# Is there such a thing as discrete Riemannian geometry?

General relativity expresses gravity as a curvature in space time, created by the stress energy tensor.

$$T_{\mu\nu} \approx R_{\mu\nu} - \frac{R}{2} g_{\mu\nu}$$

Given I put the fact that energy is quantizied and can only change within discrete levels into the stress energy tensor, then the Ricci tensor and as a result, the Christoffel symbols, as well as the metric would have to change into some finite difference form.

For large space, this finite difference would approximate the equation above, but for small areas, it would behave differently. Is there any kind of discrete Riemannian geometry that satisfies this criteria? Any sources?

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Circle packing is the closest thing to discrete Riemannian geometry that I know of (in two dimensions only). But the body of your question goes in a more specific direction: you want a discretization of the Einstein equations... "The construction of good discretizations of these equations is an active research topic where much remains to be done" - see here where some references are given. –  user31373 Aug 23 '12 at 23:28
You are mixing quantization with discretization. They are not the same things. Also, energy does not have to be quantized to discrete levels. This is true for some cases like atoms and molecules, but in general it can have purely continuous spectrum, or more often mixed ones. –  timur Aug 23 '12 at 23:48
Here physicsforums.com/showthread.php?t=391989 is a discussion of the merits of one apparently elementary argument, by Smolin, for the discreteness of spacetime. –  Ben Crowell Aug 25 '12 at 1:51

The first thing to realize is that nobody has a working theory of quantum gravity, so nobody can really answer your question.

As timur has pointed out, quantization doesn't necessarily imply discretization. It also doesn't work the other way around: discretization doesn't imply quantization. You can certainly use finite difference approximations to solve the Einstein field equations, and people do indeed do this, but all this will give you is an approximation to classical (i.e., non-quantum-mechanical) GR. This is the kind of thing relativists do, for example, when they simulate violent classical processes like the mergers of black holes and the subsequent emission of gravitational waves.

Having said all that, there are a couple of leading candidates for a theory of quantum gravity, which may or may not be equivalent if you work out their detailed implications (which nobody has been able to do). These are string theory and loop quantum gravity (LQG). LQG does in some sense quantize spacetime, but you shouldn't take that too literal-mindedly. It's not distances that are quantized but areas and volumes. A naive way of seeing that you probably can't quantize distance by using some kind of grid is that under a Lorentz transformation, the grid spacing would undergo length contraction and time dilation. (Area and volume are preserved by a Lorentz transformation.) A theory similar to LQG is causal dynamical triangulation (CDT).

Scientific American has published a couple of popular-level articles about LQG and CDT:

Smolin, "Atoms of Space and Time," Scientific American, Jan 2004

Jerzy Jurkiewicz, Renate Loll and Jan Ambjorn, "Using Causality to Solve the Puzzle of Quantum Spacetime ," Scientific American, July 2008, https://www.scientificamerican.com/article.cfm?id=the-self-organizing-quantum-universe

Smolin has also written a nice popular-level book called Three Roads to Quantum Gravity, which is unfortunately getting to be out of date at this point.

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@Ben, what if the size of the torus depended on $G$? Not that I believe one way or the other, but it is possible? –  James S. Cook Aug 25 '12 at 13:34