Into what dimension is Spacetime Curved?

twofish-quant

There are several different ways of looking at it. The way that mathematicians look at it "curvature" is this abstract thing with a precise mathematical definition which includes "curvature" on a objects in flat space.

However, this precise definition is much more general than when layman talk about "curvature".

The way that you tell if something is "curved" is that you point in a direction, you walk in a circle and then see what direction you are pointing in. If you are pointing in the same direction then the region of space that you are walking in is "flat". If not, they you are walking in "curved" space.

Now *why* space is curved is another issue. It could be that space is "curved" because you are stuck on a surface that is embedded in a "flat" space. Or to could be that you just happen to be in a region of space in which distances and angles just behave weird.

Imax

If you take a vector and you move that vector around a loop and you end up with that vector pointing in the same direction at the point where you started from, then you’re in a space that is orientable (i.e. Minkowski space). This may not be an experiment that can explain spacetime curvature.

One experiment that can give some insight into space-time curvature is to look at a pulsar as it moves behind our Sun. For a regular pulsar, the interval between pulses should vary with time, if time has a curvature.

alt

That seems to bring it back then, to the original question. Into what dimension is spacetime curved ?

And now that I mention spacetime, I seem to recall a thread here some months ago, from a mentor no less, to the effect that 'Spacetime is a fantasy' (or a fairytale or someting like that). I will try to find it, though I fear I probably won't understand it.

Imax

One way of looking at the Universe is to think of it like a Compact Laurentian Manifold. Within this framework, there is no need for expansion into a higher dimension. The Universe doesn’t need to be static, but it can expand or contract. A Compact Laurentian Manifold can allow closed space like and closed time like curves, which may be what is happening near black holes. The Universe doesn’t need to be simply connected.

alt

I googled Compact Laurentian Manifold but got nothing. So I am none the wiser, though I appreciate your effort.

Imax

Oops, my bad.

It’s “Compact Lorentzian Manifold.”

Nabeshin

The essential point in what Imax is trying to say is the same as I expressed in my earlier post in this thread. Spacetime, i.e. our universe, can be described mathematically as a four-dimensional surface. With only these four dimensions (t,x,y,z), we can also describe fully the phenomenon of 'curvature', which, as has also been expressed in this thread, you can think of simply as angles in triangles not adding to 180 degrees, or the area of circles not being what you might think, etc. The punch line is that curvature is a property INTRINSIC to the surface, meaning you do not need to reference anything outside the surface (another dimension) in order to describe this. So the answer to the question "Into which dimension does spacetime curve?" is that "Spacetime does not curve INTO any dimension, but is itself simply intrinsically possesses the quality we call curvature."