# Mathematical aspects of General Relativity

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!

• You mention Algebraic Geometry but I am wondering if you actually meant Geometric Algebra which is a different subject and one that does have some influence in these topics (e.g. GR, Differential Geometry). – K7PEH Oct 22 '14 at 19:29
• no I meant Algebraic geometry, as I didn't know Geometric Algebra existed ... but I will look into it. Thanks – Sasha Oct 22 '14 at 19:39

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.

• Are you sure you don't mean 'geometric algebra' instead of algebraic geometry? If you really do mean AG, could you give concrete examples? – rschwieb Oct 22 '14 at 22:13
• @rschwieb, yes I do mean AG, in fact although I have seen some works on geometric algebra in genera relativity I don't think it is a widely used method. As for AG in relation to twistors I haven't worked with this, but you could take a look into Twistor Geometry and Field Theory by Ward and Wells Jr. I could not attest to the truth of it, as I haven't even read the papers, but one lad once told me that Ted Newman's work on Heaven spaces also touch a lot on AG – cesaruliana Oct 23 '14 at 4:18
• I'm well aware of the status of GA. In my past year or two of reading, the areas of math most relevant to spinor and twistor theory were Lie theory, differential geometry and GA. The operations involved are mainly noncommutative, so I haven't noticed any AG. but perhaps there is some sheaf and stack theoretic stuff I simply didn't recognize. That theory is very general, after all. I can't find much on heaven spaces: the one source I found so far is mainly differential geometry. – rschwieb Oct 23 '14 at 10:05
• My statement on GA was just a reflection of my ignorance, I wasn't trying to be assertive. Regarding twistors, some sheafs do crop, as can be seen in the table of contents from Ward's book, but I wouldn't try to say how relevant AG is for twistors. I just mentioned because the OP specifically asked about AG. As for heaven spaces, they are somewhat elusive. One reference to start is the chapter on complex general relativity by Boyer, Finley and Plebanski to be found in "General Relativity and Gravitation One Hundred Years After the Birth of Albert Einstein" vol. 2, edited by A. Held (1979) – cesaruliana Oct 23 '14 at 14:49
• Is the approach to General relativity by algebraic concepts (Geometric algebras) still studied nowadays or do people in general prefer the geometric approach by differential geometry ? – Sasha Oct 23 '14 at 16:59