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.
