# About Curvature

This may be a simple question for some of you but it has baffled me for a long time.

When we say that spacetime is curved, do we mean that from a flat space of a higher dimension, our spacetime would appear curved, in the same way that the surface of a sphere looks curved when viewed from flat 3-space?

And if so, then what is the stuff of this spacetime? In the case of the sphere, we know that the surface is made of some material which can be curved. But in spacetime, what is it that is curved? Is the curvature only observable through the effects it has on the motion of photons and the like? Or do we really mean that clocks run at different rates in the field and that rods measure dfferent lengths in the field and we call that curvature?

And a related question that is bugging me is: Do we know of any mechanism by which matter/energy/pressure actually causes the curvature - besides the recipe for calculating that using Einstein's field equations? What I am trying to understand is - how does matter and energy interact with spacetime causing it to bend?

I suppose I am getting to the following - is spacetime actually flat but endowed with a gravitational field which effects clocks and rods in such a way as to give us the impression that spacetime is curved?

Thanks a lot,

Jimbob.

## Answers and Replies

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Haelfix
Science Advisor
Google for the mathematical difference between extrinsic and intrinsic curvature. Spacetime is *not* as far as we know embedded in anything.. Instead the curvature is intrinsic, it is what it is. So, differential coordinate vectors of length can receive nonunity contributions thus warping and curving our space relative to what we would expect from a euclidean treatment.

"Do we know of any mechanism by which matter/energy/pressure actually causes the curvature - besides the recipe for calculating that using Einstein's field equations? "

No, we only know the field equations, which give us a macroscopic definition of what happens. I think maybe you're asking about the microphysics, and that is something that is simply not known yet

I suppose I am getting to the following - is spacetime actually flat but endowed with a gravitational field which effects clocks and rods in such a way as to give us the impression that spacetime is curved?
Whether space-time actually is curved cannot be determined for sure. In some ways it's just a name for a mathematical property that gives more than one definition for a straight line. There is a formulation of GR which reduces to a flat 3-D space, old fashioned time, and a field to represent gravity. It isn't GR but allows dynamical calculations to be made with GR metrics. It's called ADM after the people who invented it.

There's another thread called 'a simple question about curvature' you might want to look at.

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jtbell
Mentor
At the present time, asking "Why are Einstein's field equations the way they are?" in the context of general relativity, is like asking "Why are Newton's laws of motion the way they are?" in the context of classical mechanics, or "Why are Maxwell's equations the way they are?" in the context of classical electrodynamics.

Thanks a lot for those answers. I will check out ADM. I was aware we are dealing with intrinsic curvature. I used the flat n>4 space to get to the real question which was about the essence of spacetime itself and how it is effected by matter and energy - although it is also interesting that the 4-velocity, which we would probably consider to be a physical entity, lives in the tangent plane outside the 4-d manifold. I appreciate the frank and honest answer "that is something that is simply not known yet" when it comes to how mass interacts with spacetime (and affects clock rates etc).

If you have a large spherical mass in an otherwise empty space, and you have a test mass in the vicinity, we know that there is a relative acceleration between the two masses.

Einstein tells us through the field equations that the large mass curves the spacetime near it and that the test mass follows a straight line (geodesic) through that curved spacetime.

So the first part of what I am trying to understand is - does he literally mean that there is a *thing* - a physical entity - called spacetime that gets moulded and twisted and curved by energy and mass?

Or does he simply mean that measurements would show that clocks run slower near the larger mass than farther from it and that this time gradient shows up in the metric making it differ from the metric of the flat spacetime of special relativity and so we call it a curved spacetime?

A follow on question: Is this difference in clock rates at different heights somehow causing the relative acceleration?

Basically I'd genuinely like to better understand why an apple falls to the ground! I am fascinateld by it! I've understood so far that it is curved time in the weak field near the earth that is the cause, and I can picture the apple happily following it's geodesic - at each moment sitting in a local lorenz frame - and then suddenly getting hit by the earth's surface.

And that it is really the earth's surface which is actually doing the accelerating, not the apple, right? Or at least the earth's surface is resisting its own motion along the geodesics - and so it is in some sense accelerating, right?

I'd like to better understand if (and how) this time gradient actually causes the relative acceleration between the earth's surface and the apple.

Does anyone know of any good visualizations of the curvature of time near the earth?

Thanks a lot.

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So the first part of what I am trying to understand is - does he literally mean that there is a *thing* - a physical entity - called spacetime that gets moulded and twisted and curved by energy and mass?
In my opinion Einstein did not believe that space-time has substance. That would take us back to aether. GR has no dynamical content, so we can't say that the difference in clock speeds 'causes' the accelerations. There are no forces in GR. ADM add dynamical content and have definitions of energy etc.

This is a big topic, but I have to leave it there right now.

Whether space-time actually is curved cannot be determined for sure. In some ways it's just a name for a mathematical property that gives more than one definition for a straight line. There is a formulation of GR which reduces to a flat 3-D space, old fashioned time, and a field to represent gravity. It isn't GR but allows dynamical calculations to be made with GR metrics. It's called ADM after the people who invented it.

There's another thread called 'a simple question about curvature' you might want to look at.
No offence, but the highlighted parts are, frankly, garbage. The first part shows that you don't understand the experimental support for GR, and the second part (about the ADM formalism) is completely incorrect also.

No offence taken. I do believe in the experimental support for GR and I don't know how you can deduce from what I've written that I don't.

Can you tell me about any experiment that actually measures space-time curvature, as opposed to its effect ?

I yield on ADM, not having studied it in detail.

vanesch
Staff Emeritus
Science Advisor
Gold Member
Spacetime is *not* as far as we know embedded in anything.. Instead the curvature is intrinsic, it is what it is.
There is a theorem about n-dimensional riemannian manifolds that can always (if desired) be embedded in a 3n-dimensional Euclidean space if I'm not mistaking. Now, I don't know if this also applies to a pseudo-riemanian manifold (and normally, this is not the way people look upon GR solutions).

As an aside, I read Dirac's book on GR and he used a flat n-space (n>4) when explaining parallel transport. I liked that approach.

It is repeated here and might be of interest to some of you: http://en.wikipedia.org/wiki/Introduction_to_mathematics_of_general_relativity#Parallel_transport

I've also just read elsewhere in these forums that, at least in 1920, Einstein himself was arguing that spacetime was a physical entity, capable of elastic deformation.

http://www.tu-harburg.de/rzt/rzt/it/Ether.html [Broken]

"Recapitulating, we may say that according to the general theory of relativity space is endowed with physical qualities; in this sense, therefore, there exists an ether. According to the general theory of relativity space without ether is unthinkable; "

I had read elsewhere however that he in later life moved away from these ideas. Can anyone tell me what the consensus is today among physicists?

Is spacetime a *something* capable of being bent by matter and energy?

Thanks.

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There is a theorem about n-dimensional riemannian manifolds that can always (if desired) be embedded in a 3n-dimensional Euclidean space if I'm not mistaking. Now, I don't know if this also applies to a pseudo-riemanian manifold (and normally, this is not the way people look upon GR solutions).
John Baez, wrote a few years ago on the Sci.Physics.Research forum:
Baez said:
First of all, you're being a bit optimistic if you think any Lorentzian 4-manifold can be isometrically embedded in 10d Minkowski spacetime. Any Riemannian 4-manifold can be isometrically embedded in 10d Euclidean spacetime, but you need a lot more extra dimensions to handle the Lorentzian case.

The last time we talked about this, Robert Low mentioned this paper:

Clarke, C. J. S., "On the global isometric embedding of pseudo-Riemannian manifolds," Proc. Roy. Soc. A314 (1970) 417-428

which apparently proves that any Lorentzian 4-manifold can be embedded
in 91-dimensional flat spacetime with two time dimensions.
But I am not sure if the 91 dimensional space remains Lorentzian or is now supposed to be Euclidean. And two time dimensions, how interesting!

I would not be surprised it is impossible to embed any Lorentzian manifold in some higher and not infinite dimensional Euclidean manifold. Any takers who actually know?

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I recall that Kip Thorne also briefly mentioned this (91-dimensional) complexity while lecturing on GR - http://elmer.tapir.caltech.edu/ph237/CourseOutlineA.html - see lectures 4 and 5.

Its also interesting that none of that complexity arises in Dirac's work to a synopsis of which I provided a link above. He sails through it beautifully in a few lines. Dirac simply asserts "Physical spacetime forms a four dimensional "surface" in the flat N-dimensional space. " and continues from there. Are his assumptions incorrect?

(And yes two time dimensions are indeed interesting - three would be even funkier! Nice symmetry with the 3 spacial dimensions... does anyone know if the physical consequences of a 3+3 space-time have been studied?)