Should the 4D normalization constant $8\pi$ in Einstein field equations (EFE) be changed to $(n-2)S_{n-2}$, where $S_{n-2}$ denotes the area of a $n-2$-sphere, in higher dimensions? The reason is that the factor $8\pi$ essentially comes from the Poisson equation and the Green function of Laplacian in 3D.

Or is this just a matter of convention?

  • 2
    $\begingroup$ It's a matter of convention. Yes, $\kappa = 8 \pi G_D$ (for all $D \ge 3$) is an implicit reference to our $D = 4$ spacetime. Take also note that $G_D \sim \mathrm{L}^{D - 2}$. For more details, see this topic : physics.stackexchange.com/q/450840 $\endgroup$ – Cham Sep 11 at 12:57

It is, indeed, just a matter of convention since Einstein equations in higher dimension are just mathematical, they have no currently known physical application. This means you could even drop any reference to units in them. The typical choice is to place $\kappa$ in front of the stress-energy tensor as a catch-all coupling constant as Cham mentioned in the comments. As a result, the equations are $$R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} + \Lambda g_{\mu\nu} = \kappa T_{\mu\nu}\,.$$ But the truth is most mathematicians researching Einstein equations in $D$ dimensions are either way interested in ($\Lambda$-)vacuum space-times, which leads to equations without reference to $\kappa$ $$R_{\mu\nu} = \frac{2 \Lambda}{D-2} g_{\mu\nu} \,.$$ If you find solutions such as black hole space-times, where the "matter" is essentially a boundary condition at the singularity, then in does appear in the metric as a parameter, but you just define it to get the simplest form of the metric and usually do not care about "physical" numerical factors.

But yes, if you wanted to have a mass $M$ (defined by a $D$-volume integral of $T^{00}$) exerting a gravitational acceleration $\approx G_{\rm D} M/r^{D-2}$ in the Newtonian limit, then $\kappa = (D-2)S_{D-2} G_D$ is the right convention.


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