\Pi\\!+\\!(\\!-1\\!-\gamma)P/\rho^{*}\right]\partial_{j}U\\!+\\!(4\Delta-\chi)\rho^{*}v^{k}V^{k}\partial_{j}U$
$\displaystyle\qquad+\rho^{*}v^{2}\partial_{j}\left(-\frac{1}{4}\Psi_{1}-\frac{1}{2}\Psi_{3}\right)+\rho^{*}U\partial_{j}\left(-\frac{1}{2}\Psi_{1}-\frac{1}{2}\Psi_{2}\right)+\rho^{*}\Pi\partial_{j}\left(-\frac{1}{2}\Psi_{1}\right)+P\partial_{j}\left(-\frac{1}{2}\Psi_{1}+\Psi_{3}-\Psi_{4}\right)$
$\displaystyle\qquad-8\lambda\rho^{*}V^{k}\partial_{j}V^{k}+\rho^{*}v^{k}\left[2\Delta
v^{2}\\!+(4\Delta-\chi)U\\!+4\Delta\Pi\\!+4\Delta
P\right]\partial_{j}V^{k}-\rho^{*}v^{k}\partial_{j}\mathcal{V}^{k}-\frac{1}{2}\rho^{*}v^{k}v^{l}\partial_{j}\mathcal{M}^{kl}$
$\displaystyle\qquad+(-1-2B+2\beta-10\gamma+15\gamma^{2}+10\omega)\rho^{*}Uv^{j}\partial_{t}U+(-4\Delta+\chi)\rho^{*}V^{j}\partial_{t}U+\rho^{*}v^{j}\partial_{t}\left(-\frac{1}{2}\Psi_{1}+\frac{5}{2}\Psi_{3}-\Psi_{4}+\frac{1}{2}\mathcal{M}\right)$
$\displaystyle\qquad\left.+\rho^{*}v^{k}\partial_{t}\mathcal{M}^{jk}+\rho^{*}\left[-2\Delta
v^{2}\\!-(4\Delta-\chi)U\\!-4\Delta\Pi\right]\partial_{t}V^{j}+\rho^{*}\partial_{t}\mathcal{V}^{j}\right\\}.$

Here $\supset$ denotes that only the terms at 2PN order are included. To make
the transformation procedure easier, we can first determine several 2PN
parameter constraints by requiring the symmetry of $t^{0j}$ at the 2PN order.
For example,
$\partial_{j}\left[\left(1+c^{-2}AU+c^{-4}(BU^{2}+\Psi_{4})\right)t^{0j}\right]$
in Eq. (30) contains the terms $(4\pi
c^{3})^{-1}\partial_{j}\\{-1/2(3+A-3\gamma)\partial_{[k}[2V_{1{j]}}+4(1-2\tau)V_{2{j]}}+4V_{3{j]}}+4V_{4{j]}}-8(1-\tau)\phi_{2{j]}}+2\ddot{X}_{j]}+16(1-\tau)K_{1{j]}}+12(1-4/3\tau)K_{2{j]}}]\partial_{k}U\\}$
and $(4\pi
c^{3})^{-1}\partial_{j}\\{8\Delta\partial_{k}V^{l}\partial_{[k}(\Phi_{1{{j}{]}{l}}}+P_{2{{j}{]}{l}}})\\}$,
and then the solution of
$\partial_{0}\left[\left(1+c^{-2}AU+c^{-4}(BU^{2}+\Psi_{4})\right)t^{j0}\right]$
in Eq. (37) should correspondingly contain $(4\pi
c^{4})^{-1}\partial_{t}\\{-1/2(3+A-3\gamma)\partial_{[k}[2V_{1{j]}}+4(1-2\tau)V_{2{j]}}+4V_{3{j]}}+4V_{4{j]}}-8(1-\tau)\phi_{2{j]}}+2\ddot{X}_{j]}+16(1-\tau)K_{1{j]}}+12(1-4/3\tau)K_{2{j]}}]\partial_{k}U\\}$
and $(4\pi
c^{4})^{-1}\partial_{t}\\{8\Delta\partial_{k}V^{l}\partial_{[k}(\Phi_{1{{j}{]}{l}}}+P_{2{{j}{]}{l}}})\\}$,
which can be obtained respectively from $\rho^{*}\partial_{t}\mathcal{V}^{j}$
and $\rho^{*}v^{k}\partial_{t}\mathcal{M}^{jk}$ in Eq. (37). Specifically, we
take the transformations as follows,

$\displaystyle\rho^{*}\partial_{t}\mathcal{V}^{j}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\partial_{t}(\partial_{k}U\partial_{[k}\mathcal{V}_{j]})+\frac{1}{4\pi}\partial_{k}\left[-\partial_{k}U\partial_{t}\mathcal{V}^{j}-\partial_{j}U\partial_{t}\mathcal{V}^{k}+\delta_{jk}(\partial_{l}U\partial_{t}\mathcal{V}^{l})\right]$
(38)
$\displaystyle+\frac{1}{4\pi}\left(-\partial_{t}\partial_{k}U\partial_{k}\mathcal{V}^{j}+\partial_{t}\partial_{k}U\partial_{j}\mathcal{V}^{k}+\partial_{j}U\partial_{k}\partial_{t}\mathcal{V}^{k}\right),$
$\displaystyle\rho^{*}v^{k}\partial_{t}\mathcal{M}^{jk}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\partial_{t}(\partial_{k}V^{l}\partial_{[k}\mathcal{M}_{j]l})+\frac{1}{4\pi}\partial_{k}\left[-\partial_{k}V^{l}\partial_{t}\mathcal{M}^{jl}-\partial_{j}V^{l}\partial_{t}\mathcal{M}^{kl}+\delta_{jk}(\partial_{l}V^{m}\partial_{t}\mathcal{M}^{lm})\right]$
(39)
$\displaystyle+\frac{1}{4\pi}(-\partial_{k}\partial_{t}V^{l}\partial_{k}\mathcal{M}^{jl}+\partial_{t}\partial_{k}V^{l}\partial_{j}\mathcal{M}^{kl}+\partial_{j}V^{l}\partial_{k}\partial_{t}\mathcal{M}^{kl}),$

and then the match between $t^{0j}$ and $t^{j0}$ provides constraints on the
parameters in $\mathcal{V}^{j}$ and $\mathcal{M}^{jk}$,

$\displaystyle\chi_{1}\\!=\\!\chi_{3}\\!=\\!\chi_{4}\\!=\\!\chi_{0}\\!=\\!\frac{1}{2}(\gamma\\!+\\!1),\quad\chi_{2}\\!=\\!\frac{1}{2}(\gamma+1)(1\\!-\\!2\tau)$
$\displaystyle,\quad\chi_{5}\\!=\\!\frac{1}{2}(\gamma\\!+\\!1)(1\\!-\\!\tau),\quad\iota_{1}\\!=\\!\frac{1}{2}(\gamma\\!+\\!1)(1\\!-\\!\tau),\quad\iota_{2}\\!=\\!\frac{1}{2}(\gamma\\!+\\!1)(1\\!-\\!\frac{4}{3}\tau);\\!\\!\\!\\!\\!$
$\displaystyle\theta_{1}$
$\displaystyle=\theta_{2}=\Delta=\frac{1}{2}(\gamma+1).$ (40)

Since $(4\pi)^{-1}\partial_{j}U\partial_{k}\partial_{t}\mathcal{V}^{k}$ and
$(4\pi)^{-1}\partial_{j}V^{l}\partial_{k}\partial_{t}\mathcal{M}^{kl}$ on the
right-hand sides of Eq. (38) and Eq. (39) satisfying the above constraints can
be related to the identities of potentials in Eqs. (26), (27) and (28), for
convenience, we introduce two new expressions $\bar{\mathcal{V}}^{j}$ and
$\bar{\mathcal{M}}^{jk}$ as follows,

$\displaystyle\bar{\mathcal{V}}^{j}\\!$ $\displaystyle\equiv$
$\displaystyle-8(1-\tau)UV^{j}-2V_{1}^{j}-4(1-2\tau)V_{2}^{j}-4V_{3}^{j}-4V_{4}^{j}+8(1-\tau)\phi_{2}^{j}-2\ddot{X}^{j}-16(1-\tau)K_{1}^{j}-12(1-\frac{4}{3}\tau)K_{2}^{j},$
$\displaystyle\bar{\mathcal{M}}^{jk}\\!$ $\displaystyle\equiv$
$\displaystyle\\!4\Phi_{1}^{jk}+4P_{2}^{jk}.$ (41)

Using Eq. (40) and $\bar{\mathcal{V}}^{j}$ and $\bar{\mathcal{M}}^{jk}$, after
a tedious calculation, we obtain

$\displaystyle\partial_{\beta}\left[\left(1+\frac{1}{c^{2}}AU+\frac{1}{c^{4}}(BU^{2}+\Psi_{4})\right)t^{j\beta}\right]$
$\displaystyle\supset\,$
$\displaystyle\frac{1}{c^{4}}\frac{1}{4\pi}\partial_{t}\left\\{\Delta\left[2\partial_{k}((7\\!-\\!8\tau)U^{2}\\!+2\Phi_{1}\\!-\\!2\Phi_{2}\\!+4\Phi_{3}\\!+2\ddot{X})\partial_{[j}V_{k]}+2\partial_{(t}((7-8\tau)U^{2}+2\Phi_{1}-2\Phi_{2}+4\Phi_{3}+2\ddot{X})\partial_{j)}U\right.\right.$
(42)
$\displaystyle\qquad\qquad\quad\,\,\,\left.+2\partial_{k}U\partial_{[k}\bar{\mathcal{V}}_{j]}+32(1-\tau)U\partial_{k}U\partial_{[k}V_{j]}+4(-V^{k}\partial_{k}U\partial_{j}U+\frac{1}{2}V^{j}\partial_{k}U\partial_{k}U-\partial_{t}V^{k}\partial_{k}V^{j})\right.$
$\displaystyle\qquad\qquad\quad\,\,\,\left.-\partial_{k}V^{j}\partial_{k}(2\Phi_{2}-U^{2}-4\Phi_{4})-\partial_{t}U\partial_{j}(2\Phi_{2}-U^{2}-4\Phi_{4})-20(1-\frac{4}{5}\tau)U\partial_{t}U\partial_{j}U+2\partial_{k}V^{l}\partial_{[k}\bar{\mathcal{M}}_{j]l}\,\right]$
$\displaystyle\qquad\qquad\left.+4\partial_{(t}U\partial_{j)}\left(-\frac{3}{4}\mu_{1}\Phi_{1}+\frac{1}{2}\mu_{2}\Phi_{2}-\frac{1}{2}\mu_{3}\Phi_{3}-\frac{3}{2}\mu_{4}\Phi_{4}-\frac{1}{4}\mu_{0}\ddot{X}\right)\right\\}$
$\displaystyle\\!+\frac{1}{c^{4}}\\!\frac{1}{4\pi}\\!\partial_{k}\\!\left\\{\\!\Delta\\!\left[\\!-(\partial_{k}V^{l}\partial_{l}\bar{\mathcal{V}}^{j}\\!+\\!\partial_{j}V^{l}\partial_{l}\bar{\mathcal{V}}^{k}\\!+\\!\partial_{k}\bar{\mathcal{V}}^{l}\partial_{l}V^{j}\\!+\\!\partial_{j}\bar{\mathcal{V}}^{l}\partial_{l}V^{k}\\!-\\!2\partial_{(k}V^{l}\partial_{j)}\bar{\mathcal{V}}^{l}\\!-\\!2\partial_{l}V^{(k}\partial_{l}\bar{\mathcal{V}}^{j)}\\!+\\!\partial_{k}U\partial_{t}\bar{\mathcal{V}}^{j}\\!+\\!\partial_{j}U\partial_{t}\bar{\mathcal{V}}^{k})\right.\right.$
$\displaystyle\qquad\qquad\quad\,\left.\left.+2\partial_{(k}((7\\!-\\!8\tau)U^{2}\\!+\\!2\Phi_{1}\\!-\\!2\Phi_{2}\\!+\\!4\Phi_{3}\\!+\\!2\ddot{X})\partial_{t}V_{j)}\\!+\\!\frac{1}{4}\left(\partial_{m}\bar{\mathcal{M}}^{kl}\partial_{m}\bar{\mathcal{M}}^{jl}\\!\\!-\\!\partial_{k}\bar{\mathcal{M}}^{ml}\partial_{m}\bar{\mathcal{M}}^{jl}\\!\\!-\\!\partial_{j}\bar{\mathcal{M}}^{ml}\partial_{m}\bar{\mathcal{M}}^{kl}\right.\right.\right.$
$\displaystyle\qquad\qquad\quad\,\,\left.\left.\\!+\frac{1}{2}\partial_{k}\bar{\mathcal{M}}^{ml}\partial_{j}\bar{\mathcal{M}}^{ml}\\!+\\!2\bar{\mathcal{M}}^{jk}\partial_{l}U\partial_{l}U\\!-\\!4\bar{\mathcal{M}}^{jm}\partial_{k}U\partial_{m}U\\!-\\!4\bar{\mathcal{M}}^{km}\partial_{m}U\partial_{j}U\right)\\!-\\!\partial_{k}V^{l}\partial_{t}\bar{\mathcal{M}}^{jl}\\!-\\!\partial_{j}V^{l}\partial_{t}\bar{\mathcal{M}}^{kl}\right.$
$\displaystyle\qquad\qquad\quad\,\,\left.+\frac{1}{2}(\partial_{(j}\bar{\mathcal{M}}_{k)l}\\!-\\!\partial_{l}\bar{\mathcal{M}}_{jk})\partial_{l}(2\Phi_{2}\\!\\!-\\!U^{2}\\!\\!-\\!4\Phi_{4})\\!+\\!2\partial_{(k}V_{j)}\partial_{t}(2\Phi_{2}\\!\\!-\\!U^{2}\\!+\\!4\Phi_{4})\\!\\!-\\!2\partial_{t}V_{(k}\partial_{j)}(2\Phi_{2}\\!\\!-\\!U^{2}\\!\\!-\\!4\Phi_{4})\right.$
$\displaystyle\qquad\qquad\quad\,\,\left.+2(2\Phi_{2}\\!\\!-\\!U^{2}\\!\\!-\\!4\Phi_{4})\partial_{k}U\partial_{j}U\\!-\\!\frac{1}{8}\partial_{j}(2\Phi_{2}\\!\\!-\\!U^{2}\\!\\!-\\!4\Phi_{4})\partial_{k}(2\Phi_{2}\\!-\\!U^{2}\\!-\\!4\Phi_{4})\\!+\\!\frac{1}{4}\partial_{(k}U^{2}\partial_{j)}(2\Phi_{2}\\!\\!-\\!\frac{1}{2}U^{2}\\!\\!-\\!4\Phi_{4})\right.$
$\displaystyle\qquad\qquad\quad\,\,\left.+\\!16\\!\left(\\!-\frac{1}{2}V_{(k}\partial_{j)}U\partial_{t}U-2(1-\tau)U\partial_{(k}V^{l}\partial_{l}V_{j)}+(1-\tau)U\partial_{k}V^{l}\partial_{j}V^{l}+(1-\tau)U\partial_{l}V^{k}\partial_{l}V^{j}\right.\right.$
$\displaystyle\qquad\qquad\quad\,\,\left.\left.-2(1-\tau)U\partial_{(k}U\partial_{t}V_{j)}+(1-\tau)V^{l}\partial_{(k}V^{l}\partial_{j)}U+\frac{1}{4}\partial_{t}V^{k}\partial_{t}V^{j}\right)\right]$
$\displaystyle\qquad\quad\quad+\partial_{(k}U\partial_{j)}(\mathcal{G}+\mathcal{H}+\mathcal{E}+\mathcal{X})-2\partial_{(k}\ddot{X}\partial_{j)}\left(-\frac{3}{4}\mu_{1}\Phi_{1}+\frac{1}{2}\mu_{2}\Phi_{2}-\frac{1}{2}\mu_{3}\Phi_{3}-\frac{3}{2}\mu_{4}\Phi_{4}-\frac{1}{8}\mu_{0}\ddot{X}\right)$
$\displaystyle\qquad\quad\quad-2\partial_{(k}\Phi_{1}\partial_{j)}\left[\frac{1}{2}(\\!-\\!\frac{1}{4}\\!-\\!\frac{1}{2}\gamma\\!+\\!\frac{1}{2}\omega_{1})\Phi_{1}\\!+(\\!-\\!\frac{1}{2}\\!+\\!\beta\\!+\\!\omega_{2})\Phi_{2}\\!+(\\!-\\!\frac{1}{2}\\!-\omega_{3})\Phi_{3}\\!+(\\!-\\!\frac{3}{2}\gamma\\!+\\!\omega_{4})\Phi_{4}\right]\\!$
$\displaystyle\qquad\quad\quad-2\partial_{(k}\Phi_{3}\partial_{j)}\left[(\\!-1\\!+\\!2\beta)\Phi_{2}\\!-\\!\frac{1}{2}\Phi_{3}\\!-\\!3\gamma\Phi_{4}\right]\\!-\\!2\partial_{(k}\Phi_{2}\partial_{j)}\left[\frac{1}{2}(\\!-1\\!+2\beta\\!-2\delta_{2}\\!+\frac{1}{2}\Delta)\Phi_{2}\\!+(\\!-3\gamma+6\delta_{4}\\!-\Delta)\Phi_{4}\right]$
$\displaystyle\qquad\quad\quad\left.-2\partial_{(k}\Phi_{4}\partial_{j)}\left[\frac{1}{2}(-B4-3\gamma-2\omega_{4}-6\Delta)\Phi_{4}\right]\right\\}$
$\displaystyle\\!+\frac{1}{c^{4}}\\!\frac{1}{4\pi}\\!\partial_{j}\\!\left\\{\\!\Delta\\!\left[\\!-(\partial_{l}V^{m}\partial_{l}\bar{\mathcal{V}}^{m}-\partial_{l}V^{m}\partial_{m}\bar{\mathcal{V}}^{l}-\partial_{l}U\partial_{t}\bar{\mathcal{V}}^{l})-\partial_{l}((7-8\tau)U^{2}+2\Phi_{1}-2\Phi_{2}+4\Phi_{3}+2\ddot{X})\partial_{t}V^{l}\right.\right.$
$\displaystyle\qquad\qquad\quad\,-\partial_{t}((7-8\tau)U^{2}+2\Phi_{1}-2\Phi_{2}+4\Phi_{3}+2\ddot{X})\partial_{t}U+10(1-\frac{4}{5}\tau)U\partial_{t}U\partial_{t}U+\\!\partial_{t}U\partial_{t}(2\Phi_{2}\\!-\\!U^{2}+4\Phi_{4})$
$\displaystyle\qquad\qquad\quad\,+\frac{1}{4}\\!\left(\\!\frac{1}{2}\partial_{m}\bar{\mathcal{M}}^{ln}\partial_{n}\bar{\mathcal{M}}^{ml}\\!\\!-\\!\frac{1}{4}\partial_{m}\bar{\mathcal{M}}^{nl}\partial_{m}\bar{\mathcal{M}}^{nl}\\!\\!+\\!2\bar{\mathcal{M}}^{ml}\partial_{m}U\\!\partial_{l}U\\!\\!+\\!4\partial_{l}V^{m}\\!\partial_{t}\bar{\mathcal{M}}^{lm}\\!\\!\right)\\!\\!-\\!\frac{1}{8}\partial_{l}U^{2}\partial_{l}(2\Phi_{2}\\!-\\!\frac{1}{2}U^{2}\\!\\!-\\!\\!4\Phi_{4})$
$\displaystyle\qquad\qquad\quad\,+\\!16\\!\left(\\!(1\\!-\\!\tau)U\\!\partial_{l}U\\!\partial_{t}V^{l}\\!\\!+\\!\frac{1}{4}V^{l}\partial_{l}U\\!\partial_{t}U\\!\\!-\\!\\!\frac{1}{2}(1\\!-\\!\tau)V^{l}\partial_{m}\\!V^{l}\\!\partial_{m}U\\!\\!-\\!\frac{1}{2}(1\\!-\\!\tau)U\\!\partial_{l}V^{m}\partial_{l}\\!V^{m}\\!\\!+\\!\frac{1}{2}(1\\!-\\!\tau)U\\!\partial_{l}V^{m}\partial_{m}\\!V^{l}\\!\right)\\!\\!$
$\displaystyle\qquad\qquad\quad\,\left.+\partial_{t}V^{l}\partial_{l}(2\Phi_{2}\\!-\\!U^{2}\\!-\\!4\Phi_{4})\\!-\\!(2\Phi_{2}\\!-\\!U^{2}\\!-\\!4\Phi_{4})\partial_{l}U\partial_{l}U\\!+\\!\frac{3}{16}\partial_{l}(2\Phi_{2}\\!-\\!U^{2}\\!-\\!4\Phi_{4})\partial_{l}(2\Phi_{2}\\!-\\!U^{2}\\!-\\!4\Phi_{4})\right]$
$\displaystyle\qquad\quad\quad-\frac{1}{2}\partial_{l}U\partial_{l}(\mathcal{G}+\mathcal{H}+\mathcal{E}+\mathcal{X})-2\partial_{t}U\partial_{t}(-\frac{3}{4}\mu_{1}\Phi_{1}+\frac{1}{2}\mu_{2}\Phi_{2}-\frac{1}{2}\mu_{3}\Phi_{3}-\frac{3}{2}\mu_{4}\Phi_{4}-\frac{1}{4}\mu_{0}\ddot{X})$
$\displaystyle\qquad\quad\quad+\partial_{l}\ddot{X}\partial_{l}(-\frac{3}{4}\mu_{1}\Phi_{1}+\frac{1}{2}\mu_{2}\Phi_{2}-\frac{1}{2}\mu_{3}\Phi_{3}-\frac{3}{2}\mu_{4}\Phi_{4}-\frac{1}{8}\mu_{0}\ddot{X})$
$\displaystyle\qquad\quad\quad+\partial_{l}\Phi_{1}\partial_{l}\left[\frac{1}{2}(\\!-\frac{1}{4}\\!-\frac{1}{2}\gamma\\!+\\!\frac{1}{2}\omega_{1})\Phi_{1}\\!+(\\!-\frac{1}{2}\\!+\\!\beta\\!+\\!\omega_{2})\Phi_{2}\\!+(\\!-\frac{1}{2}\\!-\omega_{3})\Phi_{3}+(\\!-\frac{3}{2}\gamma\\!+\\!\omega_{4})\Phi_{4}\right]$
$\displaystyle\qquad\quad\quad\left.+\partial_{l}\Phi_{3}\partial_{l}\left[(\\!-1\\!+\\!2\beta)\Phi_{2}\\!-\frac{1}{2}\Phi_{3}\\!-3\gamma\Phi_{4}\right]+\partial_{l}\Phi_{2}\partial_{l}\left[\frac{1}{2}(-1+2\beta-2\delta_{2}+\frac{1}{2}\Delta)\Phi_{2}+(-3\gamma+6\delta_{4}-\Delta)\Phi_{4}\right]\right.$
$\displaystyle\qquad\quad\quad\left.+\partial_{l}\Phi_{4}\partial_{l}\left[\frac{1}{2}(-B4-3\gamma-2\omega_{4}-6\Delta)\Phi_{4}\right]\right\\}+\frac{1}{c^{4}}Q_{3}^{j},$

where

$\displaystyle Q_{3}^{j}=$
$\displaystyle\frac{1}{4\pi}\partial_{j}U\left[6(\Delta(1-\frac{4}{3}\tau)-\xi_{1})\partial_{t}U\partial_{t}U+4(\Delta-\xi_{2})U\partial_{t}\partial_{t}U+8(\Delta-\xi_{3})V^{k}\partial_{k}\partial_{t}U-8(\Delta(1-\tau)-\xi_{4})\partial_{k}V^{l}\partial_{l}V^{k}\right.$
$\displaystyle\qquad\qquad\,\left.+\frac{1}{2}\xi_{5}\partial_{t}V^{k}\partial_{k}U+4(\Delta-\eta_{1})\Phi^{kl}\partial_{k}\partial_{l}U+4(\Delta-\eta_{2})P_{2}^{kl}\partial_{k}\partial_{l}U\right]+8(\Delta(1-\tau)-\lambda)\rho^{*}V^{k}\partial_{j}V^{k}$
$\displaystyle+(\chi-8\Delta\tau)\left[-\rho^{*}v^{k}U(\partial_{j}V^{k}-\partial_{k}V^{j})+\rho^{*}V^{j}\partial_{t}U+\rho^{*}U\partial_{t}V^{j}+\rho^{*}v^{k}V^{j}\partial_{k}U\right]$
$\displaystyle+\rho^{*}v^{j}v^{k}\partial_{k}\left[-\frac{1}{2}\Psi_{1}+\frac{5}{2}\Psi_{3}-\Psi_{4}+\frac{1}{2}\bar{\mathcal{M}}+\Delta(2\Phi_{2}-U^{2}-4\Phi_{4})+\frac{1}{2}(-1-2B+2\beta-10\gamma+15\gamma^{2}+10\omega)U^{2}\right]$
$\displaystyle+\rho^{*}v^{j}\partial_{t}\left[-\frac{1}{2}\Psi_{1}+\frac{5}{2}\Psi_{3}-\Psi_{4}+\frac{1}{2}\bar{\mathcal{M}}+\Delta(2\Phi_{2}-U^{2}-4\Phi_{4})+\frac{1}{2}(-1-2B+2\beta-10\gamma+15\gamma^{2}+10\omega)U^{2}\right]$
$\displaystyle+\partial_{j}U\left\\{\rho^{*}\left[(-1-B_{1}-2\gamma+3\delta_{1}-\frac{5}{2}\omega_{1}+2\Delta-\frac{3}{2}\kappa_{1})\Phi_{1}+(-2-B_{2}+4\beta-2\delta_{2}-5\omega_{2}+6\Delta-\kappa_{2})\Phi_{2}\right.\right.$
$\displaystyle\qquad\qquad\quad\left.\left.+(\\!-\\!2\\!-\\!B_{3}\\!+2\delta_{3}\\!+5\omega_{3}\\!-\\!\kappa_{3})\Phi_{3}+(\\!-\\!B_{4}\\!-\\!6\gamma\\!+6\delta_{4}\\!-\\!5\omega_{4}\\!+\kappa_{4}\\!-\\!8\Delta)\Phi_{4}+(\\!-\\!B_{0}\\!-1\\!+\delta_{0}\\!+\frac{5}{2}\omega_{0}\\!-\\!\frac{1}{2}\kappa_{0})\ddot{X}\right]\right.$
$\displaystyle\qquad\qquad\rho^{*}v^{2}\left[(\\!-\frac{3}{8}\\!-\frac{1}{2}\gamma+\frac{7}{8}\sigma_{1})v^{2}+(\\!-\frac{3}{2}+\beta\\!-2\gamma\\!-2\omega+\frac{9}{2}\sigma_{2})U+(\\!-\frac{1}{2}\\!-\gamma+\frac{3}{2}\sigma_{4})\Pi+(\\!-1\\!-\gamma+2\sigma_{5})P/\rho^{*}\right]$
$\displaystyle\qquad\qquad+\rho^{*}U\left[(-2-B+4\beta-5\gamma+\frac{15}{2}\gamma^{2}-2\delta+5\omega-3\Delta+\frac{3}{2}\kappa)U+(-1+2\beta-\sigma_{6})\Pi\right.$
$\displaystyle\qquad\qquad\qquad\quad\left.\left.+(-1-2B+2\beta-7\gamma+12\gamma^{2}+4\omega+6\sigma_{7}-4\Delta)P/\rho^{*}\right]+(4\Delta-\chi-4\sigma_{3})\rho^{*}v^{l}V^{l}\right\\}$
$\displaystyle+\rho^{*}v^{2}\partial_{j}\left[(-\frac{1}{4}-\frac{1}{2}\omega_{0}+\frac{3}{4}\mu_{1})\ddot{X}\right]+\rho^{*}\Pi\partial_{j}\left[(-\gamma+\omega_{3})\Phi_{1}+(-\frac{1}{2}+\frac{1}{2}\mu_{3})\ddot{X}\right]$
$\displaystyle+\rho^{*}U\partial_{j}\left[(-\gamma+3\delta_{1}-\beta-\omega_{2})\Phi_{1}+(2\delta_{3}-2\beta)\Phi_{3}+(-\frac{1}{2}+\delta_{0}-\frac{1}{2}\mu_{2})\ddot{X}\right]$
$\displaystyle\\!+\\!P\partial_{j}\\!\\!\left[\\!(\\!-\frac{1}{2}\\!\\!-\\!B_{1}\\!\\!-\\!\gamma\\!\\!-\\!\omega_{1}\\!\\!+\\!\frac{3}{2}\gamma\\!\\!-\\!\\!\omega_{4}\\!)\Phi_{1}\\!\\!+\\!\\!(\\!-\\!1\\!\\!-\\!\\!B_{2}\\!+\\!\\!2\beta\\!\\!-\\!\\!2\omega_{2}\\!\\!+\\!4\Delta\\!\\!+\\!3\gamma\\!\\!-\\!\\!6\delta_{4}\\!)\Phi_{2}\\!\\!+\\!\\!(\\!-1\\!\\!-\\!\\!B_{3}\\!+\\!\\!2\omega_{3}\\!\\!+\\!3\gamma\\!)\Phi_{3}\\!\\!+\\!(\\!-\frac{1}{2}\\!-\\!\\!B_{0}\\!\\!+\\!\\!\omega_{0}\\!\\!+\\!\frac{3}{2}\mu_{4}\\!)\ddot{X}\\!\right]$

Here $\supset$ also denotes that only the terms at 2PN order are included, and
we guarantee the symmetry of $t^{jk}$ during transformation. Similarly,
requiring that all of the coefficients in $Q_{3}^{j}$ vanish and matching the
time derivative part of Eq. (42) with the spatial derivative part of Eq. (30)
at the 2PN order, we finally obtain

$\displaystyle\chi=4(1+\gamma)\tau,$
$\displaystyle\omega_{0}=\gamma,\qquad\omega=-2(\gamma+1)\tau+\beta,\qquad\omega_{1}=\gamma,\qquad\omega_{2}=(-2\beta+2)+\gamma,\qquad\omega_{3}=\gamma,\qquad\omega_{4}=2\gamma-1;$
$\displaystyle\mu_{0}=1,\qquad\mu_{1}=\frac{1}{3}(2\gamma+1),\qquad\mu_{2}=2\beta-1,\qquad\mu_{3}=1,\qquad\mu_{4}=\gamma,$
$\displaystyle\delta_{0}=\beta,\qquad\delta_{1}=\frac{1}{3}(2\gamma-\beta+2),\qquad\delta_{3}=\beta,\qquad\delta_{4}=\gamma-\beta+1,\qquad\lambda=\frac{1}{2}(\gamma+1)(1-\tau),$
$\displaystyle\sigma_{1}=\frac{1}{7}(4\gamma+3),\qquad\sigma_{2}=\frac{1}{9}(4\gamma-8(1+\gamma)\tau+2\beta+3),\qquad\sigma_{3}=\frac{1}{2}(\gamma+1)(1-2\tau),\qquad\sigma_{4}=\frac{1}{3}(2\gamma+1),$
$\displaystyle\sigma_{5}=\frac{1}{2}(\gamma+1),\qquad\sigma_{6}=2\beta-1,\qquad\sigma_{7}=\frac{1}{2}(\gamma^{2}-\gamma-4(\gamma+1)\tau+2\beta),$
$\displaystyle\kappa_{0}=2\beta-1,\quad\kappa=\frac{1}{3}(\gamma\\!-6\beta\\!+4\delta\\!+4),\quad\kappa_{1}=\frac{1}{3}(2\gamma\\!-2\beta\\!+3),\quad\kappa_{2}=\gamma\\!+2\beta\\!-2\delta_{2},\quad\kappa_{3}=2\beta\\!-1,\quad\kappa_{4}=\\!-\gamma\\!+6\beta\\!-4,$
$\displaystyle\xi_{1}=\frac{1}{2}(\gamma+1)(1-\frac{4}{3}\tau),\qquad\xi_{2}=\xi_{3}=\frac{1}{2}(\gamma+1),\qquad\xi_{4}=\frac{1}{2}(\gamma+1)(1-\tau),\qquad\xi_{5}=0,\qquad\eta_{1}=\eta_{2}=\frac{1}{2}(\gamma+1),$
(44)

and

$\displaystyle B_{0}=\frac{1}{2}(5\gamma-1),\quad
B=\frac{1}{2}(15\gamma^{2}+12\beta-12\gamma-20(\gamma+1)\tau-3),\quad
B_{1}=-\frac{1}{2}(5\gamma-1),\quad$ $\displaystyle
B_{2}=3(-\gamma+4\beta-3),\quad B_{3}=5\gamma-1,\quad B_{4}=-3(5\gamma-\\!1).$
(45)

After taking all of the constraints, there remain three independent 2PN
parameters which are chosen as $\delta$, $\delta_{2}$ and $\omega$.

To summarize, there are two independent 1PN parameters ($\gamma$ and $\beta$)
and three independent 2PN parameters ($\delta$, $\delta_{2}$ and $\omega$) in
the P2PN metric Eq. (9) with conservation laws. Here we make a list of all the
other parameters in Eq. (9) in terms of the independent parameters,

$\displaystyle\beta_{1}=\frac{1}{3}(2\gamma+1),\qquad\beta_{2}=2\beta-1,\qquad\beta_{3}=1,\qquad\beta_{4}=\gamma,\qquad\beta_{0}=1,\qquad\Delta=\frac{1}{2}(\gamma+1),$
$\displaystyle\delta_{1}=\frac{1}{3}(2\gamma-\beta+2),\qquad\delta_{3}=\beta,\qquad\delta_{4}=\gamma-\beta+1,\qquad\delta_{0}=\beta,\qquad\lambda=\frac{1}{4}(2\gamma-\beta+\omega+2),$
$\displaystyle\kappa=\frac{1}{3}(\gamma-6\beta+4\delta+4),\qquad\kappa_{1}=\frac{1}{3}(2\gamma-2\beta+3),\qquad\kappa_{2}=\gamma+2\beta-2\delta_{2},\quad\kappa_{3}=2\beta-1,\qquad\kappa_{4}=-\gamma+6\beta-4,$
$\displaystyle\kappa_{0}=2\beta-1,\qquad\sigma_{1}=\frac{1}{7}(4\gamma+3),\qquad\sigma_{2}=\frac{1}{9}(4\gamma-2\beta+4\omega+3),\qquad\sigma_{3}=\frac{1}{2}(\gamma-\beta+\omega+1),\qquad\sigma_{4}=\frac{1}{3}(2\gamma+1),$
$\displaystyle\sigma_{5}=\frac{1}{2}(\gamma+1),\qquad\sigma_{6}=2\beta-1,\qquad\sigma_{7}=\frac{1}{2}(\gamma^{2}-\gamma+2\omega),$
$\displaystyle\mu_{1}=\frac{1}{3}(2\gamma+1),\qquad\mu_{2}=2\beta-1,\qquad\mu_{3}=1,\qquad\mu_{4}=\gamma,\qquad\mu_{0}=1,$
$\displaystyle\xi_{1}=\frac{1}{6}(3\gamma-2\beta+2\omega+3),\qquad\xi_{2}=\frac{1}{2}(\gamma+1),\qquad\xi_{3}=\frac{1}{2}(\gamma+1),\qquad\xi_{4}=\frac{1}{4}(2\gamma-\beta+\omega+2),\qquad\xi_{5}=0,$
$\displaystyle\eta_{1}=\frac{1}{2}(\gamma+1),\qquad\eta_{2}=\frac{1}{2}(\gamma+1),$
$\displaystyle\chi=2(\beta-\omega),\qquad\chi_{1}=\frac{1}{2}(\gamma+1),\qquad\chi_{2}=\frac{1}{2}(\gamma-\beta+\omega+1),\qquad\chi_{3}=\frac{1}{2}(\gamma+1),\qquad\chi_{4}=\frac{1}{2}(\gamma+1),$
$\displaystyle\chi_{5}=\frac{1}{4}(2\gamma-\beta+\omega+2),\qquad\chi_{0}=\frac{1}{2}(\gamma+1),\qquad\iota_{1}=\frac{1}{4}(2\gamma-\beta+\omega+2),\qquad\iota_{2}=\frac{1}{6}(3\gamma-2\beta+2\omega+3),$
$\displaystyle\omega_{1}=\gamma,\qquad\omega_{2}=(-2\beta+2)+\gamma,\qquad\omega_{3}=\gamma,\qquad\omega_{4}=2\gamma-1,\qquad\omega_{0}=\gamma,$
$\displaystyle\theta_{1}=\frac{1}{2}(\gamma+1),\qquad\theta_{2}=\frac{1}{2}(\gamma+1).$
(46)

and all the constants in the total energy-momentum pseudotensor
$\Theta^{\alpha\beta}$ Eq. (17) read

$\displaystyle A$ $\displaystyle=5\gamma-1,$ $\displaystyle
B=\frac{1}{2}(15\gamma^{2}+2\beta-12\gamma+10\omega-3),\quad B_{1}$
$\displaystyle=-\frac{1}{2}(5\gamma-1),\qquad B_{2}=3(-\gamma+4\beta-3)$
$\displaystyle B_{3}=5\gamma-1,\quad B_{4}$
$\displaystyle=-3(5\gamma-\\!1),\qquad B_{0}=\frac{1}{2}(5\gamma-1).$ (47)

Before closing this subsection, we take some concrete examples into account.
(1) In GR, we have $\gamma=\beta=\omega=\delta=\delta_{2}=1$. And the solution
for
$\left(1+\frac{1}{c^{2}}AU+\frac{1}{c^{4}}(BU^{2}+\Psi_{4})\right)t^{\alpha\beta}$
in Eqs. (30), (33) and (42) is exactly $(-\tilde{g})t_{LL}^{\alpha\beta}$ to
the 2PN order given in Pati and Will (2000).
(2) In the scalar tensor theories Mirshekari and Will (2013), the action is
given by

$\displaystyle
S=(16\pi)^{-1}\int[\phi\hat{R}-\phi^{-1}\hat{\omega}(\phi)\hat{g}^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi]\sqrt{-\hat{g}}d^{4}x+S_{m}(\mathfrak{m}),\hat{g}_{\mu\nu}),$
(48)

where $\hat{R}$ is the Ricci scalar of the spacetime metric
$\hat{g}_{\mu\nu}$, $\hat{\omega}(\phi)$ is an arbitrary function of the
scalar field $\phi$, $S_{m}$ only involves the matter fields $\mathfrak{m}$
and the metric, and $\phi$ does not couple to the matter directly. We
introduce the notation $\phi=\phi_{0}(1+c^{-2}f)$, where $\phi_{0}$ is
asymptotic value of the scalar field far away from the system and $f$ measures
the variation in $\phi$ from $\phi_{0}$. Then $\hat{\omega}(\phi)$ can be
expanded as
$\hat{\omega}(\phi)=\hat{\omega}_{0}+\frac{1}{c^{2}}\phi_{0}\left(\frac{d\hat{\omega}}{d\phi}\right)_{0}f+\frac{1}{2}\frac{1}{c^{4}}\phi_{0}^{2}\left(\frac{d^{2}\hat{\omega}}{d\phi^{2}}\right)_{0}f^{2}$,
where the subscript “$0$” denotes the variable takes value at $\phi_{0}$. With
other notations defined by

$\displaystyle\hat{\zeta}\equiv\frac{1}{4+2\hat{\omega}_{0}},\quad\hat{\lambda}_{1}\equiv\frac{(d\hat{\omega}/d\phi)_{0}\phi_{0}}{(3+2\hat{\omega}_{0})(4+2\hat{\omega}_{0})},\quad\hat{\lambda}_{2}\equiv\frac{(d^{2}\hat{\omega}/d\phi^{2})_{0}\phi_{0}^{2}}{(3+2\hat{\omega}_{0})(4+2\hat{\omega}_{0})^{2}},$
(49)

we find the independent parameters given in this paper are

$\displaystyle\gamma=1-2\hat{\zeta},\qquad\beta=1+\hat{\zeta}\hat{\lambda}_{1},\quad$
$\displaystyle\omega=1+\hat{\zeta}(-4+4\hat{\zeta}+\hat{\lambda_{1}}),\quad$
$\displaystyle\delta=1-\hat{\zeta}(-3\hat{\lambda}_{1}+\hat{\zeta}\hat{\lambda}_{1}-4\hat{\lambda}_{1}^{2}+\hat{\lambda}_{2}),\quad\delta_{2}=1+\hat{\lambda}_{1}\hat{\zeta}(3+2\hat{\lambda}_{1}).$
(50)

(3) In the tensor-multiscalar theories Damour and Esposito-Farese (1996),
Damour and Esposito-Farèse found there are two independent 2PN parameter
labeled as $\epsilon$ and $\zeta$ via a field-theory approach, and the 2PN
deviations from GR caused by two of our new parameters $\delta$ and
$\delta_{2}$ and their related potentials $U^{3}$, $U\Phi_{2}$,
$\Sigma(U^{2})$ and $\Sigma(\Phi_{2})$ are given by

$\displaystyle\Delta g_{00}$ $\displaystyle=$
$\displaystyle\frac{1}{c^{6}}\left[\frac{4(\delta-1)}{3}U^{3}+4(\delta-1)\Sigma(U^{2})\right]+\frac{1}{c^{6}}\left[4(\delta_{2}-1)U\Phi_{2}+4(\delta_{2}-1)\Sigma(\Phi_{2})\right]+\mathcal{O}(\frac{\gamma-1}{c^{6}},\frac{\beta-1}{c^{6}},\frac{\omega-1}{c^{6}}),$
$\displaystyle\Delta g_{0j}$ $\displaystyle=$
$\displaystyle\mathcal{O}(\frac{\gamma-1}{c^{5}},\frac{\beta-1}{c^{5}},\frac{\omega-1}{c^{5}}),$
$\displaystyle\Delta g_{jk}$ $\displaystyle=$
$\displaystyle\mathcal{O}(\frac{\gamma-1}{c^{4}},\frac{\beta-1}{c^{4}},\frac{\omega-1}{c^{4}});$
(51)

then the parameters $\epsilon$ and $\zeta$ adopted in Damour and Esposito-
Farese (1996) are related to $\delta$ and $\delta_{2}$ by
$4(\delta-1)=\epsilon$ and $2(\delta_{2}-1)=\zeta$.

###  III.3 Generalized harmonic gauge

After introducing these PN parameters, the harmonic gauge used in GR is no
longer valid. However, Will proposed a generalized harmonic gauge matched with
his PPN metric in Will (2018), and we suppose such a gauge can be similarly
extended to our P2PN metric in Eq. (9) with all the parameter constraints Eq.
(46).

We assume that the generalized harmonic gauge to the 2PN order takes the form,

$\displaystyle\partial_{\mu}\left\\{\left[1+\frac{1}{c^{2}}C\,U+\frac{1}{c^{4}}(D\,U^{2}+D_{1}\Phi_{1}+D_{2}\Phi_{2}+D_{3}\Phi_{3}+D_{4}\Phi_{4}+D_{0}\ddot{X})\right](\sqrt{-g}g^{\mu\nu})\right\\}=0,$
(52)

where $C$, $D$ and $D_{k}$ ($k=0,1,...,4$) are constants to be determined.
With the help of identities of potentials in Eqs. (26), (27) and (28), we find
that Eq. (52) automatically vanish if these constants satisfy

$\displaystyle C$ $\displaystyle=-\gamma+1,$ $\displaystyle
D=\frac{1}{2}(3\gamma^{2}-2\gamma-2\beta-2\omega+3),\quad D_{1}$
$\displaystyle=\frac{1}{2}(\gamma-1),\quad D_{2}=3-4\beta+\gamma,$
$\displaystyle D_{3}=-\gamma+1,\quad D_{4}$ $\displaystyle=3(\gamma-1),\quad
D_{0}=\frac{1}{2}(-\gamma+1).$ (53)

For $\gamma=\beta=\omega=1$, all the above constants equal to zero and Eq.
(52) go back to the genuine harmonic gauge.

##  IV Summary and Discussion

In this paper, going along the way which Will parameterize the 1PN metric and
obtain the parameter constraints by requiring some conservation laws, we
extend to parameterize the 2PN metric and obtain the corresponding parameter
constraints. We choose to parameterize the 2PN metric in the harmonic gauge,
and the final metric meeting the constraints is proven to be in a generalized
harmonic gauge. It turns out that three 2PN parameters, $\omega$, $\delta$ and
$\delta_{2}$, appear independently under the restriction of conservation laws,
which enables our framework to encompass, for example, the scalar-tensor
theories properly. In addition, our calculations are also in consistency with
the tensor-multiscalar theories when considering the deviation from GR caused
by $\delta$ and $\delta_{2}$.

Another important issue we need to note is about the gravitational tests.
Within the PPN framework, the solar system tests put tight bounds on
parameters $\gamma$ and $\beta$: $|\gamma-1|\leq{\cal O}(10^{-5})$ (by time
delay) and $|\beta-1|\leq{\cal O}(10^{-5})$ (by perihelion shift of Mercury)
Will (2014). When it comes to the 2PN order, Damour and Esposito-Farèse
investigated the experimental tests for the only two independent parameters
($\epsilon$ and $\zeta$) in the tensor-multiscalar theories. Since these two
parameters only appear in $g_{00}$ at $\mathcal{O}(c^{-6})$, they concluded
that the light-deflection and time-delay experiments to second order cannot
probe any 2PN deviation from GR Damour and Esposito-Farese (1996). However,
the appearance of another 2PN parameter $\omega$ in our present framework
makes the problem more complicated, because $\omega$ is involved in $g_{ij}$
at $\mathcal{O}(c^{-4})$ and possibly enter into the equation of motion of
light Epstein and Shapiro (1980); Klioner and Zschocke (2010). Damour and
Esposito-Farese (1996) also pointed out other weak-field tests, like
perihelion shift and Nordtvedt effect, are difficult to give effective access
to 2PN parameters because the 2PN contribution blends in with the high-
precision of 1PN parameters. On the other hand, the binary-pulsar experiments
Damour and Esposito-Farese (1996) leads to significant limits
$\epsilon\leq{\cal O}(10^{-2})$ and $\zeta\leq{\cal O}(10^{-3})$ in the
tensor-multiscalar theories, and other tests in the strong-field regime, like
the gravitational-wave Carson and Yagi (2020) and black-hole-shadow Psaltis
_et al._ (2020) experiments, can also provide bounds on the 2PN parameters for
a parameterized Kerr metric. It is likely that the combination of
gravitational tests from different scales would eventually reveal us the
nature of the gravitational theory.

Moreover, the present P2PN metric meeting the constraints is constructed under
the fairly stringent conservation conditions; for a broader P2PN framework
that permits the preferred-reference frames, we need other new 2PN parameters
and potentials to describe the effects when the coordinates transform from the
universal rest-frame to the moving frames relative to it. From an intuitive
angle, there will be more than forty-nine 2PN parameters (see Eqs. (9) and
(10)) in a more general P2PN framework. Compared to the 1PN potentials, the
2PN potentials are more in quantity but smaller in magnitude, and the
corresponding parameters’ individual or joint effects remain to be
investigated. We will leave this for future work.

Acknowledgments. We acknowledge the use of HPC Cluster of ITP-CAS. This work
is supported by the National Key Research and Development Program of China
Grant No.2020YFC2201502, grants from NSFC (grant No. 11975019, 11690021,
11991052, 12047503), Strategic Priority Research Program of Chinese Academy of
Sciences (Grant No. XDB23000000, XDA15020701), and Key Research Program of
Frontier Sciences, CAS, Grant NO. ZDBS-LY-7009.

##  Appendix A The metric to 2PN order in GR with regard to perfect fluid

Pati and Will calculate the near-zone metric to 3.5 PN order via direct
integration of the relaxed Einstein equations in Pati and Will (2000). Here we
sumarize the main results for the metric to 2PN order (see Pati and Will
(2000) for details),

$\displaystyle\tilde{g}_{00}$ $\displaystyle=$
$\displaystyle-(1-\frac{1}{2}N+\frac{3}{8}N^{2}-\frac{5}{16}N^{3})+\frac{1}{2}B(1-\frac{1}{2}N)+\frac{1}{2}K^{j}K^{j}+\mathcal{O}(c^{-7}),$
$\displaystyle\tilde{g}_{0j}$ $\displaystyle=$
$\displaystyle-K^{j}(1-\frac{1}{2}N)+\mathcal{O}(c^{-6}),$
$\displaystyle\tilde{g}_{ij}$ $\displaystyle=$
$\displaystyle\delta^{ij}(1+\frac{1}{2}N-\frac{1}{8}N^{2})+B^{ij}-\frac{1}{2}B\delta^{ij}+\mathcal{O}(c^{-5}),$
$\displaystyle-\tilde{g}$ $\displaystyle=$ $\displaystyle
1+N-B+\mathcal{O}(c^{-5}),$ (54)

in which

$\displaystyle N=$
$\displaystyle\frac{4}{c^{2}}U_{\sigma}+\frac{1}{c^{4}}(7U_{\sigma}^{2}-4\Phi_{1\sigma}+2\Phi_{2\sigma}+2\ddot{X_{\sigma}})+\frac{1}{c^{6}}\left(-16U_{\sigma}\Phi_{1\sigma}+8U_{\sigma}\Phi_{2\sigma}+7U_{\sigma}\ddot{X_{\sigma}}+\frac{20}{3}U_{\sigma}^{3}-4V_{\sigma}^{j}V_{\sigma}^{j}-16\Sigma_{\sigma}(\Phi_{1\sigma})\right.$
$\displaystyle\left.+\Sigma_{\sigma}(\ddot{X_{\sigma}})+8\Sigma_{\sigma}^{j}(V^{j}_{\sigma})-2\ddot{X}_{1\sigma}+\ddot{X}_{2\sigma}+\frac{1}{6}\mathop{\mathrm{Y}_{\sigma}}\limits^{(4)}-4G_{1\sigma}-16G_{2\sigma}+32G_{3\sigma}+24G_{4\sigma}-16G_{5\sigma}-16H_{1\sigma}-16H_{2\sigma}\right),$
$\displaystyle B=$
$\displaystyle\frac{1}{c^{4}}(U_{\sigma}^{2}+4\Phi_{1\sigma}-2\Phi_{2\sigma})+\frac{1}{c^{6}}(U_{\sigma}\ddot{X_{\sigma}}+4V_{\sigma}^{j}V_{\sigma}^{j}-\Sigma_{\sigma}(\ddot{X_{\sigma}})-8\Sigma_{\sigma}^{j}(V^{j}_{\sigma})+16\Sigma_{\sigma}^{ii}(U_{\sigma})+2\ddot{X}_{1\sigma}-\ddot{X}_{2\sigma}-20G_{1\sigma}$
$\displaystyle+8G_{4\sigma}+16G_{5\sigma}),$ $\displaystyle K^{j}=$
$\displaystyle\frac{4}{c^{3}}V_{\sigma}^{j}+\frac{1}{c^{5}}(8V_{2\sigma}^{j}-8\phi_{2\sigma}^{j}+8U_{\sigma}V_{\sigma}^{j}+16K_{1\sigma}^{j}+12K_{2\sigma}^{j}+2\ddot{X}_{\sigma}^{j}),$
$\displaystyle B^{ij}=$
$\displaystyle\frac{1}{c^{4}}\left[4\Phi_{1\sigma}^{ij}+4P_{2\sigma}^{ij}-\delta^{ij}(2\Phi_{2\sigma}-U_{\sigma}^{2})\right].$
(55)

Here the potentials with the subscript “$\sigma$” are defined by the
provisional “densities” for unspecified matter fields
$\tilde{T}^{\alpha\beta}$, namely

$\displaystyle\sigma\equiv
c^{-2}(\tilde{T}^{00}+\tilde{T}^{kk}),\qquad\sigma^{j}\equiv
c^{-1}\tilde{T}^{0j},\qquad\sigma^{ij}\equiv\tilde{T}^{ij},$ (56)

through the integrals

$\displaystyle\Sigma_{\sigma}(f)$ $\displaystyle\equiv$
$\displaystyle\int\frac{\sigma{{}^{\prime}}f^{\prime}}{|\mathbf{x}-\mathbf{x^{\prime}}|}d^{3}x^{\prime}=\mathrm{P}(4\pi\sigma
f),\qquad\qquad\quad\Sigma_{\sigma}^{j}(f)\equiv\int\frac{\sigma^{j}{{}^{\prime}}f^{\prime}}{|\mathbf{x}-\mathbf{x^{\prime}}|}d^{3}x^{\prime}=\mathrm{P}(4\pi\sigma^{j}f),$
$\displaystyle\Sigma_{\sigma}^{ij}(f)$ $\displaystyle\equiv$
$\displaystyle\int\frac{\sigma^{ij}{{}^{\prime}}f^{\prime}}{|\mathbf{x}-\mathbf{x^{\prime}}|}d^{3}x^{\prime}=\mathrm{P}(4\pi\sigma^{ij}f),$
$\displaystyle\mathrm{X}_{\sigma}(f)$ $\displaystyle\equiv$
$\displaystyle\int\sigma{{}^{\prime}}f^{\prime}|\mathbf{x}-\mathbf{x^{\prime}}|d^{3}x^{\prime}=\mathrm{S}(4\pi\sigma
f),\qquad\quad\,\mathrm{X}_{\sigma}^{j}(f)\equiv\int\sigma^{j}{{}^{\prime}}f^{\prime}|\mathbf{x}-\mathbf{x^{\prime}}|d^{3}x^{\prime}=\mathrm{S}(4\pi\sigma^{j}f),$
$\displaystyle\mathrm{X}_{\sigma}^{ij}(f)$ $\displaystyle\equiv$
$\displaystyle\int\sigma^{ij}{{}^{\prime}}f^{\prime}|\mathbf{x}-\mathbf{x^{\prime}}|d^{3}x^{\prime}=\mathrm{S}(4\pi\sigma^{ij}f),$
$\displaystyle\mathrm{Y}_{\sigma}(f)$ $\displaystyle\equiv$
$\displaystyle\int\sigma{{}^{\prime}}f^{\prime}|\mathbf{x}-\mathbf{x^{\prime}}|^{3}d^{3}x^{\prime}=\mathrm{SD}(4\pi\sigma
f);$ (57)

specifically, they are given by

$\displaystyle U_{\sigma}\equiv\Sigma_{\sigma}(1),$ $\displaystyle
V^{j}_{\sigma}\equiv\Sigma_{\sigma}^{j}(1),\quad\Phi_{1\sigma}\equiv\Sigma_{\sigma}^{ii}(1),\quad\Phi_{2\sigma}\equiv\Sigma_{\sigma}(U_{\sigma}),\quad
X_{\sigma}\equiv\mathrm{X}_{\sigma}(1),$ $\displaystyle
V_{2\sigma}^{j}\equiv\Sigma_{\sigma}^{j}(U),\\!\quad\\!\phi_{2\sigma}^{j}\equiv\Sigma_{\sigma}(V_{\sigma}^{j}),\\!\quad\\!\\!K_{1\sigma}^{j}\equiv\mathrm{P}(U_{\sigma}^{,k}V_{\sigma}^{k,j}),\\!\quad\\!\\!K_{2\sigma}^{j}\equiv\mathrm{P}(U_{\sigma}^{,j}\dot{U}_{\sigma}),\\!\quad\\!\\!X_{\sigma}^{j}\equiv\mathrm{X}_{\sigma}^{j}(1),\\!\quad\\!\\!\Phi_{1\sigma}^{ij}\equiv\Sigma_{\sigma}^{ij}(1),\\!\quad\\!\\!P_{2\sigma}^{ij}\equiv
P(U_{\sigma}^{,i}U_{\sigma}^{,j}),\\!\\!\\!\\!\\!\\!$ $\displaystyle
X_{1\sigma}\equiv\mathrm{X}_{\sigma}^{ii}(1),\quad
X_{2\sigma}\equiv\mathrm{X}(U),\quad Y_{\sigma}\equiv Y_{\sigma}(1),\quad
G_{1\sigma}\equiv\mathrm{P}(\dot{U}_{\sigma}^{2}),\quad
G_{2\sigma}\equiv\mathrm{P}(U_{\sigma}\ddot{U}_{\sigma}),\quad
G_{3\sigma}\equiv-\mathrm{P}(\dot{U}_{\sigma}^{,k}V_{\sigma}^{k}),$
$\displaystyle
G_{4\sigma}\equiv\mathrm{P}(V_{\sigma}^{i,j}V_{\sigma}^{j,i}),\quad\,G_{5\sigma}\equiv-\mathrm{P}(\dot{V}_{\sigma}^{k}U_{\sigma}^{,k}),\quad
H_{1\sigma}\equiv\mathrm{P}(\Phi_{1\sigma}^{ij}U_{\sigma}^{,ij}),\quad\,H_{2\sigma}\equiv\mathrm{P}(P_{2}^{ij}U_{\sigma}^{,ij}).$
(58)

When the matter field is specified to be perfect fluid with the energy-
momentum tensor Eq. (2), we usually convert the potentials from integrals over
$\sigma$, $\sigma^{j}$ and $\sigma^{ij}$ to integrals over the conventional
conserved mass density $\rho^{*}=\sqrt{-\tilde{g}}\rho u^{0}/c$. In terms of
$\rho^{*}$, the energy-momentum tensor takes the form

$\displaystyle\tilde{T}^{\alpha\beta}=\rho^{*}(1+\Pi/c^{2})(\sqrt{-\tilde{g}})^{-1}(u^{0}/c)v^{\alpha}v^{\beta}+P(u^{0}/c)^{2}/c^{2}v^{\alpha}v^{\beta}+P\tilde{g}^{\alpha\beta},$
(59)

where $v^{\alpha}=(c,\mathbf{v})$ and
$u^{0}/c=(\sqrt{-\tilde{g}_{00}-2\tilde{g}_{0j}v^{j}/c-\tilde{g}_{ij}v^{i}v^{j}/c^{2}})^{-1}$.
Combining Eqs. (56) and (59) as well as the metric Eq. (54) and the functions
in Eq. (55), we express the provisional “densities” in terms of $\rho^{*}$ to
the required order,

$\displaystyle\sigma=$
$\displaystyle\rho^{*}\left[1+\frac{1}{c^{2}}\left(\frac{3}{2}v^{2}-U_{\sigma}+\Pi+3P/\rho^{*}\right)+\frac{1}{c^{4}}\left(\frac{7}{8}v^{4}+\frac{1}{2}U_{\sigma}v^{2}+\frac{3}{2}\Pi
v^{2}+2P/\rho^{*}v^{2}-4v^{i}V_{\sigma}^{i}+\frac{3}{2}U_{\sigma}^{2}-U_{\sigma}\Pi\right.\right.$
$\displaystyle\quad\,\left.\left.-6U_{\sigma}P/\rho^{*}+4\Phi_{1\sigma}-2\Phi_{2\sigma}-\frac{1}{2}\ddot{X}_{\sigma}\right)\right],$
$\displaystyle\sigma^{j}=$
$\displaystyle\rho^{*}v^{j}\left[1+\frac{1}{c^{2}}\left(\frac{1}{2}v^{2}-U_{\sigma}+\Pi+P/\rho^{*}\right)\right],$
$\displaystyle\sigma^{ij}=$ $\displaystyle\rho^{*}v^{i}v^{j}+\delta^{ij}P,$
$\displaystyle\sigma^{ii}=$
$\displaystyle\rho^{*}v^{2}+3P+\frac{1}{c^{2}}\left[\rho^{*}v^{2}\left(\frac{1}{2}v^{2}-U_{\sigma}+\Pi+P/\rho^{*}\right)-6U_{\sigma}P\right].$
(60)

Substituting these formulas into the definitions in Eq. (58) and iterating
successively, we obtain the conversion relationships between the old
potentials defined by $\sigma$, $\sigma^{j}$ and $\sigma^{ij}$ and the new
potentials defined by $\rho^{*}$ to required order: for the Newtonian
potential and 1PN potentials,

$\displaystyle U_{\sigma}=$ $\displaystyle
U+\frac{1}{c^{2}}\left(\frac{3}{2}\Phi_{1}-\Phi_{2}+\Phi_{3}+3\Phi_{4}\right)+\frac{1}{c^{4}}\left(\frac{7}{8}\Sigma(v^{4})+\frac{1}{2}\Sigma(v^{2}U)-4\Sigma(v^{i}V^{i})+\frac{3}{2}\Sigma(U^{2})+\frac{5}{2}\Sigma(\Phi_{1})-\Sigma(\Phi_{2})\right.$
$\displaystyle\left.-\Sigma(\Phi_{3})+9\Sigma(\Phi_{4})-\frac{1}{2}\Sigma(\ddot{X})+\frac{3}{2}\Omega(v^{2})+2T(v^{2})-\Omega(U)-6T(U)\right),$
$\displaystyle V_{\sigma}^{j}=$ $\displaystyle
V^{j}+\frac{1}{c^{2}}\left(\frac{1}{2}V_{1}^{j}-V_{2}^{j}+V_{3}^{j}+V_{4}^{j}\right),$
$\displaystyle\Phi_{1\sigma}=$
$\displaystyle\Phi_{1}+3\Phi_{4}+\frac{1}{c^{2}}\left(\frac{1}{2}\Sigma(v^{4})-\Sigma(v^{2}U)+\Omega(v^{2})+T(v^{2})-6T(U)\right),$
$\displaystyle\Phi_{2\sigma}=$
$\displaystyle\Phi_{2}+\frac{1}{c^{2}}\left(\frac{3}{2}\Sigma(\Phi_{1})-\Sigma(\Phi_{2})+\Sigma(\Phi_{3})+3\Sigma(\Phi_{4})+\frac{3}{2}\Sigma(v^{2}U)-\Sigma(U^{2})+\Omega(U)+3T(U)\right),$
$\displaystyle X_{\sigma}=$ $\displaystyle
X+\frac{1}{c^{2}}\left(\frac{3}{2}X_{1}-X_{2}+X_{3}+3X_{4}\right);$ (61)

for some 2PN potentials,

$\displaystyle\Phi_{1\sigma}^{ij}=\Phi_{1}^{ij}+\delta^{ij}\Phi_{4},\quad\,U_{\sigma}\Phi_{1\sigma}=U\Phi_{1}+3U\Phi_{4},\quad\,\Sigma_{\sigma}(\Phi_{1\sigma})=\Sigma(\Phi_{1})+3\Sigma(\Phi_{4}),$
$\displaystyle
X_{1\sigma}=X_{1}+3X_{4},\quad\,H_{1\sigma}=H_{1}-\Sigma(\Phi_{4}),\quad\,\Sigma_{\sigma}^{ii}(U)=\Sigma(v^{2}U)+3T(U),$
(62)

and the remaining 2PN potentials in Eq. (58) keep their forms when they are
defined by $\rho^{*}$, for example, $\phi_{2\sigma}^{j}=\phi_{2}^{j}$,
$X_{2\sigma}=X_{2}$. Substituting the conversions into Eq. (58) for the metric
Eq. (54), we finally obtain the 2PN metric with regard to perfect fluid, i.e.
Eq. (9) with $\chi=\xi_{5}=0$ and the other unspecified parameters are all
equal to 1.

##  Appendix B Transformation tricks

Some skillful transformations are needed in solving Eq. (20), and we list
representative examples with regard to the equation for $\alpha=0$ in Eq.
(30):
(1) the transformation most frequently used

$\displaystyle-4\pi\rho^{*}\partial_{t}\Phi_{1}=\nabla^{2}U\partial_{t}\Phi_{1}=-\partial_{t}(\partial_{j}\Phi_{1}\partial_{j}U)+2\partial_{j}(\partial_{(t}\Phi\partial_{j)}U)+4\pi\rho^{*}v^{2}\partial_{t}U;$
(63)

(2) different transformations for the same component

$\displaystyle\begin{cases}-4\pi\rho^{*}U\partial_{t}U&=\nabla^{2}\Phi_{2}\partial_{t}U=-\partial_{t}(\partial_{j}\Phi_{2}\partial_{j}U)+\partial_{j}(\partial_{t}U\partial_{j}\Phi_{2})+\partial_{j}U\partial_{j}\partial_{t}\Phi_{2},\\\
-4\pi\rho^{*}U\partial_{t}U&=\frac{1}{2}\nabla^{2}U\partial_{t}U^{2}=-\partial_{t}(U\partial_{j}U\partial_{j}U)+\partial_{j}(U\partial_{t}U\partial_{j}U)+U\partial_{j}U\partial_{j}\partial_{t}U,\end{cases}$
(64)
$\displaystyle\begin{cases}-4\pi\rho^{*}V^{j}\partial_{j}U&=\nabla^{2}\Phi_{2}^{j}\partial_{j}U=-\partial_{j}(\partial_{k}U\partial_{[k}\phi_{2j]})+\partial_{k}U\partial_{k}\partial_{j}\phi_{2}^{j},\\\
-4\pi\rho^{*}V^{j}\partial_{j}U&=V^{j}\nabla^{2}U\partial_{j}U=\partial_{k}(V^{j}\partial_{k}U\partial_{j}U)-\frac{1}{2}\partial_{j}(V^{j}\partial_{k}U\partial_{k}U)-\frac{1}{2}\partial_{t}U\partial_{k}U\partial_{k}U-\partial_{k}V^{j}\partial_{k}U\partial_{j}U;\end{cases}$
(65)

(3) synthetical transformation by use of identities of potentials

$\displaystyle-4\pi\rho^{*}V^{j}\partial_{j}U-4\pi\rho^{*}v^{j}v^{k}\partial_{k}V^{j}$
$\displaystyle=$
$\displaystyle\left[\partial_{k}(V^{j}\partial_{k}U\partial_{j}U)-\frac{1}{2}\partial_{j}(V^{j}\partial_{k}U\partial_{k}U)-\frac{1}{2}\partial_{t}U\partial_{k}U\partial_{k}U\right]+\partial_{l}\partial_{l}P_{2}^{jk}\partial_{k}V^{j}+\partial_{l}\partial_{l}\Phi_{1}^{jk}\partial_{k}V^{j}$
$\displaystyle=$
$\displaystyle\left[\partial_{k}(V^{j}\partial_{k}U\partial_{j}U)-\frac{1}{2}\partial_{j}(V^{j}\partial_{k}U\partial_{k}U)-\frac{1}{2}\partial_{t}U\partial_{k}U\partial_{k}U\right]-2\partial_{l}\left[\partial_{k}V^{j}\partial_{[k}(P_{2l]j}+\Phi_{1l]j})\right]+\partial_{l}V^{j}\partial_{l}\partial_{k}(\Phi_{1}^{jk}+P_{2}^{jk})$
$\displaystyle=$
$\displaystyle\left[\partial_{k}(V^{j}\partial_{k}U\partial_{j}U)-\frac{1}{2}\partial_{j}(V^{j}\partial_{k}U\partial_{k}U)-\frac{1}{2}\partial_{t}U\partial_{k}U\partial_{k}U\right]-2\partial_{l}\left[\partial_{k}V^{j}\partial_{[k}(P_{2l]j}+\Phi_{1l]j})\right]$
$\displaystyle-\partial_{l}V^{j}\partial_{l}\partial_{t}V^{j}+\frac{1}{4}\partial_{l}V^{j}\partial_{l}\partial_{j}(2\Phi_{2}-U^{2}-4\Phi_{4})$
$\displaystyle=$
$\displaystyle\left[\partial_{k}(V^{j}\partial_{k}U\partial_{j}U)-\frac{1}{2}\partial_{j}(V^{j}\partial_{k}U\partial_{k}U)-\frac{1}{2}\partial_{t}U\partial_{k}U\partial_{k}U\right]-2\partial_{l}\left[\partial_{k}V^{j}\partial_{[k}(P_{2l]j}+\Phi_{1l]j})\right]$
(66)
$\displaystyle-\frac{1}{2}\partial_{t}(\partial_{l}V^{j}\partial_{l}V^{j})+\frac{1}{4}\partial_{j}\left[\partial_{l}V^{j}\partial_{l}(2\Phi_{2}-U^{2}-4\Phi_{4})\right]+\frac{1}{4}\partial_{l}\partial_{t}U\partial_{l}(2\Phi_{2}-U^{2}-4\Phi_{4}),$
$\displaystyle-4\pi[(2v^{2}\\!+\\!4U\\!+4\Pi\\!+4P/\rho^{*})\rho^{*}v^{j}\partial_{j}U\\!-\\!8\rho^{*}V^{j}\partial_{j}U\\!+8\rho^{*}V^{j}\partial_{j}U\\!+8\rho^{*}v^{j}U\partial_{j}U]$
$\displaystyle=$
$\displaystyle-4\pi[(2v^{2}\\!+\\!4(1\\!-\\!2\tau)U\\!+4\Pi\\!+4P/\rho^{*})\rho^{*}v^{j}\partial_{j}U\\!-\\!8(1\\!-\\!\tau)\rho^{*}V^{j}\partial_{j}U\\!+8(1\\!-\\!\tau)\rho^{*}V^{j}\partial_{j}U\\!+8(1\\!-\\!\tau)\rho^{*}v^{j}U\partial_{j}U]$
$\displaystyle-64\pi\tau\rho^{*}v^{j}U\partial_{j}U$ $\displaystyle=$
$\displaystyle\nabla^{2}[2V_{1}^{j}\\!+4(1\\!-\\!2\tau)V_{2}^{j}\\!+4V_{3}^{j}\\!+4V_{4}^{j}\\!-\\!8(1\\!-\\!\tau)\phi_{2}^{j}\\!+8(1\\!-\\!\tau)(UV)^{j}\\!+16(1\\!-\\!\tau)K_{1}^{j}\\!+12(1\\!-\\!\frac{4}{3}\tau)K_{2}^{j}\\!+2\ddot{X}^{j}]\partial_{j}U\\!$
$\displaystyle+\\!12(1\\!-\\!\frac{4}{3}\tau)\partial_{k}U\partial_{k}U\partial_{t}U\\!-\\!4\partial_{t}\partial_{t}V^{j}\partial_{j}U-16\tau\left[2\partial_{j}(U\partial_{k}U\partial_{[k}V_{j]})+U\partial_{k}U\partial_{k}\partial_{t}U\right]$
$\displaystyle=$
$\displaystyle-2\partial_{j}\\{\partial_{[k}[2V_{1{j]}}\\!+\\!4(1\\!-\\!2\tau)V_{2{j]}}\\!+\\!4V_{3{j]}}\\!+\\!4V_{4{j]}}\\!-\\!8(1\\!-\\!\tau)\phi_{2{j]}}\\!+\\!8(1\\!-\\!\tau)(V_{{j}{]}}U)\\!+\\!16(1\\!-\\!\tau)K_{1{j]}}\\!+\\!12(1\\!-\\!\frac{4}{3}\tau)K_{2{j]}}\\!+\\!2\ddot{X}_{j]}]\partial_{k}U\\}$
$\displaystyle+\partial_{k}\partial_{j}[2V_{1}^{j}\\!+4(1\\!-\\!2\tau)V_{2}^{j}\\!+4V_{3}^{j}\\!+4V_{4}^{j}\\!-\\!8(1\\!-\\!\tau)\phi_{2}^{j}\\!+8(1\\!-\\!\tau)(UV^{j})\\!+16(1\\!-\\!\tau)K_{1}^{j}\\!+12(1\\!-\\!\frac{4}{3}\tau)K_{2}^{j}\\!+2\ddot{X}^{j}]\partial_{k}U$
$\displaystyle+12(1-\frac{4}{3}\tau)\partial_{k}U\partial_{k}U\partial_{t}U-4\partial_{t}\partial_{t}V^{j}\partial_{j}U-16\tau\left[2\partial_{j}(U\partial_{k}U\partial_{[k}V_{j]})+U\partial_{k}U\partial_{k}\partial_{t}U\right]$
$\displaystyle=$
$\displaystyle-2\partial_{j}\\{\partial_{[k}[2V_{1{j]}}\\!+\\!4(1\\!-\\!2\tau)V_{2{j]}}\\!+\\!4V_{3{j]}}\\!+\\!4V_{4{j]}}\\!-\\!8(1\\!-\\!\tau)\phi_{2{j]}}\\!+\\!8(1\\!-\\!\tau)(V_{{j}{]}}U)\\!+\\!16(1\\!-\\!\tau)K_{1{j]}}\\!+\\!12(1\\!-\\!\frac{4}{3}\tau)K_{2{j]}}\\!+\\!2\ddot{X}_{j]}]\partial_{k}U\\}$
$\displaystyle-\partial_{k}\partial_{t}((7-8\tau)U^{2}\\!+\\!2\Phi_{1}\\!-\\!2\Phi_{2}\\!+\\!4\Phi_{3}\\!+\\!2\ddot{X})\partial_{k}U\\!+\\!12(1-\frac{4}{3}\tau)\partial_{k}U\partial_{k}U\partial_{t}U\\!-\\!4\partial_{t}(\partial_{t}V^{j}\partial_{j}U)\\!+\\!2\partial_{t}(\partial_{j}V^{k}\partial_{k}V^{j})\\!$
$\displaystyle-\\!4\partial_{j}(\partial_{k}V^{j}\partial_{t}V^{k})-32\tau\partial_{j}(U\partial_{k}U\partial_{[k}V_{j]})-16\tau
U\partial_{k}U\partial_{k}\partial_{t}U$ $\displaystyle=$
$\displaystyle-2\partial_{j}\\{\partial_{[k}[2V_{1{j]}}\\!+\\!4(1\\!-\\!2\tau)V_{2{j]}}\\!+\\!4V_{3{j]}}\\!+\\!4V_{4{j]}}\\!-\\!8(1\\!-\\!\tau)\phi_{2{j]}}\\!+\\!8(1\\!-\\!\tau)(V_{{j}{]}}U)\\!+\\!16(1\\!-\\!\tau)K_{1{j]}}\\!+\\!12(1\\!-\\!\frac{4}{3}\tau)K_{2{j]}}\\!+\\!2\ddot{X}_{j]}]\partial_{k}U\\}$
(67)
$\displaystyle\\!-\\!4\partial_{t}(\partial_{t}V^{j}\partial_{j}U)\\!+\\!2\partial_{t}(\partial_{j}V^{k}\partial_{k}V^{j})-\\!4\partial_{j}(\partial_{k}V^{j}\partial_{t}V^{k})-32\tau\partial_{j}(U\partial_{k}U\partial_{[k}V_{j]})$
$\displaystyle-\partial_{k}\partial_{t}(7U^{2}\\!+\\!2\Phi_{1}\\!-\\!2\Phi_{2}\\!+\\!4\Phi_{3}\\!+\\!2\ddot{X})\partial_{k}U\\!$

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