1. Dec 23, 2007

### Helios

Yes, general relativity is out of my depths for now. Now I've often seen drawings of a gravitational source represented by a dimple ( downward ) on a surface. Yet GR never speaks of a fifth dimension. Nor have I ever seen a dimple upwards that I would suppose would represent repulsive gravity which I've never heard to exist. Is gravity more like an optical distortion. A gradient in the index of refraction will curve light and matter too. Surely someone has formulated this, given the "gravitational lens" effect. Is curvature misleading in that it alludes to a higher dimension?

2. Dec 23, 2007

### A.T.

GR speaks about longer distances between coordinates, in some areas compared to other areas. Apply this to a simple 2D grid, and demand that one point has a longer distance to its neighbor points, than all the other points to their neighbors. One way to visualize this is to offest this point into an additional 3rd dimension. This is called embedding.

A dimple upward on a 2D surface produces the same distances as a dimple downward, and thats all that matters for those living in the the 2D world. They know nothing about the 3rd dimension and that downward-upward-stuff. Free falling objects moving on geodesics (straight forward without steering left or right) take the same path around a upward and downward dimple.

Thats another way to visualize intrinsic curvature. Instead of moving the point into a higher dimension you could say that the area around it is "denser" which results in the increased distances.

Mathematically it doesn't matter. It is just a question how to envision curvature. Here my post in a similar recent thread:

3. Dec 24, 2007

### pervect

Staff Emeritus
I believe that you are already familiar with space-time diagrams from another post I saw of yours.

You can think of GR as drawing your space-time diagrams on a curved surface. Exactly what surface for the case of the Schwarzschild geometry is described in a paper by Marolf.

http://arxiv.org/abs/gr-qc/9806123

Note that it is not really space that is curved, but space-time. The point is that one does standard special relativity, but draws the space-time diagrams on a curved surface rather than a flat one. In this example, we envision this curved surface by the usual visual aid of an embedding diagram, i.e. we visualize the curved surface as an actual surface of some object in a higher dimensional space.

Some of the paper may be a bit hard to follow, there's some discussion at
https://www.physicsforums.com/showthread.php?t=149932&page=3 which also has some color plots. (Marolf's paper also has plots, but they are black & white).

here.

Time runs upwards in this plot, so timelike worldlines must move primarily upwards. The region external to the event horizon, our universe, is colored green. The interior of the black hole is colored red. The event horizon is where the red and green surfaces intersect. Of course this is only a 2-d subset of the full Schwarzschild geometry - it is the radial r-t plane.

Because this includes the kruskal extensions, the picture also includes a "white hole" region (pink) and a second asymptotically flat space-time (blue).

Last edited: Dec 24, 2007