Majid said:
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.
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.
