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Probably another question without an answer! ;-)

In most books/papers I saw on General Relativity, everybody writes $\kappa = 8 \pi G_D$ in the right part of Einstein's equation, even for spacetimes of dimensionality $D \ne 4$ (the minus sign is just a matter of convention): \begin{equation}\tag{1} G_{\mu \nu} + \Lambda_D \, g_{\mu \nu} = -\, 8 \pi G_D \, T_{\mu \nu}. \end{equation} I understand that we could always define $G_D$ so we can write $8 \pi$ in front of it. But it's not "natural", since $\Omega_3 = 4 \pi$ is the solid angle in flat space of dimension 3. In general: \begin{align}\tag{2} \Omega_{2 n} &= \frac{1}{(n - 1)!} \, 2 \pi^n, \qquad &\Omega_{2 n + 1} &= \frac{2^{2 n} \, n!}{(2 n)!} \, 2 \pi^n. \end{align} For examples: $\Omega_1 = 2$, $\Omega_2 = 2 \pi$, $\Omega_3 = 4 \pi$, $\Omega_4 = 2 \pi^2$. So it would be more natural to write this, in Einstein's equation (I know, it looks awefull, but it has more sense!): \begin{equation}\tag{3} G_{\mu \nu} + \Lambda_D \, g_{\mu \nu} = -\, 2 \, \Omega_{D - 1} \, G_D \, T_{\mu \nu}. \end{equation} Notice that Newton's constant has a dimension that depends on $D$ (since $G_{\mu \nu} \sim \mathrm{L}^{-2}$ and $T_{\mu \nu} \sim \mathrm{L}^{-D}$ in cartesian-style coordinates): \begin{equation}\tag{4} G_D \sim \mathrm{L}^{D - 2}. \end{equation} That constant has a conformal weight! (it's also true for the fine structure constant by the way: $\alpha \sim \mathrm{L}^{D - 4}$, if you write Maxwell equation in $D$ dimensional spacetime).

So my questions are these:

Why use $G_D = \kappa / 8 \pi$ instead of $G_D = \kappa / 2 \, \Omega_{D-1}$? (it obviously doesn't give the same constant when $D \ne 4$!). $\kappa = 8 \pi G_D$ may be simpler than $\kappa = 2 \, \Omega_{D - 1} \, G_D$, but it doesn't make physical sense!

What is the meaning of the factor $2$ in $\kappa = 2 \, \Omega_{D - 1} \, G_D$ (for any $D$)? Is it related to the graviton's spin? (it is not given by $2 s + 1$ or $s (s + 1)$). In the special case of $D = 4$, what is the interpretation of that factor in $8 \pi G_4 = 2 \cdot 4 \pi G_4$?

In - very - hypothetical spacetimes with many timelike dimensions, what should be the natural expression for $\kappa$? For example, for $D = 4$ and metric signature $\eta = (1, 1, -1, -1)$ (two timelike dimensions, two spacelike dimensions), should we write $\kappa = 2 \cdot 2 \pi G_4$ instead of the usual $\kappa = 8 \pi G_4$?

I'm pretty sure nobody would have an answer to these "idiotic" questions. I'm asking them anyway, in case someone else is crazy enough to scratch his/her head like me! ;-)

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    $\begingroup$ I don't see any real physics here. Which prefactor we assign to $G_D$ is entirely conventional, and the factors involving $\pi$ are simply due to mismatched conventions between Newtonian physics and GR, cf. physics.stackexchange.com/q/71636/50583. The belief that any of the other choices for $G_D$ is objectively "more natural" does not strike me as useful, especially since people are still free to set (by choice of units) whatever multiple of $G$ to unity they like. There is no deep physics in the choices for dimensionful constants, cf. also arxiv.org/abs/1412.2040. $\endgroup$ – ACuriousMind Dec 28 '18 at 15:46
  • $\begingroup$ @ACuriousMind, I don't agree. In flat space, the elliptic equation part of Poisson's equation do make the $4 \pi$ apparent, in front of $G$. That solid angle is important. In curved space, that $4 \pi$ can be defined from the tangent space, as a limit of a very small region in curved space (the solid angle doesn't depend on the size of the region considered). In $D > 4$, the solid angle is different than in $D = 4$ and should be taken into account, in the elliptic part of Poisson's equation (weak field case). $\endgroup$ – Cham Dec 28 '18 at 15:50
  • $\begingroup$ I don't want to pick sides here, just wanted to point to a footnote in Ortin's Gravity and Strings, that seems to share the discomfort for this convention. In the second edition of the book, in page 73, at the end of section 3.1.1, footnote 6: "This is an unfortunate convention in the literature in which the factor $4\pi$, which is appropriate for rationalized units in four dimensions, is indiscriminately used in all dimensions.". This book also discusses the implications for Newton's constant in arbitrary dimensions, if only with the books own conventions which you may or may not dislike. $\endgroup$ – secavara Dec 28 '18 at 16:02
  • $\begingroup$ @ACuriousMind, an example that shows the relative importance of the solid angle. In Schwarzschild coordinates, we have this metric component: $g_{00} = 1 - \frac{2G M}{r}$. Absorbing the solid angle in $G$ would give $g_{00} = 1 - \frac{2 G' M}{4 \pi r}$, which is horrible (and $4 \pi r$ is not even an area)! The $4 \pi$ factor in Einstein's equation is a "chauvinistic" reference to our 3 space dimensions. $\endgroup$ – Cham Dec 28 '18 at 16:04
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    $\begingroup$ This paper might be of interest: "Normalization conventions for Newton’s constant and the Planck scale in arbitrary spacetime dimension," arxiv.org/abs/gr-qc/0609060. Notice the word "conventions" in the title. $\endgroup$ – Chiral Anomaly Dec 28 '18 at 16:15

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