Constraining energy-momentum-squared gravity by binary pulsar observations

To obtain the Landau-Lifshitz formulation of the EMSG field equations, we utilize the similar procedure introduced in Poisson and Will (2014). The following identity is the main relation in this approach:

$\displaystyle\partial_{\mu\nu}H^{\alpha\mu\beta\nu}=2(-g)G^{\alpha\beta}+\frac{16\pi G}{c^{4}}(-g)t_{\text{LL}}^{\alpha\beta},$

(10)

where $H^{\alpha\mu\beta\nu}$ is the pseudotensor constructed from the gothic metric $\mathfrak{g}^{\alpha\beta}=\sqrt{-g}g^{\alpha\beta}$ as follows

$\displaystyle H^{\alpha\mu\beta\nu}=\mathfrak{g}^{\alpha\beta}\mathfrak{g}^{\mu\nu}-\mathfrak{g}^{\alpha\nu}\mathfrak{g}^{\beta\mu},$

(11)

and it has the same symmetries as the Riemann tensor. In the above identity, $(-g)t_{\text{LL}}^{\alpha\beta}$ is the Landau-Lifshitz pseudotensor given by Eq. (6.5) of Poisson and Will (2014)³³3 In Appendix. B, for the sake of convenience, the definition of this pseudotensor is also given by Eq. (B).. It should be noted that the identity (10) is valid for every spacetime. Inserting the modified EMSG field equations, i.e., Eq. (4), within Eq. (10), we obtain

$\displaystyle\partial_{\mu\nu}H^{\alpha\mu\beta\nu}=\frac{16\pi G}{c^{4}}(-g)\Big{(}T_{\text{eff}}^{\alpha\beta}+t_{\text{LL}}^{\alpha\beta}\Big{)}.$

(12)

This relation is the nontensorial form of the EMSG field equations in the Landau-Lifshitz formalism. In the next step toward this derivation, we consider that $\mathfrak{g}^{\alpha\beta}=\eta^{\alpha\beta}-h^{\alpha\beta}$. Here, $h^{\alpha\beta}$ is a gravitational potential and $\eta^{\alpha\beta}$ is the Minkowski metric. Furthermore, we use the harmonic gauge condition $\partial_{\beta}h^{\alpha\beta}=0$. Applying this new definition of $\mathfrak{g}^{\alpha\beta}$ and imposing the harmonic gauge condition in Eq. (12), after some manipulations, one can arrive at

$\displaystyle\square h^{\alpha\beta}=-\frac{16\pi G}{c^{4}}\tau^{\alpha\beta}_{\text{eff}},$

(13)

where $\square=\eta^{\alpha\beta}\partial_{\alpha\beta}$ is the Minkowski wave operator and

$\displaystyle\tau^{\alpha\beta}_{\text{eff}}=(-g)\Big{(}T^{\alpha\beta}_{\text{eff}}+t_{\text{LL}}^{\alpha\beta}+t_{\text{H}}^{\alpha\beta}\Big{)},$

(14)

is the effective energy-momentum pseudotensor. Here, $(-g)t_{\text{H}}^{\alpha\beta}$ is given by

$\displaystyle(-g)t_{\text{H}}^{\alpha\beta}=\frac{c^{4}}{16\pi G}\big{(}\partial_{\mu}h^{\alpha\nu}\partial_{\nu}h^{\beta\mu}-h^{\mu\nu}\partial_{\mu\nu}h^{\alpha\beta}\big{)}.$

(15)

The wave equation (13) is the cornerstone to derive the PM approximation to the EMSG field equations. By applying the systematic method in the PM theory, we can approximately solve the highly nonlinear wave equation (13) in the wave zone where the radiation effects are significant. As expected, by dropping $T_{\mu\nu}^{\text{\tiny EMS}}$ in this relation, the standard Landau-Lifshitz formulation of the Einstein field equations in GR is recovered, cf. Eq. (6.51) in Poisson and Will (2014). In this case, the general form of the solution of the wave equation is comprehensively introduced in chapter 6 in Poisson and Will (2014). Regarding the mathematical similarity of Eq. (13) and that of in GR, we can use the same general from of the solutions here. Therefore, we drop the relevant calculations. Instead, the general form of the solutions are presented in Appendix. B.

As the final point in this subsection, we should mention that the pseudotensor $(-g)t_{\text{H}}^{\alpha\beta}$ satisfies the following identity: $\partial_{\beta}\big{(}(-g)t_{\text{H}}^{\alpha\beta}\big{)}=0$. Keeping this fact in mind and also using the harmonic gauge condition, we deduce from Eq. (13) that

$\displaystyle\partial_{\mu}\Big{(}(-g)(T^{\mu\nu}_{\text{eff}}+t_{\text{LL}}^{\mu\nu})\Big{)}=0.$

(16)

In fact, this is a statement of the conservation equations in this formalism. The wave equation (13) and the conservation equations (16) are the main relations to find the PM approximation to the EMSG field equations.

Let us introduce the energy-balance equation in the Landau-Lifshitz formalism of EMSG. As we know, by studying this equation, one can obtain the relation between the time dependent physical properties of the systems and the amount of energy carried away by the GW emissions. To do so, in a similar fashion to the method introduced in the Landau-Lifshitz formalism of GR, we first introduce the total four-vector momentum as

$\displaystyle P^{\alpha}[V]=\frac{1}{c}\int_{V}(-g)\big{(}T^{\alpha 0}_{\text{eff}}+t_{\text{LL}}^{\alpha 0}\big{)}d^{3}x.$

(17)

Here, $V$ is the volume of the three-dimensional region where $P^{\alpha}[V]$ is defined. The time and space components of this four-vector momentum represent the total energy $P^{0}[V]=c^{-1}E[V]$ and three-vector momentum $P^{j}[V]$ associated with $V$, respectively. It is also considered that there is no matter field on the boundary of this region. In other words, $V$ is large enough so that the surface of $V$, indicated by $S$, does not intersect the matter distribution. Bearing this fact in mind and also using the conservation equations (16), one can arrive at

$\displaystyle\dot{P}^{\alpha}[V]=-\oint_{S}(-g)t_{\text{LL}}^{\alpha k}dS_{k},$

(18)

for the conservation statement of the total four-vector momentum. Here, the overdot stands for the derivative with respect to time $t$. At the first glance, this relation reveals that not only the standard energy-momentum tensor but also the EMSG portion $T_{\mu\nu}^{\text{\tiny EMS}}$ have no contribution in this conservation statement and the mathematical form of this equation is the same as in GR Poisson and Will (2014). However, it should be mentioned that the Landau-Lifshitz pseudotensor is not necessarily similar in these two theories. In the following section, we obtain this pseudotensor in EMSG and illustrate the differences.

As we aim to study the rate at which the energy is removed from the GW sources, we focus our attention on the time component of Eq. (18). Then for $\alpha=0$, we have

$\displaystyle\dot{E}[V]=-c\oint_{S}(-g)t_{\text{LL}}^{0k}dS_{k}.$

(19)

Here,

$\displaystyle E[V]=\int_{V}(-g)\big{(}T^{00}_{\text{eff}}+t^{00}_{\text{LL}}\big{)}d^{3}x,$

(20)

is the total energy of the system. Regarding Eq. (19), one can define the energy-balance equation in EMSG as

$\displaystyle\frac{dE}{dt}=-\mathcal{P},$

(21)

where

$\displaystyle\mathcal{P}=c\oint_{S}(-g)t_{\text{LL}}^{0k}dS_{k},$

(22)

is the flux of energy across the surface $S$.

The next step toward studying the energy-balance equation is to find the Landau-Lifshitz pseudotensor in terms of the gravitational potential $h^{\alpha\beta}$. It is shown in the shortwave approximation where the characteristic wavelength of the GW signals $\lambda_{c}$ is much smaller than the distance to the center of the mass of the GW system $r=|\mathbf{x}|$, i.e., where $\lambda_{c}/r\ll 1$, one can simplify the definition (B) as ⁴⁴4 To see more detailed discussions on this issue, we refer the reader to Sec. 12.2.3 of Poisson and Will (2014).

$\displaystyle(-g)t_{\text{LL}}^{\alpha\beta}=\frac{c^{2}}{32\pi G}\Big{(}\partial_{\tau}h_{\text{TT}}^{jk}\partial_{\tau}h^{\text{TT}}_{jk}\Big{)}k^{\alpha}k^{\beta},$

(23)

where $\partial_{\tau}$ shows the derivative with respect to retarded time $\tau=t-r/c$ and four vector $k^{\alpha}$ is defined by $k^{\alpha}=(1,\bm{n})$. Here $h^{jk}_{\text{TT}}=\Big{(}q_{l}^{j}q_{m}^{k}-\frac{1}{2}q^{jk}q_{lm}\big{)}h^{lm}$ represents the transverse-tracefree (T-T) portion of the gravitational potential $h^{jk}$ in which $q_{jk}=\delta_{jk}-n_{j}n_{k}$ and $n_{j}=\partial_{j}r$. As seen, the mathematical appearance of Eqs. (21) and (22) is similar to the GR one. However, $h_{\text{TT}}^{jk}$ can be basically different in EMSG. So, it is not trivial how GW systems will behave in the context of this modified theory of gravity.

To sum up, in order to obtain $(-g)t_{\text{LL}}^{\alpha\beta}$ and consequently the flux of energy in EMSG, we should find T-T part of the gravitational potential in this theory. So, we turn to solve the EMSG field equations (13). In the next section, we attempt to solve these equations by utilizing the PM approximation and obtain $h_{\text{TT}}^{jk}$ to the leading order in the EMSG theory. Those readers who are unfamiliar with the PM approximation, or have no interest in the detailed derivation, can directly skip to Sec. V. In this section, applying the results of Sec. IV, we study the flux of energy due to the GW emissions in EMSG.

Before starting the main calculations, let us approximately obtain the GW field $h_{\text{TT}}^{jk}$ for a specific distribution of the matter in EMSG. We consider the famous quadrupole formula for $h_{\text{TT}}^{jk}$, i.e., $h_{\text{TT}}^{jk}=(2G/c^{4}r)\partial_{\tau\tau}\mathcal{I}^{jk}$, in which

$\displaystyle\mathcal{I}^{jk}(\tau)=\int c^{-2}\tau^{00}(\tau,\mathbf{x}^{\prime})x^{\prime j}x^{\prime k}d^{3}x^{\prime},$

(24)

is the quadrupole-moment tensor of the matter distribution. In the EMSG case, one may replace $\tau^{00}$ with $\tau_{\text{eff}}^{00}$. To find $h_{\text{TT}}^{jk}$, we should then derive the time-time component of the effective energy-momentum pseudotensor (14). To do so, we choose a perfect fluid for describing the matter distribution in the Minkowski spacetime as $T^{\alpha\beta}=\big{(}\rho+\epsilon/c^{2}+p/c^{2}\big{)}u^{\alpha}u^{\beta}+p\eta^{\alpha\beta}$. Here, $\epsilon$ is the proper internal energy density and $p$ is the pressure. We also consider that the velocity of the fluid elements is slow compared to the speed of light and the following set of conditions is established in this fluid system ($v^{2}/c^{2}\ll 1,\,p/\rho c^{2}\ll 1,\,\text{and}\,\epsilon/\rho c^{2}\ll 1$). With this in mind, regarding Eqs. (5), (B), and (15), one can show that $\tau^{00}_{\text{eff}}$ up to the leading order is given by

$\displaystyle\tau^{00}_{\text{eff}}=c^{2}\rho_{\text{\tiny EMS}}^{\text{eff}}+O(1),$

(25)

where $\rho_{\text{\tiny EMS}}^{\text{eff}}=\rho\left(1+c^{2}f_{0}^{\prime}\rho\right)$ represents the effective mass density in EMSG. So, we can conclude that finding the quadrupole moment

$\displaystyle\mathcal{I}^{jk}(\tau)=\int\rho_{\text{\tiny EMS}}^{\text{eff}}(\tau,\mathbf{x}^{\prime})x^{\prime j}x^{\prime k}d^{3}x^{\prime},$

(26)

will be the main task toward deriving $h_{\text{TT}}^{jk}$ in EMSG. This crude estimation clearly shows that not only does $\rho$ play a role but $\rho^{2}$ also has a contribution to the GW field $h_{\text{TT}}^{jk}$. On the other hand, it should be emphasized that as this estimation is based on the behavior of the perfect fluid in the flat spacetime, the matter field indeed does not respond to gravity. In other words, enforcing Eq. (6) for the matter field in the Minkowski spacetime reveals that the matter is not subjected to the gravitational interaction in this estimation. Moreover, the Landau-Lifshitz and harmonic pseudotensors vanish here. Therefore, to incorporate the role of the gravity into the result, it is necessary to obtain $\tau^{00}_{\text{eff}}$ more precisely. Toward this goal, in the next section, we introduce the PM expansion of the gravitational potentials in EMSG.

As it will be shown, we also need to indicate the order of magnitude of the free parameter of this theory. To do so, let us recall the time-time component of the metric up to the Newtonian order: $g_{00}=-1+2U/c^{2}$ where $U$ is the gravitational potential and given by $\nabla^{2}U=-4\pi G\rho$. One may deduce that this metric component will be $g_{00}=-1+2U_{\text{\tiny N}}/c^{2}+2f_{0}^{\prime}U_{\text{\tiny EMS}}$ after changing $\rho$ to the effective mass density $\rho_{\text{\tiny EMS}}^{\text{eff}}$. Here, $\nabla^{2}U_{\text{\tiny N}}=-4\pi G\rho$ and $\nabla^{2}U_{\text{\tiny EMS}}=-4\pi G\rho^{2}$ are the Poisson equations for the Newtonian and EMSG potentials $U_{\text{\tiny N}}$ and $U_{\text{\tiny EMS}}$, respectively. Considering the weak field limit and comparing the first, second, and third terms in this component of the metric reveal that $U_{\text{\tiny N}}/c^{2}$ and $f_{0}^{\prime}U_{\text{\tiny EMS}}$ should be very small, i.e., $U_{\text{\tiny N}}/c^{2}\ll 1$ and $f_{0}^{\prime}U_{\text{\tiny EMS}}\ll 1$. So, we consider that $f_{0}^{\prime}U_{\text{\tiny EMS}}$ is at most of the order of $U_{\text{\tiny N}}/c^{2}$ ⁵⁵5In the following, for the sake of simplification, we drop the index “N ”..

Regarding the estimation of the PN order of the $h^{\alpha\beta}$ components stated before as well as utilizing Eq. (88), we find the PM expansion of the metric components in terms of the gravitational potentials as

		$\displaystyle g_{00}=-1+\frac{1}{2}h^{00}-\frac{3}{8}\big{(}h^{00}\big{)}^{2}+\frac{1}{2}h^{kk}\big{(}1-\frac{1}{2}h^{00}\big{)}$
		$\displaystyle-\frac{1}{8}\big{(}h^{kk}\big{)}^{2}+O(c^{-6}),$		(29a)
		$\displaystyle g_{0j}=-h^{0j}+O(c^{-5}),$		(29b)
		$\displaystyle g_{ij}=\delta_{ij}\Big{(}1+\frac{1}{2}h^{00}\Big{)}+h^{ij}-\frac{1}{2}\delta_{ij}h^{kk}+O(c^{-4}).$		(29c)

These equations provide sufficient PN corrections for the metric components that we need in the following computation. In an equivalent manner, using the general equation (89), we arrive at

$\displaystyle(-g)=1+h^{00}-h^{kk}+O(c^{-4}),$

(30)

for the PM expansion of $g$ containing the PN corrections to $O(c^{-4})$ order. As expected from the high order of $h^{kk}$, in this modified theory of gravity, the PM expansions of the metric and its determinant are more complicated than the GR ones.

After choosing the matter portion of the system, we are now in the position to solve the EMSG version of the field equations in the Landau-Lifshitz formalism. To do so, we utilize the iteration procedure. We recall that although our aim is to obtain $h^{jk}$ in the wave zone, enough information about the metric in the near zone, especially in the first iteration step is required. Let us mention that, in the PM theory, the near zone is restricted to a three-dimension sphere with the radius $\mathcal{R}\approx\lambda_{\text{c}}$. Beyond this region is called the wave zone. Given the position of the source point in the solution of the wave equations, the gravitational potential is separated into two parts. In a similar fashion to Poisson and Will (2014), we call these parts the near-zone and wave-zone portions of the gravitational potential when the source point is situated in the near and wave zones, respectively. Of course, the solution of the wave equations also depends on the position of the field point. In each case, we specify the field point position.

We now launch the iteration procedure. In the zeroth iterated step, we assume that $h_{\alpha\beta}^{(0)}=0$ and $g_{\alpha\beta}^{(0)}=\eta_{\alpha\beta}$. It should be emphasized that the assumption $h_{\alpha\beta}^{(0)}=0$ is equivalent to what we have considered to derive Eq. (3). In fact, here, it is assumed that there is no matter field in the background. Therefore, we have $\sqrt{-g^{(0)}}=1$ and $\gamma=1+\frac{1}{2}\frac{v^{2}}{c^{2}}+O(c^{-4})$. By considering the definition of rescaled mass density, one can simply deduce that $\rho=\big{(}1-\frac{1}{2}\frac{v^{2}}{c^{2}}+O(c^{-4})\big{)}\rho^{*}$ in the zeroth iteration. We then turn to find $h_{\alpha\beta}^{(1)}$ in the first iterated step. As we will see in the second iterated step, in order to extract the correct and required PN order for $h_{\alpha\beta}^{(2)}$, we need to find $h_{(1)}^{00}$ to $O(c^{-2})$, $h_{(1)}^{0j}$ to $O(c^{-3})$, and $h_{(1)}^{ij}$ up to $O(c^{-4})$ in the first iterated step. So, we should derive the sources of these gravitational potentials, i.e., the components of the effective energy-momentum tensor, as follows: $\tau^{00}_{\text{(0)eff}}$ to $O(c^{2})$, $\tau^{0j}_{\text{(0)eff}}$ to $O(c)$, $\tau^{ij}_{\text{(0)eff }}$ up to $O(1)$. In the following, we introduce each part of $\tau^{\alpha\beta}_{\text{(0)eff}}$ separately. For the usual energy-momentum tensor, $T^{\alpha\beta}_{(0)}$, we have

		$\displaystyle c^{-2}T^{00}_{(0)}=\rho^{*}+O(c^{-2}),$		(31a)
		$\displaystyle c^{-1}T^{0j}_{(0)}=\rho^{*}v^{j}+O(c^{-2}),$		(31b)
		$\displaystyle T^{ij}_{(0)}=O(1).$		(31c)
Also, for the EMSG energy-momentum tensor (28) up to the similar PN order, we find that

		$\displaystyle c^{-2}T^{00}_{\text{\tiny(0)EMS}}=2c^{2}f_{0}^{\prime}{\rho^{*}}^{2}+O(c^{-2}),$		(32a)
		$\displaystyle c^{-1}T^{0j}_{\text{\tiny(0)EMS}}=2c^{2}f_{0}^{\prime}{\rho^{*}}^{2}v^{j}+O(c^{-2}),$		(32b)
		$\displaystyle T^{ij}_{\text{\tiny(0)EMS}}=c^{4}f_{0}^{\prime}{\rho^{*}}^{2}\delta^{ij}+O(1).$		(32c)

Furthermore, since in the zeroth iterated step $h^{\alpha\beta}_{(0)}=0$, we arrive at $t_{(0)\text{LL}}^{\alpha\beta}=0=t_{(0)\text{H}}^{\alpha\beta}$. Hence, we obtain the main materials desired to construct the gravitational potentials in the next iteration of the gravitational field equations. Now, we attempt to find the gravitational potentials where both the field and source points are situated in the near zone in the first iterated step, i.e., $h_{(1)\mathcal{N}}^{\alpha\beta}$. To do so, we insert Eqs. (31a)-(32c) into integral (90). For the time-time component of $h^{\alpha\beta}_{(1)}$, we have

$\displaystyle h_{(1)\mathcal{N}}^{00}=\frac{4}{c^{2}}U+4f_{0}^{\prime}U_{\text{\tiny EMS}}+O(c^{-3}),$

(33)

in which $U$ is the Newtonian potential in terms of $\rho^{*}$ and $U_{\text{\tiny EMS}}$ is the EMSG gravitational potential. The Poisson integrals of $U$ and $U_{\text{\tiny EMS}}$ are given by

		$\displaystyle U=G\int_{\mathcal{M}}\frac{{\rho^{*}}^{\prime}}{\rvert{\bm{x}-\bm{x}^{\prime}}\rvert}d^{3}x^{\prime},$		(34a)
		$\displaystyle U_{\text{\tiny EMS}}=G\int_{\mathcal{M}}\frac{{{\rho^{*}}^{\prime}}^{2}}{\rvert\bm{x}-\bm{x}^{\prime}\rvert}d^{3}x^{\prime},$		(34b)

respectively. Here, $\mathcal{M}$ represents the three-dimension region which in fact separates the near zone from the wave zone. Similarly, for the time-space components of $h^{\alpha\beta}_{(1)}$ in the required order, we find that

$\displaystyle h^{0j}_{(1)\mathcal{N}}=\frac{4}{c^{3}}U^{j}+\frac{8}{c}f_{0}^{\prime}U^{j}_{\text{\tiny EMS}}+O(c^{-4}),$

(35)

where $U^{j}$ is the well-known PN vector potential

$\displaystyle U^{j}=G\int_{\mathcal{M}}\frac{{\rho^{*}}^{\prime}v^{\prime j}}{\rvert{\bm{x}-\bm{x}^{\prime}}\rvert}d^{3}x^{\prime},$

(36)

while $U^{j}_{\text{\tiny EMS}}$ is the new vector potential in the EMSG framework defined as

	$\displaystyle U^{j}_{\text{\tiny EMS}}=G\int_{\mathcal{M}}\frac{{{\rho^{*}}^{\prime}}^{2}v^{\prime j}}{\rvert{\bm{x}-\bm{x}^{\prime}}\rvert}d^{3}x^{\prime}.$		(37a)

Finally, for the space-space components, we arrive at

$\displaystyle h^{ij}_{(1)\mathcal{N}}=4f_{0}^{\prime}\delta^{ij}\Big{(}U_{\text{\tiny EMS}}-\frac{G}{c}\frac{d}{dt}\mathfrak{M}\Big{)}+O(c^{-4}),$

(38)

in which

$\displaystyle\mathfrak{M}=\int_{\mathcal{M}}{{\rho^{*}}^{\prime}}^{2}d^{3}x^{\prime},$

(39)

is a new quantity in EMSG. It should be noted that all integrands in the above relations are a function of $t$ and ${\bm{x}^{\prime}}$. So far, we have found the near-zone portion of the gravitational potential. To complete our derivation in this iterated step, we should also find the wave-zone part of the gravitational potential, i.e., $h_{(1)\mathcal{W}}^{\alpha\beta}$ where the source point is in the wave zone and the field point is situated in the near zone. In this case, as mentioned before, the source term of the potential $h_{(1)}^{\alpha\beta}$ is equal to $T_{(0)\text{eff}}^{\alpha\beta}$; and $t_{(0)\text{LL}}^{\alpha\beta}$ and $t_{(0)\text{H}}^{\alpha\beta}$ have no portion. Therefore, the source term is completely constructed from the matter part in this step. On the other hand, the matter part of the system is completely restricted to the near zone when the slow-motion condition is established Poisson and Will (2014). As we consider a gravitationally bound system, this assumption is also true throughout our derivation. This fact reveals that the wave-zone part of the gravitational potential in the first iterated step is zero. Subsequently, we have $h_{(1)\mathcal{W}}^{\alpha\beta}=0$ and $h_{(1)}^{\alpha\beta}=h_{(1)\mathcal{N}}^{\alpha\beta}$.

To sum up, we obtain the PN expansion of the EMSG potential’s components required to construct the near-zone metric in the first iterated step. Substituting Eqs. (33), (35), and (38) within Eqs. (29a), (29b),(29c), and (30), we finally arrive at

		$\displaystyle g_{00}^{(1)}=-1+\frac{2U}{c^{2}}+8f_{0}^{\prime}U_{\text{\tiny EMS}}+O(c^{-3}),$		(40a)
		$\displaystyle g_{0j}^{(1)}=-\frac{4U^{j}}{c^{3}}-\frac{8}{c}f_{0}^{\prime}U^{j}_{\text{\tiny EMS}}+O(c^{-4}),$		(40b)
		$\displaystyle g_{ij}^{(1)}=\Big{(}1+\frac{2U}{c^{2}}\Big{)}\delta_{ij}+O(c^{-4}),$		(40c)
		$\displaystyle(-g_{(1)})=1+\frac{4U}{c^{2}}-8f_{0}^{\prime}U_{\text{\tiny EMS}}+O(c^{-3}).$		(40d)

Here, we discard some extra PN orders.

As seen, the components and determinant of the near-zone metric are constructed from the usual potentials $U$ and $U^{j}$ as well as the EMSG potentials $U_{\text{\tiny EMS}}$ and $U^{j}_{\text{\tiny EMS}}$. Notice that in order to describe a relativistic system to the first PN (1PN ) order, one should find the PN corrections of $g^{00}$, $g^{0j}$, and $g^{jk}$ to $O(c^{-4})$, $O(c^{-3})$, and $O(c^{-2})$, respectively. It is obvious from the above terms, in this iterated step, the required PN orders for the time-time metric component are not completely evaluated. It means that one should continue the journey toward the next iterated step and find $g^{(2)}_{\alpha\beta}$. It is worth noting that the odd order $c^{-3}$ that appears in the determinant and time-time component of the metric is entirely built of the EMSG term that is proportional to $d\mathfrak{M}/dt$. In Appendix. C, we simplify this term. However, because in this step, the required PN orders of the metric to describe a relativistic system are not built, we are not concerned about this odd-power order. In other words, after finding the metric components at least to the 1PN order, one can truly interpret the role of the odd PN orders and examine the time-reversal invariance of solutions.

In the following, we attempt to obtain the appropriate PN corrections for $\tau^{\alpha\beta}_{\text{(1)eff}}$ that are necessary to derive $h^{\alpha\beta}_{(2)}$. Regarding the relationship between the effective energy-momentum pseudotensor and gravitational potential components, we deduce that the source terms of the gravitational potential should be built to $O(c^{2})$ for $\tau^{00}_{\text{(1)eff}}$, $O(c)$ for $\tau^{0j}_{\text{(1)eff}}$, and $O(1)$ for $\tau^{jk}_{\text{ (1)eff}}$. This fact will be clarified in the following computations.

By inserting Eqs. (40a)-(40c) into the normalization condition $g_{\alpha\beta}u^{\alpha}u^{\beta}=-c^{2}$, we find that $\gamma^{(1)}=1+4f_{0}^{\prime}U_{\text{\tiny EMS}}+\frac{1}{c^{2}}\big{(}\frac{1}{2}v^{2}+U\big{)}+O(c^{-3})$. Then, using $\gamma^{(1)}$ and $g_{(1)}$, we have $\rho^{*}=\Big{[}1+\frac{1}{c^{2}}\big{(}\frac{1}{2}v^{2}+3U\big{)}\Big{]}\rho+O(c^{-4})$ for the rescaled mass density. Now, by applying these parameters and the components of $g_{\alpha\beta}^{(1)}$ in Eq. (27), we arrive at

		$\displaystyle T^{00}_{(1)}=\rho^{*}c^{2}+O(1),$		(41a)
		$\displaystyle T^{0j}_{(1)}=\rho^{*}v^{j}c+O(c^{-1}),$		(41b)
		$\displaystyle T^{jk}_{(1)}=p\delta^{jk}+\rho^{*}v^{j}v^{k}+O(c^{-2}),$		(41c)

for the contravariant components of the energy-momentum tensor in the first iterated step. We keep terms up to the desired PN order mentioned earlier. It is worth mentioning that in Eq. (41a), the $O(1)$ term is entirely constructed from the EMSG term and it is given by $\rho^{*}c^{2}f_{0}^{\prime}U_{\text{\tiny EMS}}$.

We now obtain the EMSG part of the energy-momentum tensor $T^{\alpha\beta}_{\text{\tiny(1)EMS}}$. According to the original definition of this tensor, i.e., Eq. (8), we first obtain ${\bm{T}}^{2}_{(1)}$ for the perfect fluid up to $O(1)$. This is the appropriate PN order to find the required PN expansion of $T^{\alpha\beta}_{\text{\tiny(1)EMS}}$’s components. In this step, one can deduce

$\displaystyle{\bm{T}}^{2}_{(1)}={\rho^{*}}^{2}c^{4}-{\rho^{*}}^{2}c^{2}\big{(}v^{2}-2\Pi+6U\big{)}+O(1).$

(42)

Here, $\Pi=\epsilon/\rho^{*}$. Then using the above relation and the PN expansion of the metric and energy-momentum tensor components in the first iterated step, we arrive at

		$\displaystyle T^{00}_{\text{\tiny(1)EMS}}={\rho^{*}}^{2}c^{4}f_{0}^{\prime}+O(1),$		(43a)
		$\displaystyle T^{0j}_{\text{\tiny(1)EMS}}=2{\rho^{*}}^{2}c^{3}f_{0}^{\prime}v^{j}+O(c^{-1}),$		(43b)
		$\displaystyle T^{jk}_{\text{\tiny(1)EMS}}={\rho^{*}}^{2}c^{4}f_{0}^{\prime}\Big{[}\delta^{jk}+\frac{1}{c^{2}}\Big{(}2v^{j}v^{k}-\big{(}v^{2}-2\Pi$
		$\displaystyle+8U\big{)}\delta^{jk}\Big{)}\Big{]}+O(c^{-2}),$		(43c)

after some simplifications. Here, we again truncate the PN expansion of these components to the required accuracy. It should be mentioned that the $c^{0}$ order in $T^{00}_{\text{\tiny(1)EMS}}$ is built of the EMSG term that is proportional to ${\rho^{*}}^{2}c^{4}{f_{0}^{\prime}}^{2}U_{\text{\tiny EMS}}$.

The remaining source terms of the gravitational potential $h_{(2)}^{\alpha\beta}$ are $(-g^{(1)})t_{(1)\text{LL}}^{\alpha\beta}$ and $(-g^{(1)})t_{(1)\text{H}}^{\alpha\beta}$. As we have mentioned before, these pseudotensors themselves are constructed from the gravitational potential. In the first iterated step, they are built of $h_{(1)}^{\alpha\beta}$. We keep those PN orders in the expansion of $h_{(1)}^{\alpha\beta}$ which construct the order $c^{2}$ for $(-g^{(1)})t_{(1)\text{LL,H}}^{00}$, order $c^{1}$ for $(-g^{(1)})t_{(1)\text{LL,H}}^{0j}$, and order $c^{0}$ for $(-g^{(1)})t_{(1)\text{LL,H}}^{jk}$. We start with the Landau-Lifshitz pseudotensor. After inserting Eqs. (33), (35), and (38) within the definition (B), we find

		$\displaystyle(-g^{(1)})t_{(1)\text{LL}}^{00}=O(1),$		(44a)
		$\displaystyle(-g^{(1)})t_{(1)\text{LL}}^{0j}=O(c^{-1}),$		(44b)
		$\displaystyle(-g^{(1)})t_{(1)\text{LL}}^{jk}=\frac{1}{4\pi G}\bigg{[}\partial^{j}U\partial^{k}U-\frac{1}{2}\partial^{n}U\partial_{n}U\delta^{jk}\bigg{]}$		(44c)
		$\displaystyle+\frac{c^{4}f_{0}^{\prime}}{\pi G}\bigg{[}f_{0}^{\prime}\Big{(}\partial^{n}U_{\text{\tiny EMS}}\partial_{n}U_{\text{\tiny EMS}}\delta^{jk}-\partial^{j}U_{\text{\tiny EMS}}\partial^{k}U_{\text{\tiny EMS}}\Big{)}$
		$\displaystyle+\frac{1}{c^{2}}\Big{(}2\partial^{j}U_{\text{\tiny EMS}}\partial^{k}U-\partial^{n}U_{\text{\tiny EMS}}\partial_{n}U\delta^{jk}\Big{)}\bigg{]}+O(c^{-1}),$

for the PN expansion of the Landau-Lifshitz pseudotensor components. The last task here is to evaluate $(-g^{(1)})t_{(1)\text{H}}^{\alpha\beta}$. To do so, we substitute Eqs. (33), (35), and (38) into Eq. (15). After some simplifications, we deduce that

		$\displaystyle(-g^{(1)})t_{(1)\text{H}}^{00}=O(1),$		(45a)
		$\displaystyle(-g^{(1)})t_{(1)\text{H}}^{0j}=O(c^{-1}),$		(45b)
		$\displaystyle(-g^{(1)})t_{(1)\text{H}}^{jk}=\frac{c^{4}f_{0}^{\prime 2}}{\pi G}\bigg{[}\partial^{j}U_{\text{\tiny EMS}}\partial^{k}U_{\text{\tiny EMS}}$		(45c)
		$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-U_{\text{\tiny EMS}}\partial_{n}\partial^{n}U_{\text{\tiny EMS}}\delta^{jk}\bigg{]}+O(c^{-1}).$

Gathering together Eqs. (41a)-(41c), and (43a)-(45c), one can construct the components of the EMSG effective pseudotensor as follows:

		$\displaystyle c^{-2}\tau^{00}_{(1)\text{eff}}=\rho^{*}+c^{2}f_{0}^{\prime}{\rho^{*}}^{2}+O(c^{-2}),$		(46a)
		$\displaystyle c^{-1}\tau^{0j}_{(1)\text{eff}}=\rho^{*}v^{j}+2c^{2}f_{0}^{\prime}{\rho^{*}}^{2}v^{j}+O(c^{-2}),$		(46b)
		$\displaystyle\tau^{jk}_{(1)\text{eff}}=\rho^{*}v^{j}v^{k}+p\delta^{jk}+\frac{1}{4\pi G}\bigg{[}\partial^{j}U\partial^{k}U-\frac{1}{2}\partial_{n}U\partial^{n}U\delta^{jk}\bigg{]}$
		$\displaystyle+c^{4}f_{0}^{\prime}\bigg{[}{\rho^{*}}^{2}\delta^{jk}+\frac{1}{c^{2}}\Big{(}{\rho^{*}}^{2}\delta^{jk}\big{(}2\Pi-v^{2}-4U\big{)}+2{\rho^{*}}^{2}v^{j}v^{k}$
		$\displaystyle+\frac{1}{\pi G}\big{(}2\partial^{j}U_{\text{\tiny EMS}}\partial^{k}U-\partial_{n}U_{\text{\tiny EMS}}\partial^{n}U\delta^{jk}\big{)}\Big{)}$		(46c)
		$\displaystyle+f_{0}^{\prime}\delta^{jk}\Big{(}\frac{1}{\pi G}\partial_{n}U_{\text{\tiny EMS}}\partial^{n}U_{\text{\tiny EMS}}-4{\rho^{*}}^{2}U_{\text{\tiny EMS}}\Big{)}\bigg{]}+O(c^{-1}).$

Here, to simplify the last component, we utilize the Poisson equation $\nabla^{2}U_{\text{\tiny EMS}}=-4\pi G{\rho^{*}}^{2}$. As seen, except for the space-space component, the standard and EMSG parts of $\tau^{00}_{(1)\text{eff}}$ and those of $\tau^{0j}_{(1)\text{eff}}$ have the same order of the magnitude. Only the term $c^{4}f_{0}^{\prime}{\rho^{*}}^{2}$ in $\tau^{jk}_{(1)\text{eff}}$ has an unusual PN order, $O(c^{2})$, compared to the standard terms in this component. In the following computations, we investigate the possible role of this term in the final result.