Curvature in Spacetime

[quixata]

I have read that spacetime has curvature in it.This is due to the presence of masses or whatever.
I can understand when one talks about the curvature of a surface. I can follow when they talk about a rubbersheet with masses scattered on it.
To be more specific, I can understand the curvature of a body at its surface.
But to be frank, the term 'curvature in spacetime' is not totally understood by me. Is it the different density of matter in space in one region to another?
Someone please explain in some more detail.
Q

[jinydu]

Admittedly, it's not really possible to fully answer your question without knowing the mathematics of General Relativity, something I haven't studied in detail yet. Here's what I do know:

In relativity, we regard the entire universe as being describable using something we call spacetime. An event, that is, a point in spacetime where something happens (in a particular frame of reference) is described using four coordinates, three for spatial position and one for time. The collection of all events (that we can observe) is the entire history of the Universe.

We define something called a distance between two points in spacetime. In Special Relativity, Einstein derived a formula for this distance; if I remember correctly, it was something like:

D = sqrt((x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2 - (c^2)(t2-t1)^2))
where (x1, y1, z1, t1) and (x2, y2, z2, t2) are the spacetime of the two events.

(I may have misremembered the precise formula here, but I'm quite sure it's a simple algebraic formula like that).

However, this formula is valid only in Special Relativity, when there is no gravitational field and spacetime is "flat".

In General Relativity though, it is found that the prescence of a gravitational field causes this distance formula to change (admittedly, I don't know precisely what it changes into, but it surely depends on the specifics of the particular gravitational field). It is in this sense that spacetime is "curved" by gravity.

[wendy]

It is useful to understand what the rubber sheet model is, and is not.

Firstly, it is pure space: it is a billiard-table, with some nasty pockets in it. One shoots balls in real time, the path of the balls represent how things would move under an initial speed and some gravity.

Because gravity is represented by a stretching of space, it is not dependent on the exchange of particles (gravitons) between two masses, but rather one mass stretching space, and the other experiencing it.

Whatever the billiard-table is warped in, it is not time.

Secondly, curvature of space is considerably nastier concept than the billard ball would suggest. All space is curved, even Euclidean space.

To understand how this magic works, imagine a context where the circle has a radius of 57.3 mms. In euclidean space, the circumference yields 1 mm per degree. In non-euclidean geometries like the spheric (ie the one that the surface of the earth represents), it is less, eg 0.999 mms./degree.

The kind of geometries that one studies have a fixed circumference for a circle of radius 57.3 mms. It it is longer than 360 mms, then it's negative curvature, or hyperbolic, and if it's shorter than 360 mms, it's positive curvature, or spheric.

In the curved space model, the circle around a point does not have an equal conversion into degrees, but some degrees are longer than others. For example, near a large mass, the degree might be 1.001 mms., while opposite it might be 0.999 mms.

Space is then in tension, with equal tugging per mm. of perimeter. This means that the degrees with the larger number of mms. are going to win. The force of this tension, then, is gravity.

For the model of gravity, the nature of space is that we have regions of negative curvature. These then to be like looking into a horn-shape opening, rather like what one makes when one sits on a matress, or what one has when a large mass sits in a rubber sheet.

W

[bsaucer]

People tend to think of "curvature" as though the surface (or hypersurface) were "bending" into a higher dimension. This is called "extrinsic" curvature. General Relativity deals with "intrinsic" curvature. This does not require an extra dimension.

Imagine you exist at some point (event) in 4D space-time. You have in your posession a ruler (for measuring distance) and a compass (for measuring direction, NOT for drawing circles!). The ruler contains a clock for measuring time, as well, but we'll treat it like it's another spatial dimension.

You pick a set of four perpendicular lines (axes) meeting at your location. You measure out a unit of distance along all four axes. This distance is the same in all four eight directions (two directions on each axis).

Now move yourself along a path to another point, making sure to keep the compass pointing in the same direction as you go. When you get there, measure the coordinate lengths again.

If the spacetime is flat, the measurements of distance and direction is still the same. If spacetime is not flat, then you might notice that the x axis has been "squeezed" by some amount, and the y axis has been "stretched" by some other amount. The z axis may be squeezed or stetched, as well. Time may be "flowing" faster or slower than it was at the other point.

You can recalibrate your ruler to make the new coordinates "orthonormal" again, but if you go back to the original point, they're no longer "ortho-normal".

If you move from point A along one path, and then return to point B along another path, you might find your original coordinates rotated and/or "boosted" (lorentz rotation).

Is it meaningful for time to speed up or slow down over the course of time?

You to go to any place in spacetime and measure the speed of light when you get there. It's always a constant. However, if you could measure the speed of light at a remote location or time using your local coordinates (without actually going there), you could get a different answer...

[darthbadass]

I find it helpful to think of curved spacetime as the 2D surface of an expanding 3D sphere (the universe), only in relality the 2D surface is 3D and the sphere is actually a hypersphere. So you see how 3D space curves around 4D. After all, if one of the the presently accepted models is true and the universe is like an expanding sphere and we live on its outer surface (as described by Stephen Hawking in "The Universe in a Nutshell", great book by the way), then because we live in 3D, the Universe would have to be 4D.