The gravitational equation in higher dimensions

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. The thermodynamical parameters bear the same relation to horizon radius, for example entropy always goes as $r_h^{d-2n}$ and so for the critical dimensions it always goes as $r_h, \, r_h^2$. In terms of the area, it would go as $A^{1/n}$. The generalized analogues of the Nariai and Bertotti-Robinson solutions arising from the product of two constant curvature spaces, also bear the same relations between the curvatures $k_1=k_2$ and $k_1=-k_2$ respectively.