# The abstract space of metrics in GR

I know this is a general (har har) question, but has any work been done understanding the mathematical space the allowed metrics in GR form? (I guess it'd be called a tensor space???)

• Differential geometry has been one of the major topics in mathematics since, I believe, the late 19th century. By the time Einstein began using it for GR it was already a well developed discipline. Einstein was also not alone in his pursuit of general relativity, David Hilbert probably discovered the main equations at the same time, but there is some historic dispute about the details. Jan 1, 2015 at 4:10
• I know differential geometry isn't new, but I was asking if any progress has been accomplished w.r.t. to understanding the abstract space GR metrics form. Jan 1, 2015 at 4:19
• Well, the space of a certain class of manifolds is normally a moduli space; the branch which deals with such spaces is normally called 'special geometry' in the literature. I have only learned about that in the context of string theory and instantons, not general relativity. Jan 1, 2015 at 9:39

Sure, this kind of thing is well studied. The "space of metrics" of a manifold is usually called its moduli space. More precisely, if we have a manifold $M$, Let $\text{Met}(M)$ be the set of all possible metrics on $M$ and define an equivalence relation $\thicksim$ on it, where metrics $g, g'$, are considered the same if there is an automorphism (diffeomorphism, biholomorphism etc depending on what structure on $M$ we are interested in preserving) of $M$ that takes $g$ to $g'$. The moduli space of $M$ is then defined to be $\text{Met}(X)/\thicksim$, and in most cases it can be given at least the structure of a "manifold with corners". I don't think this full-fledged space is that important in GR, since in GR we are for the most part only interested in metrics that obey Einstein's equations. However, its of fundamental importance in string theory, where one integrates over moduli space of Riemann surfaces.