Are there consistent theories of GR where our Universe is embedded in higher dimensional space?

Question

In GR we have a metric space given by the metric $g^{\mu\nu}(x)$. This allows us to deetermine the curvature $R^{\mu\nu}$ at each point in space-time.

This curved 4D manifold is not required to be embedded in some higher dimensional space. But it is perfectly possible to do so. The Nash embedding theorems and related theorems say that an Einsteinian manifold can be embedded in a flat space given enough dimensions (probably provided it is simply connected). Let $\phi^N$ be the coordinates of some higher dimensional space. In which case we could write the metric as:

$$g_{\mu\nu}(x) = \sum_N \partial_\mu \phi^N(x) \partial_\nu \phi^N(x)$$

Where $N$ ranges over a suitably high enough number. If we substitute this back into the GR equations, nothing would change except we could solve for $\phi^N$ (where there would be some freedom of choice).

But where it would become interesting is if the fields $\phi^N(x)$ were observable by themselves. e.g. had an action (although it looks like the action would cancel itself out in a way):

$$\sqrt{g} (g^{\mu \nu}(x) \partial_\mu \phi^N(x) \partial_\nu \phi^N(x) + m^2 \phi^N(x)\phi^N(x)) = \sqrt{g} (1 + m^2) \phi^N(x)\phi^N(x) $$

Although perhaps there would be better actions for the $\phi$ fields maybe if they were non-abelian.

But anyway, I have heard about ideas that the Universe we live in is a 4-dimensional membrane in a higher dimensional space but I don't know if the above math is how that is realised. The question is, are there theories of gravity such that the metric desrcibes the embedding of a manifold in a higher dimensional space? And what would be the evidence for or against such an idea?

Answer 1

Philosophically speaking, one has to ask about the ontology of the higher dimensional space that the Einsteinian universe is embedded in. Merely because there is a mathematical theorem demonstrating such an embedding is no evidence of any physical reality of a higher dimensional space. This is why it was so important to physics that an intrinsic geometry of curved spaces was developed by mathematicians. In fact soon after that invention, William Clifford, the British physicist and geometer, declared that reality was nothing other than curvature in spacetime and this fifty years before Einstein was able to treat Clifford’s vision mathematically with his discovery of GR in 1915.

Nordstrom, the Finnish physicist and rival to Einstein, pioneered the idea of higher dimensional spaces which was later picked up by Kaluza and Klien and this is essentially what is used to non-commutative geometry models of the universe and also in string theory.

What you are thinking of are instead brane world cosmologies. This is peculiar to string theory. Here, the universe is located on a brane in a higher dimensional space and all the forces apart from gravity are also confined to the brane. However, gravity leaks out into the bulk and so is weaker than the other forces. Its explanatory motivation is that it resolves the hierarchy problem in physics - why the forces all have different strengths - and it also means that gravity should be stronger than expected on microscopic levels. Testing this prediction will be a tall order given just how weak gravity is. So it's likely to remain speculative physics for quite some time.

Even if the prediction were to be found to be correct, the philosophical question of what reality we should give the higher dimensional bulk space will still rear its head. Its quite likely, given the fertile inventiveness of physicists, that some other interpretation of the bulk space may become mooted and become fashionable. After all, even a century after the discovery of QM, we are still arguing over the right ontological interpretation of QM, with at least twenty or so being current.

Personally speaking, although some physicists take interpretations not be quite physics, they are at the root of what physics are. After all, physics began as the question as to what physical reality actually is. It was a question about it's ontological dimensions. But given the intractability of these questions - since the advent of QM - and the many methods physicists have for doing physics, it's perhaps no surprise they're been hived of to the philosophy of physics. In fact, they are merely physics by another name. This is especially evident when contrasted to other traditional philosophical concerns - for example, the question of the good life, for one man as well as a community and questions of political authority - questions that 'philosophers of physics' hardly touch upon - and which they should, especially since Eisenhowers warning of the industrial-military complex - and in his mind, that would have included the scientific enterprise and whose pre-eminent representative is physics.