Lower dimensional gravity

Many of the fundamental conceptual issues in quantum gravity involve general features of the theory, which don't depend on "details" like the number of dimensions of spacetime. In fewer than four dimensions, though, the mathematics of general relativity becomes vastly simpler. Lower dimensional models therefore become useful testing grounds for quantum gravity.

One way to understand the simplification is the following. In n dimensions, the phase space of general relativity -- the space of generalized positions and momenta, or equivalently the space of initial data -- is characterized by a spatial metric on a constant-time hypersurface, which has n(n-1)/2 independent components, and its time derivative (or conjugate momentum), which adds another n(n-1)/2 degrees of freedom per spacetime point. It is a standard result of general relativity, however, that n of the Einstein field equations are constraints on initial conditions rather than dynamical equations. These constraints eliminate n degrees of freedom per point. Another n degrees of freedom per point can be removed by using the freedom to choose n coordinates. We are thus left with n(n-1)-2n = n(n-3) physical degrees of freedom per spacetime point.

In four spacetime dimensions, this gives the four phase space degrees of freedom of ordinary general relativity, two gravitational wave polarizations and their time derivatives. If n=3, on the other hand, there are no field degrees of freedom: up to a finite number of possible global degrees of freedom, the geometry is completely determined by the constraints.

Equivalently, it can be shown from basic Riemannian geometry that the full Riemann curvature tensor in 2+1 dimensions is algebraically determined by the Einstein tensor. The Einstein tensor, in turn, is fixed uniquely, through the Einstein field equations, by the distribution of matter. As a result, there are no propagating gravitational degrees of freedom -- the geometry of spacetime at a point is (almost) entirely determined by the amount and type of matter at that point. In particular, if there is no matter, the Einstein field equations in 2+1 dimensions imply that spacetime is flat.

At first sight, this is too strong a restriction. It's not much of a test to be able to quantize a theory with no degrees of freedom, and general relativity is supposed to be about curved spacetimes, not flat ones. But this sort of counting argument can miss a finite number of "global" degrees of freedom. The simplest example of such degrees of freedom is the following:

Consider a flat, square piece of paper, with the following "gluing" rule: a point at any edge is to be considered the same as the corresponding point at the opposite edge. Such a space is topologically a torus, and is sometimes called the "video game model" of the torus. (When you reach one edge, you automatically pop back in at the opposite edge, as in many video games.) Geometrically, this space is flat -- all of the rules of Euclidean geometry hold in any small finite region -- because, after all, any small region looks just like a region of the piece of paper you started with. It's a fun exercise to convince yourself that this is true even for regions that contain an "edge."

Now change this model a little bit by starting with a parallelogram rather than a square. This gives you a different manifold for each different choice of parallelogram, up to some subtle symmetries (the "mapping class group"). Each of these manifolds is flat, but they are geometrically distinguishable. In fact, this construction gives you a three-parameter family of flat spaces with torus topology: there's one parameter for the length of each side of the parallelogram, plus one for the angle between two adjacent sides.

One of the simplest nontrivial solutions to (2+1)-dimensional general relativity is precisely such a torus universe. The constraints fix the overall scale in terms of two parameters (say, one side length and one angle), but these two parameters have an interesting and nontrivial evolution. More complicated topologies give more parameters. So do point particles, which can be represented as conical "defects" in space. With a negative cosmological constant, (2+1)-dimensional general relativity even admits black hole solutions, which behave almost like ordinary (3+1)-dimensional black holes.

Note that this does not contradict the earlier counting argument. There are still only finitely many total degrees of freedom, rather than one or more degrees of freedom per point. But this finite number of degrees of freedom still has a very interesting dynamics, and the theory is rich enough to test many standard approaches to quantum gravity.

Two-dimensional spacetimes also provide an interesting testing ground for quantum gravity. As you might guess from the earlier counting argument, ordinary general relativity does not make sense in two spacetime dimensions -- the count gives "-2 degrees of freedom per point." But there are simple modifications that lead to interesting models of what is called "dilaton gravity."

For a beautiful nontechnical introduction to topologies like the "video game torus," see Jeff Weeks' book, The Shape of Space.

Contributors and Attributions

• Steve Carlip (Physics, UC Davis)