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.