How does Einstein Field Equations (EFE) work with more than 4 infinitely large spacetime dimensions?

Question

How does Einstein Field Equations (EFE) work with more than 4 infinitely large spacetime dimensions?

But is gravity in Newtonian physics valid with more than 3 infinitely large spatial dimensions?

Answer 1

When you derive the EFE from the Einstein-Hilbert action, $$ S_{EH}= \frac{1}{2 \kappa} \int R \sqrt{-g} \, d^d x \ , $$ at no point do you need to restrict the number of dimenions $d$ to 4 (but remember that the coupling constant $\kappa$ depends on $d$). Variation with respect to the metric leads to the usual field equations, $$ G_{\mu \nu} = \kappa T_{\mu \nu} \ . $$ Nowhere in the derivation do we need to explicitly fix $d$. So this is valid in whatever $d>3$ you were asking in the question.

If you're asking about compactifications from some number of higher dimensions down to 4 dimensions (which you'd want to do to get results that correspond to the physical world), then that's a different question which crops up more often in string theory or Kaluza-Klein theories.

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As for your questions about Newtonian gravity, the inverse square law is for 3 spatial (or 4 spacetime) dimensions only. If you want to think about generalising this to higher dimensions, which obviously doesn't correspond to our universe, then you can look at Gauss's law which implies the power of $r$ goes like $n-1$ where $n$ is the number of spatial dimensions, i.e. $n=3 \rightarrow F \propto \frac{1}{r^2} $ , $n=4 \rightarrow F \propto \frac{1}{r^3} $, etc. See https://arxiv.org/abs/astro-ph/0104026 for details about this.

Lastly, you ask whether the EFE in higher dimensions can be made 'consistent with Newtonian gravity'. I'm not too sure what you mean, but clearly for $d \neq 4$, assuming no compactification procedures, we're no longer describing the universe we observe. So this wouldn't agree with Newtonian gravity in any limit. It might also be worth taking a look at Gravitational constant in higher dimensions?.