5 Newtonian and Anti-Newtonian Fields

We can associate a Newtonian model with purely gravitoelectric ($H_{ab}=0$ ). Without the gravitomagnetism, the nonlocal nature of the Newtonian force cannot be retrieved from relativistic models. It also excludes gravitational waves. The Newtonian model is a limited model to show the characteristics of the gravitoelectric. We can also consider an anti-Newtonian model; a model with purely gravitomagnetic ($E_{ab}=0$ ). The anti-Newtonian model obstructs sounding solutions. In [43], it has been proven that the anti-Newtonian model shall include either shear or vorticity.

5.1 Newtonian Model

Let us consider the Newtonian model ($H_{ab}=0$ ) in an irrotational static spacetime ( $\omega_{a}=\dot{u}_{a}=0$ ) and a perfect-fluid model ( $q^{a}=\pi_{ab}=0$ ). The constraints and propagations shall be

$$\begin{array}[c]{cc} {C^{1}{}_{a}=(\mathrm{div}E)_{a}-{\texts... ...m{D}_{a} \rho=0,} & {C^{2}{}_{a}=[\sigma,E]_{a}=0,} \end{array}$$ (65)

$$\begin{array}[c]{c} {P^{1}{}_{ab}=-\dot{E}_{\left\langle {ab}... ...p)=0,}\\ \\ {P^{2}{}_{ab}=\mathrm{curl}(E)_{ab}=0,} \end{array}$$ (66)

$$\begin{array}[c]{cc} {C^{6}{}_{a}={\textstyle{\frac{2}{3}}}\m... ...0,} & {C^{7}{}_{ab}=-\mathrm{curl}(\sigma)_{ab}=0.} \end{array}$$ (67)

To the first order, divergence and evolution of Eq. (66b) are

$$\mathrm{D}^{b}P^{2}{}_{ab} ={\textstyle{\frac{1}{2}}}\varepsilon _{abc}\mathrm{D}^{b}\mathrm... ...+{\textstyle{\frac{1}{3}}} \Theta\lbrack\sigma,E]_{a}-\sigma_{ab}[\sigma,E]^{b}$$

$$={\textstyle{\frac{1}{2}}}\mathrm{curl}(C^{1}){}_{a}+{\textstyle{... ...}-\sigma_{a}{}^{b}C^{2}{}_{b}+{\textstyle{\frac{1} {3}}}\omega{}_{a}\dot{\rho},$$ (68)

$$\dot{P}^{2}{}_{ab} =-{\textstyle{\frac{1}{3}}}\Theta{\mathrm{curl} (E)_{ab}}-\sigma_... ...c}\varepsilon_{cd(a}\mathrm{D}^{e}E_{b)}{} ^{d}+{\mathrm{curl}(\dot{E})}_{{ab}}$$

$$=-{\textstyle{\frac{3}{2}}}\varepsilon^{cd}{}_{(a}\sigma_{b)c}C^{... ...}_{ab}+{\textstyle{\frac{3}{2}} }\varepsilon^{c}{}_{d(a}C^{6}{}_{c}E_{b)}{}^{d}$$

$$-{\textstyle{\frac{1}{2}}}(\rho+p)C^{7}{}_{ab}-\mathrm{curl}(P^{1... ...mathrm{curl(}\sigma_{c\left\langle a\right. }E_{\left. b\right\rangle }{}^{c}).$$ (69)

The last parameter ( ${\textstyle{\frac{1}{3}}}\omega_{a}\dot{\rho}$ ) in Eq. (68) vanishes because of irrotational condition. Eq. (68) then conserves the constraints. Eq. (69) must be consistent with Eqs. (65) and (67). Thus, the last parameters in Eq. (69) has to vanish:

$$\mathrm{curl}(\sigma_{c\left\langle a\right. }E_{\left. b\right\rangle } {}^{c})=0.$$ (70)

It is a necessary condition for the consistent evolution of propagation. This condition is satisfied with irrotational product of gravitoelectric and shear, but it is a complete contrast to Eq. (65b). Thus, the Newtonian model is generally inconsistent with generic relativistic models. Moreover, the temporal evolution of propagation shows no wave solutions.

5.1.1 Newtonian Limit

The Newtonian model obstructs wave solution, due to the instantaneous interaction. Following [28,29,30], we consider a model whose action propagates at infinite speed ( $c\rightarrow \infty$ ). This is compatible with $\mathop{\lim }\limits_{c\rightarrow\infty }E_{ab}=E_{ab}(t)\vert _{\infty}$ , where $E_{ab}(t)\vert _{\infty}$ is an arbitrary function of time.

We define the Newtonian potential as

$$E_{ab}\equiv\mathrm{D}_{\left\langle a\right. }\mathrm{D}_{\left.... ...rm{D}_{a}\mathrm{D}_{b}\Phi-{\textstyle{\frac{1}{3} }}h_{ab}\mathrm{D}^{2}\Phi.$$ (71)

On substituting into Eq. (65a), we get

$$C^{1}{}_{a}=\mathrm{D}_{a}\mathrm{D}^{2}\Phi-{\textstyle{\frac{1}... ...{D}^{b}h_{ab}\mathrm{D}^{2}\Phi-{\textstyle{\frac{1}{3}}} \mathrm{D}_{a}\rho=0.$$ (72)

In a spatial infinity, we obtain the Poisson equation of the Newtonian potential:

$$C^{1}\equiv\mathrm{D}^{2}\Phi-{\textstyle{\frac{1}{2}}}\rho =0.$$ (73)

Eq. (65a) generalizes the gravitoelectric as the Newtonian force in the gradient of the relativistic energy density.

Moreover, Eq. (66a) gives

$$P^{1}{}_{ab}=-\mathrm{D}_{a}\mathrm{D}_{b}\dot{\Phi}+{\textstyle{... ...b} \Phi+{\textstyle{\frac{1}{3}}}(\dot{h}_{ab}+\Theta h_{ab}\mathrm{)D}^{2}\Phi$$

$$+3\sigma_{c\left\langle a\right. }\mathrm{D}_{\left. b\right\rang... ...{}^{\mathrm{c}}\mathrm{D}^{2}\Phi-{\textstyle{\frac{1}{2}}}\sigma_{ab} (\rho+p) =0.$$ (74)

In the Newtonian theory, we could not find the temporal evolution of the Newtonian potential.

5.1.2 Acceleration Potential

In an irrotational spacetime, Eq. (46) is

$$P^{4}{}_{a}={\textstyle{\frac{1}{2}}}\mathrm{curl}(\dot{u}){}_{a}=0.$$ (75)

It introduces a scalar potential:

$$\dot{u}_{a}=\mathrm{D}_{a}\Phi,$$ (76)

where $\Phi$ is the acceleration potential. This scalar potential corresponds to the Newtonian potential. In the irrotational Newtonian model, the linearized acceleration is characterized as the acceleration potential.

5.2 Anti-Newtonian Model

Let us consider the anti-Newtonian model ($E_{ab}=0$ ) in a shearless static spacetime ( $\omega_{a}=\dot{u}_{a}=0$ ) and a perfect-fluid model ( $q^{a}=\pi_{ab}=0$ ). The constraints and propagations shall be

$$\begin{array}[c]{cc} {C^{1}{}_{a}=-3\omega^{b}H_{ab}-{\textst... ...2}{}_{a}=(\mathrm{div}H)_{a}+\omega_{a}(\rho+p)=0,} \end{array}$$ (77)

$$\begin{array}[c]{cc} {P^{1}{}_{ab}=\mathrm{curl}(H)_{ab}=0,} ... ...b}-[\omega,H]_{\left\langle {ab}\right\rangle }=0,} \end{array}$$ (78)

$$\begin{array}[c]{cc} {C^{6}{}_{a}={\textstyle{\frac{2}{3}}}\m... ...ngle a\right. } \omega_{\left. b\right\rangle }=0.} \end{array}$$ (79)

To linearized order, divergence and evolution of Eq. (78a) are

$$\mathrm{D}^{b}P^{1}{}_{ab} ={\textstyle{\frac{1}{2}}}\varepsilon _{abc}\mathrm{D}^{b}\mathrm{(D}_{d}H^{cd})$$

$$={\textstyle{\frac{1}{2}}}\varepsilon_{ab}{}^{c}\mathrm{D}^{b}C^{... ...2}}}(\rho+p)C^{6}{}_{a}+{\textstyle{\frac{1}{3}} }(\rho+p)\mathrm{D}_{a}\Theta,$$ (80)

$$\dot{P}^{1}{}_{ab} =-{\textstyle{\frac{1}{3}}}\Theta\mathrm{curl} {(H)_{ab}}+{\mathrm{curl}(\dot{H})_{ab}}$$

$$=-{\textstyle{\frac{4}{3}}}\Theta P^{1}{}_{ab}+\mathrm{curl}(P^{2}){} _{ab}+\mathrm{curl}([\omega,H])_{\left\langle {ab}\right\rangle }.$$ (81)

Eq. (80) is consistent only in the spacetime being free from either the gravitational mass and pressure or the gradient of expansion. According to Eqs. (77) and (79), the last term in Eq. (81) has to vanish:

$$\mathrm{curl}([\omega,H])_{\left\langle {ab}\right\rangle }=0.$$ (82)

It is a necessary condition for the consistent evolution of propagation. This condition is satisfied with irrotational vorticity products of gravitomagnetic, but it is not consistent with Eq. (79b):

$$\varepsilon^{c}{}_{d(a}C^{7}{}_{b)c}\omega^{d}-{\textstyle{\frac{... ...[\omega,\omega]_{a} -{\textstyle{\frac{1}{4}}}\mathrm{D}_{a}[\omega,\omega]_{b}$$

$$+{\textstyle{\frac{1}{6}}}\omega_{b}\mathrm{D}_{a}\Theta+{\textst... ...}_{db}\dot{u}_{\left\langle a\right. }\omega_{\left. c\right\rangle }\omega^{d} =0.$$ (83)

Thus, the anti-Newtonian model is generally inconsistent with relativistic models. Furthermore, there is not a possibility of gravitational waves.

5.2.1 Vorticity Potential

In an unexpansive spacetime, Eq. (79a) takes the following form:

$$C^{6}{}_{a}=\mathrm{curl}{(\omega)_{a}}=0.$$ (84)

It defines a vorticity scalar potential $\Psi$ as

$$\omega_{a}=\mathrm{D}_{a}\Psi.$$ (85)

In the unexpansive anti-Newtonian model, the linearized vorticity is characterized as the vorticity potential.

5.2.2 Anti-Newtonian Limit

We may consider a gravitomagnetic model whose action propagates at infinite speed. Let us define the anti-Newtonian potential as

$$H_{ab}\equiv\mathrm{D}_{\left\langle a\right. }\mathrm{D}_{\left.... ...rm{D}_{a}\mathrm{D}_{b}\Psi-{\textstyle{\frac{1}{3} }}h_{ab}\mathrm{D}^{2}\Psi.$$ (86)

We substitute Eqs. (85) and (86) into Eq. (77b):

$$C^{2}{}_{a}=\mathrm{D}_{a}\mathrm{D}^{2}\Psi-{\textstyle{\frac{1}{3}} }\mathrm{D}^{b}h_{ab}\mathrm{D}^{2}\Psi+\mathrm{D}_{a}\Psi(\rho+p)=0.$$ (87)

In a spatial infinity, we derive the Helmholtz equation:

$$C^{2}\equiv\mathrm{D}^{2}\Psi+{\textstyle{\frac{3}{2}}}(\rho+p)\Psi=0.$$ (88)

Eq. (77b) associates the gravitomagnetic with the angular momentum $\omega_{a}(\rho+p)$ .