# How is spacetime curved?

according to GR spacetime is curved coz of the matter and energy present in the universe... many of the books and tv programmes shows this using a rubber sheet being curved by a heavy ball.. the rubber sheet is 2 dimensional and it is being curved into the third dimension.. but spacetime cannot be curved into a fifth dimension.. GR deosnt claim that there is a fifth dimesion does it..?
i read in einsteins book that in curved spacetime the pythogoras theorem is no longer ds^2=du^2+dv^2 but it becomes ds^2=g11du^2+g12dudv+g21dudv+g22dv^2... but how is this possible if spacetime is not curved into a fifth dimension.?

Hurkyl
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The word "curved" is an analogy taken from the case where we really are studying surfaces embedded in Euclidean 3-space. Roughly speaking, the actual meaning of the word simply means "different from 'Euclidean' space". (or, in the more general context needed for relativity, the appropriate generalization of 'Euclidean')

Consider maps. If you have a map of the Earth, you can compute the length of some path on the map... but you can't do it with ds^2=du^2+dv^2. You have to use some better metric. Thus, the surface described by the map is curved. But the map is drawn on a sheet of paper -- a plane. It isn't living in a three-dimensional space.

In fact, if I remember my history right, manifolds were invented precisely for this reason; so that we can study a surface intrinsically as a 2-dimensional shape... so that we don't picture it as living in 3-dimensional space.

Interestingly, the Whitney embedding theorem roughly says that there is no difference between the concepts of

(1) A (possibly curved) n-dimensional space

and

(2) A (possibly curved) n-dimensional space living in 2n-dimensional Euclidean space.

(A similar theorem holds for spacetimes, but I don't recall the exact details)

(edit: whitney just gets the smooth embedding. See post #5 for a reference to the Nash embedding theorem, which gets the metric right too)

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robphy
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The word "curved" is an analogy taken from the case where we really are studying surfaces embedded in Euclidean 3-space. Roughly speaking, the actual meaning of the word simply means "different from Minkowski space". (Euclidean space is a special kind of Minkowski space)

It might be more accurate (though possibly more complicated) to say "different from an affine space" [so it can be used outside of the spacetime context].

By "(Euclidean space is a special kind of Minkowski space)", I assume that you are implicitly making reference to the metric signature. Though more complicated, it is probably better to suggest the generalized signature possibilities by using a prefix like pseudo- or semi- preceding Euclidean [or Riemannian]. Minkowski strictly refers to a signature of the form (1,n-1). In my opinion, from a physical point of view, I think that quoted parenthetical statement is subject to physical misinterpretation.

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Hurkyl
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Hrm. I've made a rewrite, which will hopefully please all. I don't like "affine", because affine space doesn't come equipped with a notion of metric: just a notion of "line" and (I think) "cross ratio".

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. Of course, this is just a mathematical technique.

http://en.wikipedia.org/wiki/Nash_embedding_theorem

Wallace
Of course, this is just a mathematical technique.

But it's all just mathematical techniques. What (if anything) do we gain by deciding which mathematically equivalent way of thinking about GR is the, in some sense, correct way? That's a question by the way, not a statement, I'm not saying there is no benefit.

What matters is that we get the correct (and identical) prediction regardless of how you choose to think about it. I guess some representations might be better at guiding your intuition than others?

The real reason not to choose the embedding is that we have no evidence for them at the moment. I believe any general Lorentzian 4D manifold can be embedded in an 87D flat spacetime, but do not quote me on that as I do not have the reference. This is, of course much more complicated than the equally adequate 4D description we currently use.

Ockham and his razor have the final say.

But it's all just mathematical techniques. What (if anything) do we gain by deciding which mathematically equivalent way of thinking about GR is the, in some sense, correct way?

Well, you always want to make the fewest assumptions in a physical theory. It's my understanding that GR can be formulated as a field theory on a flat background spacetime, but that the background "transforms away". On the other hand, the geometric view can be formulated completely intrinsically without any need for a background space or higher dimensional spaces.

From my point of view, if it makes it makes the math easier to understand or more intuitive, viewing spacetime as an embedded manifold is a valuable technique because of our intuition of embedded manifolds (specifically surfaces embedded in $\mathbf{R}^3$). See http://people.hofstra.edu/faculty/Stefan_Waner/diff_geom/Sec8.html [Broken] for an example.

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As per my info , we study the curvature of the spacetime using the Einstein Field equations , in which we insput the stress environment and get the resultant metric tensor.

daniel_i_l
Gold Member
To imagine a "curved" piece of paper in 3D imagine a piece of graph paper with curved lines - even though it looks as if the distance between the lines is different in different places really the distance between the lines is constant, so if points A and B are both on the same curved line the shortest path from A to B is along the line, not the "straight" line connecting them.
And a word of advice: don't read to carefully into the rubber sheet analogy, besides the point you mentioned it has lots of other problems that get in the way of understanding GR. the thing that has always bothered me the most with that analogy was the fact that what is actually making the ball roll down the rubber sheet is gravity! without gravity the ball would just stay were it was, regardless of how you curve the sheet (as long as you don't push the sheet into the ball) so how can you use it to explain gravity?
I prefer the analogy were two people start at opposite ends of the equator and walk straight up, even through both of them have walked straight they meet at the north pole so it as if some "force" has pulled them together.

the thing that has always bothered me the most with that analogy was the fact that what is actually making the ball roll down the rubber sheet is gravity!

You don't need gravity, the analogy still works if you imagine objects constrained to the surface of the sheet but able to move without friction.

The real reason that it's not the greatest analogy for GR is that the $g_{00}$ component of the metric usually dominates at the low mass/low velocity limit (Newtonian gravity). Even for the bending of light near the sun, the spatial curvature only accounts for half of the bending. Still, it's a pretty good analogy if you don't take it too far.

daniel_i_l
Gold Member
You don't need gravity, the analogy still works if you imagine objects constrained to the surface of the sheet but able to move without friction.
Lets say that i just place a ball on the curved sheet without gravity - it will just sit there when really it should be rolling to the middle. This is a resualt of the fact that the analogy fails to show how the passage through time causes the ball to acually roll towards the center. In the equator analogy this is easily shown if you look at the "north" direction as time and the longitude as space - then even if two bodies are placed on opposite sides of the equator since they're constantly moving "north" through time they will eventually meet at a pole.

Lets say that i just place a ball on the curved sheet without gravity - it will just sit there when really it should be rolling to the middle. This is a resualt of the fact that the analogy fails to show how the passage through time causes the ball to acually roll towards the center. In the equator analogy this is easily shown if you look at the "north" direction as time and the longitude as space - then even if two bodies are placed on opposite sides of the equator since they're constantly moving "north" through time they will eventually meet at a pole.

Yes, excellent point, I was being dense. You don't get any dynamics if it's just static spatial curvature. So let the whole rubber sheet accelerate through a 3rd dimension at a constant rate.

pervect
Staff Emeritus
I prefer the analogy were two people start at opposite ends of the equator and walk straight up, even through both of them have walked straight they meet at the north pole so it as if some "force" has pulled them together.

I also prefer this analogy. The point is that it is space-time that is curved, not space.

If one imagines drawing a 2d space-time diagram on the surface of a sphere, one gets exactly the results you describe.

The analogy isn't perfect because the 2d surface of a sphere has a finite area, one really wants to imagine an infinite plane that everywhere has an intrinsic curvature like a sphere. Unfortunately, I don't think this object can be embeded into 3d space - so we have to make do with imaging drawing our space-time diagrams on spheres.

An interesting alternative is to limit the spatial part of the space-time diagram, and then "wrap" the resulting finite-width infinte-length strip around the sphere. This way at least the time dimension is infinite.

George Jones
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As per my info , we study the curvature of the spacetime using the Einstein Field equations , in which we insput the stress environment and get the resultant metric tensor.

Or the other way 'round!

For example: Alcubierre; Morris and Thorne.

robphy
Homework Helper
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..or both simultaneously... [since the stress tensor might already require the metric].

Chris Hillman
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:

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

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 possess some energy, and the energy invested in one field can generally be transfered 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.

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but drawing a 2D surface on a sphere means there is a third dimensiion... if its similar to tat then that means there is a fifth dimension for spacetime to curve into..
or does curved space time mean that it is just different from the ordinary euclidean space time?? is it not actually curved liked the rubber sheet analogy...

so does this all mean that spacetime is curved into a higher dimension but since we cannot imagine it we r using tools to manipulate it in lower dimensions... like the surface of earth is brought onto a map..

pervect
Staff Emeritus
so does this all mean that spacetime is curved into a higher dimension but since we cannot imagine it we r using tools to manipulate it in lower dimensions... like the surface of earth is brought onto a map..

Merry christmas!

What it means is that the existence of such other dimensions in General Relativity is not testable by experiment. Therfore the existence or non-existence of such dimensions is not really a scientific question.

To go a little further, most scientists don't like to incorporate non-testable concepts into their theories, i.e. no invisible 10 feet rabbits, no "Harveys", though technically there isn't anything stopping anyone from doing this if they really want to or if it's convenient. In GR it's a lot more conveinent NOT to think about the existence or non-existence of multiple dimensions, and to concentrate on what we can actually observe directly, which is "intrinsic" curvature.

wat do u mean by "intrinsic" curvature?.. sorry i am new to the theory..

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how is it possible for a surface to curve without a higher dimension?

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 statement I marked blue confuses me.
What do you mean "locally", what distinguishes those Semi-Riemannian manifolds from the above mentioned geometries globally?

pervect
Staff Emeritus