Mathematical aspects of General Relativity

Question

I was wondering what mathematical subjects are used in the study of the theory of General Relativity (black holes ...) I assume mostly differential geometry, Riemann Geometry ... but is there also some use of algebraic geometry or any other subject ? Thanks!

Answer 1

To answer it just let me put a caveat: what is General Relativity may be open to interpretation as to the extend you would like to add astrophysics and cosmology to the definition. This more "applied" fields depend a lot on other parts of physics and would carry whatever mathematical techniques this areas involve. Also it would depend if you want to consider quantum gravity.

As such I would like to present a very tentative answer based on the restricted definition that General Relativity concerns the geometry of 4 dimensional manifolds equipped with a smooth metric $g$ of Lorentzian signature (-+++), which satisfy some "good" conditions, the major one being the Einstein Equations $Ric= 8\pi T(g)$, where $Ric$ is the Ricci tensor and $T(g)$ is some rank-2 symmetric tensor that depends on the metric. Many restrictions on $T(g)$ come from other parts of physics, so let's not dwell on that, besides noting that it should have other "good" properties which are of generic nature.

In this form it would be fair to say that Differential Geometry is the most influential subject here. Another almost equally important is Partial Differential Equations, since the Einstein Equations are a system of PDEs. More importantly they are hyperbolic PDEs whose solutions are subject to contraints in the form of other PDEs, typically of elliptic, but may be also of parabolic nature, depending on the spacetime decomposition you work with. Since you're working with PDEs I could note that Functional Analysis is not to be neglected. Lie Groups also make lots of appearances, so they're in the main toolbox. This year there was a school on Mathematical Relativity in Vienna (here the link), so maybe taking a look at the courses you can get a more detailed answer.

What I would like to note is what subjects are not particularly useful. A lot of the theory rests purely on general topology, but algebraic topology is much less prominent. Also to be noted Riemannian Geometry, as in manifolds with positive definite metrics, say at the level of Petersen's book, is not that useful. Sure some general ideas are the same, but the signature really makes a difference such that the more advanced tools do not work equally. What I'm saying is that Riemannian and Lorentzian geometries are very different. As for Algebraic Geometry it does appear when you study spinors or twistors, but besides that they are pretty much absent.

Answer 2

As far as I know, the following formulations of General Relativity exist today:

• Tensor calculus (the usual one)
• Differential forms
• Spinors
• Twistors
• Geometric (a.k.a. Clifford) algebra. See for instance "Physical Applications of Geometric Algebra" by Chris Doran and Anthony Lasenby.
• Algebraic geometry. See for instance "Arithmetic gravity and Yang-Mills theory: An approach to adelic physics via algebraic spaces" by René Schmidt.