Added emphasis, pedantic caveats, and citations
Hi all,
As others have already pointed out, the key thing to remember about embeddings is that they rather analogous to embedding a curve in euclidean space; given the intrinsic geometry of a (semi)-Riemannian manifold,
there are always infinitely ways to do it in any space of sufficiently large dimensions; studying these is a legitimate topic in differential geometry, but concerns
extrinsic geometry in the sense of Gauss.
In the physical interpretation of Lorentzian manifolds and spacetime models used in gtr (and modulo some caveats, in brane theory and so on), only the
intrinsic geometry is physically signicant.
Put in other words, embeddings should be considered more or less artificial representations of spacetime models. It turns out that they are not often as useful as you might expect from surface theory (the theory of Riemannian two-manifolds immersed in three-dimensional euclidean spaces, which was introduced by Gauss and which led to Riemannian geometry). In fact they can be rather misleading, particularly if you are not familiar with semi-Riemannian geometry in large dimensions or lack the ability to visualize four-surfaces immersed in (generally) at least six dimensions.
A word about semi-Riemannian geometry: the signature is geometrically highly significant; it turns out to be a generally bad idea (at least for beginners) to use imaginary coordinates. The nature of the geometry and isometry groups of D-dimensional flat spacetimes with distinct signature is completely different; in particular for D=4, signature 4,2,0 give quite different geometries. Semi-Riemannian manifolds are of course locally similar to these flat geometries, by definition (roughly, the metric involves "quadratic corrections" in an infinitesimal neighorhood of each point in such a manifold; these are of course encoded in the metric tensor, which is second rank).
The original poster might want to ignore the following pedantic caveats, which will probably be useful only for advanced students:
Daverz said:
You can treat spacetime as if it's embedded in a higher dimensional space. See for example Dirac's treatment of parallel propagation in his little GR book. There's a theorem of differential geometry that says that any Riemannian manifold can be embedded in \mathbf{R}^n for some n, and I assume this works for Lorentz manifolds as well.
Note that there is a crucial distinction between
local embeddings and
global embeddings. There are indeed embedding theorems for Lorentzian-four manifolds giving upper bounds for the minimal number of dimensions which suffices to embedd any given Lorentzian four-manifold. Global embeddings are much trickier and the minimal dimension is much larger in this case (I also forget the exact number, but IIRC something like ninety dimensions are needed for global embeddings of some Lorentzian four-manifolds!).
Another crucial ingredient involves "degrees of smoothness". As for the Riemannian case, it is easier to obtain embedding theorems for
analytic manifolds, but for a local field theory which is a metric theory of gravitation, we need smooth manifolds, a much more general notion.
Some easy examples:
1. The FRW models with S^3 hyperslices orthogonal to the world lines of the dust can be globally embedded in {\bold E}^{(1,4)}.
2. The Schwarzschild vacuum can be locally embedded in {\bold E}^{(1,5)} (in fact the entire exterior region can be so embedded). If you are interested in visualizing the two dimensional quotient manifold from the Carter-Penrose diagram depicting the maximal extension, this can be embedded in {\bold E}^{(1,2)}; see for example
http://www.arxiv.org/abs/gr-qc/0305102 and
http://www.arxiv.org/abs/gr-qc/9806123.
See Exact Solutions of Einstein's Field Equations by Stephani et al. for the "embedding class" of an exact solution and a list of all solutions with low embedding class.
An interesting variant on embedding Lorentzian manifolds in higher dimensional
flat manifolds is the notion of embedding them in higher dimensional
Ricci flat manifolds. (In gtr, of course, any vacuum solution is already Ricci flat.) Here, the Campbell-Magaard theorem states that any
analytic Lorentzian four-manifold can be
locally and analytically embedded in some five-dimensional Ricci flat semi-Riemannian manifold. For example,
http://www.arxiv.org/abs/gr-qc/0503122 claims to do this in the case of the Goedel lambdadust.
There are also interesting but more abstract notions of embedding; see for example
http://www.arxiv.org/abs/gr-qc/0606045
George Jones said:
Or the other way 'round!
For example: Alcubierre; Morris and Thorne.
I am pretty sure George is thinking of the following fundamental observation: start with any Lorentzian four-manifold, compute the Einstein tensor (a purely mathematical operation), divide by 8 \pi, and call the result "the stress-energy tensor of a spacetime model". If this were always a legitimate procedure, the EFE would be trivial! Needless to say, Einstein had in mind something more reasonable:
1. Gravitation has a universal character: the mass-energy associated with all forms of matter and also the field energy of physical fields such as the electromagnetic fields (and even of the gravitational field itself!) gravitate. The natural desire to capture this unique characteristic of gravitation in an elegant way was in fact one of the principle notions for introducing what we now call metric theories of gravitation, of which gtr is in various mathematical senses the most elegant.
2. This universality implies that a useful general theory of gravitation should accept theories of matter and of nongravitational fields as "input". It should include a procedure for obtaining the stress-energy contribution from each nongravitational field (fortunately this is straightforward in the case of theories which admit a Lagrangian formulation, courtesy of results of Noether). Then, the gravitation theory should admit an well-posed initial value formulation which allows one (in principle) to determine a unique gravitational field, a unique matter distribution, and unique nongravitational physical fields given suitable initial data. That is, the initial data must satisfy certain "constraint equations", and it describes the intrinstic geometry of a hyperslice and initial field values and matter distribution on that slice. Then "evolution equations" tell us how to evolve the geometry, matter distribution, and nongravitational fields "over time". (This gets a bit tricky, since we have considerable freedom to use various hyperslicings into "spaces at a time".) Finally, the result should vary continuously under small variations of "legal" initial data. I hasten to add that gtr admits many such formulations; Brans-Dicke theory and even Nordstrom's theory are also well formulated in this sense. See Wald, General Relativity for details.
3. A classical observation is that "energy" has a universal character: all matter and all physical fields possesses some energy, and the energy invested in one field can generally be transferred to another, subject to conservation of energy. In the theory of matter, we also wish to treat interconversion between heat and "useful" forms of energy, using some theory which does not depend upon choosing any detailed theory of matter. This theory is of course classical thermodynamics. This observation suggests a vague analogy between general theories of gravitation and thermodynamics. In fact, this turns out to be very deep analogy! This is particular true for gtr: the EFE can be obtained from thermodynamic principles! See
http://www.arxiv.org/abs/gr-qc/9504004 and
http://www.arxiv.org/abs/gr-qc/0612078
4. To try to rule out from the "eigenthings" of the Einstein tensor all but "well-understood" contributions to the stress-energy tensor of a Lorentzian spacetime under consideration possible spacetime model, physicists long ago introduced mathematical "eigenthing" conditions called "energy conditions". The implicit goal was to rule out everything objectionable while keeping everything legitimate, but it has turned out that energy conditions are probably not very well suited for this task after all. Nonetheless they remain in common use because of their simplicity.
5. Returning to "Lorentzian wormholes", "warp drives" and other "exotic" Lorentzian manifolds: these would usually not be considered "exact solutions" (despite bad pop science sources which call them that) because their putative stress-energy tensors do not arise from combinations from well-understood theories of matter or nongravitational fields (such as a charged dust plus an electromagnetic field possibly including EM radiation from "outside"). For one thing, they all violate most of the most reasonable energy conditions, and their stress energy tensors are very unlike anything which can arise from ordinary matter plus electromagnetism. On the other hand, it is true that corrections from quantum field theory in general lead to "effective field theories", which allow us to "fake it", to a limited extent, in the context of classical field theories. Then the Casimir effect, for example, does lead to a notion of (locally) "negative energy", so this is an example in which an experimentally well-established situation violates various energy conditions. Nonetheless, there is a large literature exploring in detail various reasons why speculative models of "warp drives" and "Lorentzian wormholes" remain highly suspect on theoretical grounds. These arguments currently tend to throw cold water on attempts to interpret various observational mysteries in cosmology in terms of some these speculative proposals.
