General relativity (gravitation) in time and one spatial dimension

Question

I don't have any idea of general relativity but intend to learn. Is it a good idea to study general relativity in 1+1D (time and a single spatial dimension) in the beginning to get good idea on the subject? If it is, then please give some references for such a treatment.

Answer 1

Depends on your reference. One thing that should be said is that 1+1 relativity theory is trivially simple in several ways.

First, the Riemann Curvature tensor $R_{abcd}$ is determined completely by one component, meaning that Einstein's equation reduces to a single PDE. Since this is generally a tensor equation involving a somewhat large number of coupled PDE's, you will likely not get appropriate intuition for the theory.

Second, there is a conformal invariance$^{1}$ built into the 1+1 Einstein's equations that further makes the theory very easy to solve, since we're now dealing with a single PDE (plus a matter equation of state) with conformal invariance that lets us control the input o that PDE.

These two complications will give a very bad intuition for how Einstein's equations work--in particular, they reduce Einstein's equation to an equation depending only on topological invariants, and not upon local properties of the theory. There can be no radiation in 1+1 gravity, for example. In 3+1 dimensions, where neither of the above simplifications apply, the qualitative behaviour of the theory is very different.

That said, a guided study of particular Minkowskian two-spaces can help one understand full Relativity theory, but that is something best done for you by an expert. Please just start with Schutz's book if you're just trying to teach yourself Relativity.

$^{1}$conformal invariance means I can map the metric $g_{ab}$ into $\bar g_{ab} = \phi g_{ab}$ and have $\bar g_{ab}$ satisfy the same equation that $g_{ab}$ did. Einstein's equations in 1+1 dimensions are unable to distinguish between the metric and a rescaling of the metric.

Answer 2

It is true that gravity in $1+1$ dimensions is pretty trivial, but if you couple it to a scalar field in the right way it becomes much richer and turns into a very nice model which allows one to study black holes, black hole formation, and black hole evaporation. I apologize for advertising some of my own work, but one of the first papers on this is C. Callan, S. Giddings, JH, and A. Strominger, hep-th/9111056. This CGHS model has been studied in great detail, both analytically and numerically. For example, here is a recent paper that contains a numerical study of black hole evaporation in this model: http://arXiv.org/abs/1012.0077. I think one can learn quite a bit about black holes and GR in this model without many of the complications of four dimensions.

Answer 3

No, not really. One of the neat things about GR is that the same theory works in any number of dimensions. In other words, when studying GR, most of what you learn is independent of the dimensionality of spacetime, so there is little to be gained from limiting yourself to 1+1D.

Of course, it can certainly be instructive to apply the theory in 1+1D spacetime or other dimensionalities, to calculate how gravity or electromagnetism would work in "flatland," for example. But those sorts of calculations are interesting mainly because of the results you get. The procedure to apply GR in different dimensionalities is essentially the same as in 3+1D.