Due to a recent discussion on Phil's Blog, I was reminded of a very famous and interesting theorem that I was first introduced to in my differential geometry class at Cal, the Nash Embedding Theorem. In a nutshell, it states:
Every \(n\) dimensional Riemannian manifold \(M\) of class \(C^k\;(k\geq 3)\) can be \(C^k\) isometrically embedded in any small portion of a Euclidean space \(\mathbb{R}^m\), where \(m=\tfrac{1}{2}n(3n+11)\) for compact \(M\) and \(m=\tfrac{1}{2}n(n+1)(3n+11)\) for noncompact \(M\).
The reason that this is so amazing is that we can take essentially any smooth manifold and find an isometric (length-preserving) embedding \(f:M\hookrightarrow\mathbb{R}^m\) into a higher dimensional Euclidean (i.e., "flat") space. For you relativity nuts, this is a big deal: curved spacetime can be embedded into a "flat" higher dimensional space.
There is, however, one caveat: the spacetime of general relativity is not Riemannian, but rather pseudo-Riemannian, wherein the metric need not be positive definite. In this case, we must appeal to Clarke's Embedding Theorem:
Any \(n\) dimensional manifold with \(C^k\) pseudo-Riemannian metric (\(k\geq 3\)) of rank \(r\) and signature \(s\) can be embedded isometrically in \(\mathbb{R}^{p,p+q}\) provided that \(p\geq n-\tfrac{1}{2}(r+s)+1\) and \(q\geq\tfrac{1}{2}n(3n+11)\) for \(M\) compact, \(q\geq\tfrac{1}{6}n(2n^2+37)+\tfrac{5}{2}n^2+1\) for \(M\) non-compact.
For our spacetime, \(s=2\) (3 positive eigenvalues of the metric [space] - one negative eigenvalue [time]) and \(r=n=4\), so that the spacetime of general relativity can be embedded isometrically in \(\mathbb{R}^{2,89}\) (our spacetime is non-compact). In reality, our spacetime is also globally hyperbolic ("stably causal", so that there are no closed time-like loops), so that we can further reduce the embedding to \(f:M\hookrightarrow\mathbb{R}^{1,88}\), that is, one time-like and 87 space-like dimensions.
Now, I'm sure we don't want to be working in 88 dimensional spacetime, and in reality we don't have to: this is merely the number of dimensions necessary to isometrically embed ANY 3+1 dimensional spacetime GLOBALLY, a very strict requirement indeed! Our particular spacetime may be embedded in a smaller space, and if we only concern ourselves with local behavior, the Euclidean space may be quite small. At any rate, 88 dimensions is the lower bound on the size of a global isometric embedding into Euclidean space.