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 $\omega_{a}(\rho+p)$.