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

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

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

Equation (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.