I answered a similar question on Quora recently, as can be read here, and the solution to the problem which is there stated (also asked on the mathematical cousin of this site), namely: if it's possible to employ one of the embedding theorems to embed a symplectic structure \((\mathcal M, \Omega)\). The answer is positive: assuming class \( C^{\omega} \), simply use the original Whitney formulation to find an embedding \(i:\mathcal M \longrightarrow \mathbb{R}^n\) and let \(\hat{\Omega} \in \bigwedge^2 \mathbb{R}^n \) be such that its pull-back obeys \(i^{*}(\hat{\Omega})=\Omega \). Since by assumption our structures are \(C^{\omega}\) , analytical continuation of the components of \( \hat{\Omega} | _{i(\mathcal{M})}\) completely fixes the 2-form \(\hat{\Omega}\) on the entire ambient space \( \mathbb{R}^n\).
Observe that in the above argument, no particular use is made of a metric, so we can use Whitney theorem that only requires that the manifold be differentiable, and moreover, \( n = 2 \mathrm{dim} (\mathcal{M})\) . The state of affairs becomes more complicated as you give more structure to the manifold which you wish to find an embedding. For instance, the famous theorem of Nash comes into play when your manifold \(\mathcal{M}\) is given a Riemannian metric, and you want to embed it into an Euclidean space.
When one thinks about embedding theorems in theoretical physics, GR provides the natural setting, since the manifolds of interest are non-compact pseudo-Riemannian spaces (carrying Lorentzian metrics), and one requires a generalization of Nash theorem. This work was performed by a relativity expert, C. S. Clarke, and can be found on his paper "On the Global Isometric Embedding of Pseudo-Riemannian Manifolds."
An instance where an embedding was used in GR was one of the paths that eventually lead to the analytical maximal extension of the Schwarzschild metric, the Kruskal-Szekeres spacetime. The first attempt to embed the Schwarzschild metric started already with Kasner in 1921, where he found an embedding in the 6-d pseudo-Euclidean space \(\mathbb{R}^{2,4}\), but his embedding suffered from a topological defect (he used trigonometric functions to parametrize the temporal coordinate, identifying the time-axis with \(S ^1\), creating a naive "time-machine" spacetime!). His mistake was later corrected by Fronsdal only in 1959 (Fronsdal replaced the trigonometric mappings by hyperbolic ones), and embedded Schwarzschild in the 6-d Minkowskian spacetime \(\mathcal{S}\subset\mathbb{R}^{1,5}\). With the embedding at hand, Fronsdal easily showed the geodesic completeness of the hypersurface \( \mathcal{S} \), finally proving that the "singularity" at the Horizon \(r=2M\) was a coordinate defect of Schwarzschild's coordinates, completing the construction of the simplest black hole spacetime in GR. (For a discussion of these matters, see my review 1403.2371, sec. 4.2).
The method above delineated to find the maximal analytical continuation of the Schwarzschild spacetime is not the simplest one to accomplish such a task. In fact, textbook derivations relies in the use of a (singular) coordinate transformation to arrive at some chart covering the entire manifold of the Schwarzschild spacetime, e.g., Eddington-Finkelstein, Painlevé-Gullstrand, Kruskal etc. However, I think that at least in principle, the method outlined here, using Clarke's embedding theorem, could be used to systematically produce analytic continuations for those Lorentzian metrics whose components are degenerate in some regions, but whose curvature invariant are nevertheless well-behaved. For instance, I would be curious to know if one can study the ergosphere living inside the outer horizon of the Kerr spacetime using an embedding like this.
Q&A (4871)
