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

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?

• The EFE have a bunch of spacetime indices, right? Those indices can run from 0 to 3 or from 0 to n if you want. Likewise, you can solve Laplace's equation in any number of dimensions. Is there something specific that you are worried about? – kaylimekay Jan 12 at 2:36
• However, is gravity in newtonian physics valid even with large/infinite space dimensions more than 3? – Samzun Jan 12 at 2:50
• Yeah that's what I mean about Laplace's equation. – kaylimekay Jan 12 at 2:51
• You seem to have three separate questions here. (1) Do the EFE generalize to more than 3+1 dimensions ? and (2) What is the equivalent of Newtonian gravity in more than 3 spatial dimensions and (3) Can the EFE for $n+1$ dimensions (\$n>3) be made consistent with the corresponding "Newtonian gravity". Would that be what you are asking ? – StephenG Jan 12 at 3:42
• I told someone "the EFE can handle many dimensions (not just 4)" and he told me "Not the EFE of GR, no. That EFE is specifically for 4 dimensions. As above, GR describes only the geometry of the 4 ordinary spacetime dimensions we observe. As far as GR is concerned, these "extra dimensions", or more precisely their effects as manifested in things like new particles or fields beyond the ones we already know of (in the Standard Model of particle physics) would appear as part of the stress-energy tensor not the spacetime geometry. – Samzun Jan 12 at 3:57

## 1 Answer

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.

--

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?.

• is it true the EFE of GR is specifically for 4 dimensions? As if EFE is not necessarily GR but more general? I told someone "the EFE can handle many dimensions (not just 4)" and he told me "Not the EFE of GR, no. That EFE is specifically for 4 dimensions. GR describes only the geometry of the 4 ordinary spacetime dimensions we observe." Objections? Clarifications tnx – Samzun Jan 12 at 22:50
• @Samzun 1) The EFE are not exclusively 4 dimensional, (as in, they can be applied in any number of dimensions and are still mathematically valid), but to describe our universe you'd work in 4 dimensions. This is probably why you and the other person were disagreeing. 2) For all intents and purposes, GR and EFE are basically synonymous (the EFE are a part of GR). 3) Standard GR works with a 4-dimensional manifold, but it holds in any number of dimensions. If you're talking about GR in our universe, one would assume you're talking about 4D spacetime. – Eletie Jan 12 at 23:07
• Is it not possible our universe have more than 4D spacetime, not compactified but large where gauss's law is working and not really inverse square laws? – Samzun Jan 12 at 23:25
• @Samzun No, it doesn't appear to be – Eletie Jan 13 at 0:10