Answerer

Hi guys, those of you who are familiar with general relativity will understand better about the question I'm going to ask.
As you all know, in order for the curvature of a two-dimensional Euclidean geometry(lets say a circle) to take place, a three-dimensional Euclidean space must be needed.
Therefore, since we are living in a four-dimensional Euclidean spacetime and spacetime curvature often take place, does this mean that s in our universe,itself ,there exists a fifth dimensional Euclidean spacetime and that we are actually living in a five dimensional rather four dimensional spacetime?

Aquila ka Hecate

Hang on, I though we were in a non Euclidean spactime?

The curvature can't be examined int erms of Euclidean geometry, surely?

Steven S

In differential and reimannian geometry there are two different types of curvature to be aware of, intrinsic and extrinsic curvature. The common way to illustrate the difference between these two forms of curvature is to take an idealized piece of paper with zero thickness. So you start with a two dimensional surface. Now take this sheet of paper and roll it into a cylinder. One thing you noticed is that distances between points are preserved since you have only 'bent' and not streched the sheet of paper. In this case since we haven't stretched our manifold (our two dimensional sheet) mathematically we say the intrinsic curvature is zero for a cylinder. But if we, as beings living in a three dimensional world, we see the sheet is bent into a cylinder and say it has curvature. This curvature is the extrinsic curvature. Extrinsic curvature depends upon how a surface or manifold is embedded into a higher dimensional manifold. In general relavity all measures of curvature are intrinsic. Intrinsic curvature doesn't make any reference to spacetime being embedded in a higher dimensional space. So as far as GR is concerned it doesn't make sense to say our universe is embedded into a higher dimensional space. I hope I've been somewhat clear.

Steven S

Undercurrent

It goes something like this. Consider a sphere with a two dimensional surface (which is curved).

1) We, being outside the sphere, see that the surface is curved because we can look at it and say "hey, that's curved."

2) What is more subtle is that a creature confined to to the surface of the sphere can also determine that the surface is curved without ever leaving the surface. For example, by accurately measuring the ratio of circumference to diameter of a circle, and finding it to be a bit less (or a lot less, for really big circles relative to the size of the sphere) than pi.

We naturally can't say whether or not space-time is curved in the first sense, because we can't leave the universe to find out. The general theory of reletivity says that space-time is curved in the second sense, and that the perception of gravity depends on some of those little, subtle things our sphere-bound person would measure to determine that his universe was bent.

In other words, GR says that mass affects space-time in such a way that it seems to someone confined to the space-time that the space-time is curved. Whether or not this corresponds to an "actual" bending of space-time that might be seen by some 5-dimensional observer outside the universe (as opposed to mass just buggering the equations in the same way as such a bending would) is unknown, and probably unknowable.

m.

Answerer

Well guys, thanks for your help, I have understood GR and curvature better. However, my question still remain(sorry) but I will be clearer this time.
In order for a circle(a two dimensional euclidean geometry) to curve inwards or downwards, a three euclidean space must exist, otherwise it is impossible for the circle to be able to curve inwards. So, since we are experiencing curvature of spacetime( a four D non-Euclidean spacetime), does this means that a fifth dimensional space must exist in order for our spacetime to undergo curvature?
Well, maybe I'm having some misconception problems.

Abacus

Actually, you should think of the circle as a line (one dimensional) curved through two dimensions. However, if you were a one dimensional being living on a line, you would never be able to tell if the line was curved into a circle, oval, square, or a triangle. This is extrinsic curvature. There is no intrinsic curvature. Using the circle as an analogy to our 4-D universe is inappropriate and misleading.

The gist of non-euclidean geometry and, as a result, general relativity, is that curvature (of the intrinsic kind) can be mathematically defined without appealing to any higher dimensions. If you can define this curvature (and thus explain gravity) using only the four dimensions, then why postulate a 5th dimension?

Starspun

We 'curve' in 4 dimensions everyday. (3 spatial, 1 time). To warp space-time, according to GR, only 4 dimensions are needed. Now, when you move past this to, say string theory, you will get different answers (up to 11 dimensions). But, according to General Relativity, 3 spatial and 1 time dimension is all that is needed.

Steven S

You're problem is that you want to say something is curved ( a sphere) by comparing it with something else (R^3--euclidean space). Don't do this--as I explained in my earlier post there are two different curvatures and want to think about an objects extrinsic curvature. In GR we measure curvature like little two dimensional people living on the sphere's surface. We notice our world is curved by making triangles and noticing they do not add up to 180 degrees (or maybe finding the Reimman curvature if our 2d people were really clever). The philosophers and mathematicians of our 2d people can certainly conjecture that they live in a 2d sphere embedded in R^3 but that's all it is, a conjecture. The 2d people can't get out of their universe (assuming there is anythplace to get to)and say, "Oh, wow, we live on a sphere" like we can't get out of our universe.

Steven S

Undercurrent

I think the point is a rather philsophical one. All we can "see" is variations in some of the mathematical properties of space. These variations are mathematically the same variations that occur when a 4-d manifold is curved (e.g. by some guy sitting in a 5-d Euclidian space poking the manifold with a stick.) Since we can't leave our manifold, we can't actually see if we are "really" on a manifold being poked by sticks, or whether the variations in the properties of space just look like that.

A similar (philisophically) example occurs in quantum mechanics. In QM, every particle can be described by a wavefunction which is complex-valued (i.e. it is a function from every point in space onto a complex number.) What has been found experimentally is that the absolute phase of the complex function doesn't matter. That is, you could multiply every wave function at every point by a unit complex number exp(i*theta) and you'd still have the same universe in that all measureable properties of the universe would be the same (observable properties tend to depend on the squared modulus of transformations of this function which are invariant under changes of phase.)

The point of the matter is, our models of the universe have parameters (absolute phase in QM, extrinsic curvature in GR) that are undetectable in the real universe. Given this, you have to face one of two conclusions:

• The universe really has these properties and conspires to hide them from us.
• Our mathematical models of the universe are in a sense richer than the actual universe is. (e.g. absolute phase doesn't matter to the universe because absolute phase is not a property of the universe, only of the model.)

Getting back GR, I believe the thought is something like: We model the universe as a manifold with an intrinsic curvature as a result of extrinsic curvature, but that doesn't mean that the universe is a manifold with an intrinsic curvature as a result of an extrinsic curvature.

m.

Answerer

Okay guys, I got some ideas already, thanks. Anyway, do anyone of you have links to a good explanation of intrinsic and extrinsic curvature?