# Direction of space curvature

1. Dec 8, 2004

### Majid

hi

how is space curvature around a mass. i mean what's its direction in real 3D space?

[ i know it is simillar to a rubber sheet with a ball on it, but it is 2 dimentional. i want its real demonstration. just like real world.]

2. Dec 8, 2004

### somy

Last edited: Dec 8, 2004
3. Dec 8, 2004

### pervect

Staff Emeritus
Consider the surface of the Earth. It's a curved 2d surface. What do you consider the direction of curvature of this 2D surface in "real" 2D space?

I don't quite see how this question makes sense - also, your talk about "real" curvature and the "real" direction and the "real" whatever raises a few red flags for me. I'm always tempted to ask the person who asks me about "real" this and "real" that to define reality, but I have a nagging fear that they will actually attempt to do so, in the process spawning a never-ending philosophical debate that will go on and on without any resolution whatsoever for a very very long time. :-(

4. Dec 9, 2004

### Majid

tnx a lot somy.

pervect i used "real" to explain my question and to distinguish 2d demonstrations that we see in books and web from 3d space.
it is difficult for me to demonstrate 3d curvature around a mass like earth or sun.

5. Dec 9, 2004

### hellfire

The notion of curvature in general relativity applies usually to spacetime (a four dimensional 'space'). A direction of curvature is not necessary because it is assumed that spacetime is not included in an embedding space with more than four dimensions. The curvature is therefore an intrinsic curvature within the four dimensional spacetime and it describes how geodesics (shortest distances) in spacetime 'move' with respect to each other.

On the other hand, there are some cases in general relativity in which the curvature of space (three dimensional space instead of four dimensional spacetime) is considered. In such a case, one can speak either about an intrinsic and an extrinsic curvature of space (since space is embedded in the four dimensional spacetime). The first one describes how the geodesics in space move with respecto to each other and the second one describes the rate of change of the tangent vector to the geodesics in space with respect to the normal vector to the spatial 'hypersurface'.

6. Dec 9, 2004

Staff Emeritus
When Riemann considered the curvature of higher dimensional spaces, he found that you have to consider the directions in pairs. There is a number that tells how much (roughly) the x-axis is bent in the y-direction and another that says how much it is bent in the z-direction and so on. In 3-dimensional space there will be nine such numbers and in four dimensional spacetime there will be sixteen. Riemann found they form the components of a special kind of mathematical thing called a tensor, what is now known as the Riemann-Christoffel tensor. It is this mathematical object which characterizes the curvature of a Riemannian manifold.

7. Dec 9, 2004

### pervect

Staff Emeritus
Suppose you were living on a 2D surface, and wanted to see if it was curved, like the Earth, or flat, like a plane.

How might you do this? Well, there are a couple of ways that come to mind. The first is some very accruate distance measurements. When you draw a circle on a plane, or a very small circle on a sphere, you will find that the circumference is pi times the diameter.

But if you draw a very large circle on the Earth, you'll find that the circumference is less than pi times the diameter.

Another way of distinguisihing the plane from the sphere via measurements made on the surface is to look at the sum of the angles of a triangle.

A very large triangle on the surface of the Earth will have the sum of its angles greater than 180 degrees. Consider for instance a triangle with a 90 degree right angle at the north pole. Both sides of the triangle go down to the equaotor along a north- south path. The equator forms the third side of the triangle. All three angles of this triangle are 90 degrees so the sum of the angles of the triiangle is 270 degrees.

This method of defnining and describing curvature is known as "intrinsic" curvature, because it's done entirely by experiments done within the curved space.

You might find a websearch for "intrinsic curvatrure" vs "extrensic curvature" helpful for more background information.

It is possible to visualize intrinsic curvature as a surface imbeded as a higher dimensional space, like the Earth's 2d surface being the surface of a 3d object. However, since we cannot directly measure such dimensons, they should be regarded as "visual aids". It is possible to visualzie curvature as an embedding of a surface (manifold) in another, higher dimension, (another higher dimensonal manifold), but it is not necessary.

According to general relativity, space (as well as space-time) would be curved, so if we had a massive enough object we could in fact do similar experiments in 3 dimensons to the ones i've outlined in 2 dimensions. Unfortunately, the amount of spatial curvature is small, and I don't think that there is any direct experimental confirmation of the GR prediction of this spatial curvature.

The curvature of space-time, though (as opposed to the curvature of space) is easily measured. There's a mathematical object called the Riemann tensor that describes the curvature of space-time, and some (though not all) of its components can be measured by measuring the tidal gradient of gravity at a point. The other components can be measured as well, though not as simply (a couple of different methods exist to do this).

There exists an actual device which can directly measure the tidal gravity at a point - it's called a Forward mass detector. So you can think of it as a curvature meter, but what it measures not the 3 dimensonal spatial curvature, but the curvature of the 4 dimensional space-time.

8. Dec 9, 2004

### pmb_phy

In general the question has no meaning. However it can take on a meaning in special circumstances. Consider, as an example, the gravitational field of the Earth; space curvature (not spacetime curvature) around the earth is radial in this case, i.e. along a line which passes through the center of the Earth. Spacetime curvature, on the otherhand, does not have a direction in general. The rubber sheet analogy is best interpreted as an embedding diagram. Such a diagram illustrates, not spacetime curvature, but spatial curvature.

The term "spatial curvature" refers to altered spatial distance relationships. Thus in the case of the Earth the mass of the Earth slightly alters radial distances, increasing them.

For details please see -- http://www.eftaylor.com/pub/chapter2.pdf page 2-25 I believe.

Pete

9. Dec 9, 2004

### pmb_phy

I've never heard of a device called a Forward mass detector. Also, spacetime curvature can't literally be measured at a single point. One must be in an $\epslon$ neighborhood of the point in question since detection of curvature requires detection of geodesic deviation which requires more than a point ... A bit of a messy nit pick though, sorry. :yuck:

The device I know of which measures spacetime curvature is called a gradiometer. It measures tidal forces, which is just another name for spacetime curvature.

E.g. See page 42 of Gravitation and Inertia, Ciufolini, Wheeler. This part is actually online at
http://www.pupress.princeton.edu/sample_chapters/ciufolini/chapter2.pdf
Pete

ps - The 3rd chapter is also online at
http://www.pupress.princeton.edu/sample_chapters/ciufolini/chapter3.pdf

Last edited: Dec 9, 2004
10. Dec 9, 2004

### Majid

thank you all.