### One and two (space-time) dimensional general relativity

#### by Willie Wong

This is a bit of a silly post, partly written because I came across this paper when doing some literature search.

To give a spoiler: the punchline is

Life is very boring in one and two space-time dimensions.

Readers familiar with Einstein’s equations (with cosmological constant) may want to pause and think for a few seconds; then they may want to skip reading this post entirely.

**One space-time dimension**

Okay, if the total space-time dimension is only 1, it is not really a space-time. It is either space, or it is time. There’s also no difference between Lorentzian and Riemannian manifolds: they are all locally isometric and are all completely flat. Hence the left hand side of Einstein’s equation is automatically 0. Since the dimension is 1, the stress-energy tensor is completely specified by one component and is in fact a scalar function. Thus the stress-energy is required to be “constant” (in time, or in space, whichever way you prefer).

In other words, if we were to interpret the single dimension as time: the matter fields are required to solve an ordinary differential equation satisfying the conservation of energy. No additional constraints are imposed.

**Two space-time dimensions**

The definition of the scalar curvature implies that the left hand side of Einstein’s equation again vanishes identically. Thus . In particular, the energy momentum tensor satisfies .

If there’s no cosmological constant, we have that the stress-energy vanishes identically. Under assumptions on the form of the stress-energy tensor (a strengthened form of dominant energy condition which requires the matter fields to vanish when the stress-energy vanishes) this will imply that the space-time admits no matter fields.

Note that an assumption of dominant energy condition will automatically impose that the the cosmological constant is negative. For specific matter models, we can say also some things about the case with cosmological constant.

*Perfect fluids*. One easily sees that the requirement requires that the mass density equals the *negative* of the pressure. This is a rather unphysical assumption that also rules out both dust and radiation fluids.

*Scalar field*. The stress-energy tensor is trace-free (in two dimensions the scalar field equation is conformal). This is only compatible with vanishing cosmological constant.

*Maxwell fields*. Maxwell’s equation in 2 space-time dimensions is no longer conformal. The Maxwell field, being a two form, is necessarily proportional to the volume form, so we can write . Taking the trace of the stress-energy tensor we have that must be constant on the space-time. So effectively there is no dynamics.

In one dimension, wouldn’t the stress energy tensor be zero?

In three space-time dimensions, life would be a bit more interesting than one and two dimensions but still boring.

If the cosmological constant is set to zero, sure. But not making that assumption, the one dimensional case can support a “dust” perfect fluid- there’s no such thing as pressure anyway when space is just a point. (note that the dust solution is not admissible in two dimensions.) Another example of a nontrivial field in 1d is the Klein-Gordon field, which is essentially the simple harmonic oscillator. In 2d, KG is compatible with a cosmological constant, but only for a everywhere constant field!

Also, your comment about 3d I assume is based on the fact that there’s no graviton (weyl field). But it is much less boring! There are nontrivial solutions of the Einstein-matter field equations, which is already much much more interesting then the 1d case where geometry is completely trivial and the 2d case where for most matter models the matter field is completely trivial.

Thanks for your reply. About the 3D case, yes I was thinking of vacuum solutions which are flat or constant curvature space-times. Of course you are right you can add matter and/or defects on lower dimensional manifolds to get interesting solutions. But still, there is the question whether the 3D GR can be considered a simplified, but not too simple version of GR, in the same way, say 2D Navier-Stokes relates to 3D Navier-Stokes. I mean, can it serve as a lower dimensional testing ground for techniques for proving hard conjectures in GR? If not, what would be a good candidate?

Lower dimensional versions of GR can, in situations, provide interesting insight to the 3+1 case. However, necessarily the lower dimensional model needs to be coupled to matter fields.

Two examples that leap to mind are: a) The series of papers by Christodoulou on the spherical symmetric collapse of a scalar field. The reduced equations look like a 1+1 dim. GR model with an external potential. The insights gained in that program lead to his recent work on nonsymmetric vacuum collapse. b) Another example is given by the work of Moncrief and collaborators in the late 90s til early 00s, where they studied U(1) symmetric vacuum spacetime via the reduction to essentially the 2+1 Einstein-scalar field system. They were able to establish interesting global existence results assuming a compact spatial topology.

As to why the 2+1 case max be interesting mathematically by itself: Einstein’s equation us highly nonlinear. To a first approximation one has the splitting that the Weyl part corresponds to free propagation, while the Ricci part corresponds to “inhomogeneous” term, if you take an analogy to the linear wave equation with source. So the 2+1 case may help understand the inhomogeneous 3+1 case as a model without backreaction. (This is very speculative, take it with a grain of salt.)

(If I were writing a funding application, I would also mention here AdS CFT and domain walls. But let us not get into that here.)