The gravitational equation in higher dimensions

Understand

Like the Lovelock Lagrangian which is a specific homogeneous polynomial in Riemann curvature, for an alternative derivation of the gravitational equation of motion, it is possible to define a specific homogeneous polynomial analogue of the Riemann curvature, and then the trace of its Bianchi derivative yields the corresponding polynomial analogue of the divergence free Einstein tensor defining the differential operator for the equation of motion.

• We propose that the general equation of motion is $G^{(n)}_{ab} = -\Lambda g_{ab} +\kappa_n T_{ab}$ for $d=2n+1, \, 2n+2$ dimensions with the single coupling constant $\kappa_n$, and $n=1$ is the usual Einstein equation.
• It turns out that gravitational behavior is essentially similar in the critical dimensions for all $n$.
• All static vacuum solutions asymptotically go over to the Einstein limit, Schwarzschild-dS/AdS.