5.1.1 Newtonian Limit

The Newtonian model obstructs wave solution, due to the instantaneous interaction. Following Refs. [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)

Equation (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.