# 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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Here's a fuzzy argument I've heard since my undergraduate days in physics: If the universe is bounded then quantum mechanics would seem to indicate that the momentum operator does not assume a continuous spectrum of eigenvalues. It seems a reasonable extension to suppose that spacetime itself would be some sort of discrete object. The continuum which we frame classical physics would just be a macroscopic approximation of some quantum plank-length geometry.

One way to implement such an idea is to replace the commutative coordinates of classical manifold theory with noncommutative coordinates where the departure from commutivity is introduced by in physics literature by the introduction of star-products. Look up the Moyal bracket, or the paper Deformation quantization of Poisson manifolds by M. Kontsevic http://arxiv.org/abs/q-alg/9709040. The literature that follows this is voluminous, see the work of Julius Wess and his collaborators in particular. The features of GR that have been recovered in this formalism are surprising. I'll leave it to someone more adept than myself to comment further on the connections with algebraic geometry and Alain Connes program of noncommutative geometry.

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I think the idea that spacetime must be discrete, is less fundamental than the idea that something about spacetime must be noncommutative (i.e., quantum). Whether or not spacetime is discrete, is supposed to be decided by the quantum theory itself. –  timur Aug 24 '12 at 14:07
The argument in the first paragraph is incorrect. If the universe was bounded (which it may or may not be), then the momentum operator, not the position operator, would have a discrete spectrum. Also, this argument can't connect to the existence of the Planck scale because its premises hold even in a flat universe with the topology of a torus, so that the result doesn't depend on the value of the gravitational constant G --- and yet the Planck length depends on G. –  Ben Crowell Aug 25 '12 at 1:36
@Ben, you're correct, I should have said momentum operator. –  James S. Cook Aug 25 '12 at 13:20
@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
A discrete spectrum for the momentum operator does not relate in any logical way to quantization of spacetime. What would relate to quantization of spacetime would be a bounded spectrum for the momentum operator. The point of the example about the torus was just to show using a specific counterexample that there can't possibly be correct reasoning behind your claim, by showing that your premises can hold while your conclusions fail. –  Ben Crowell Aug 25 '12 at 23:33