# *The* open problem in General Relativity?

Q. Is there a single, clear mathematical question that has emerged as the open problem in General Relativity?

I ask this on the ~100th anniversary of Einstein's (4-page!) 1915 paper,

"Die Feldgleichungen der Gravitation," Preussische Akademie der Wissenschaften, Sitzungsberichte, 1915 (part 2), 844–847. (Wikisource:The Field Equations of Gravitation.)

Einstein's notebook, Riemann curvature tensor.

It is difficult to surpass Willie Wong's thorough 2011 answer to the MSE question, "Open problems in General Relativity." But perhaps a core mathematical question has risen to the fore since?

Naked singularities? Beyond the Cauchy horizon? "Electrodynamics of moving bodies?" Florentin Smarandache's 2013 Unsolved Problems in Special and General Relativity (PDF download.)?

Or perhaps Willie's inventory cannot be condensed or sorted further at this point in time?

• Does "mathematical" rule out "search for unified field theory"? – The Masked Avenger Apr 3 '15 at 23:54
• That means you're not looking for mathematical problems which relate GR to QFT. In which case, I haven't heard of there being the problem on everyone's minds. But, you'll be interested in the mathematical seminar paper of Penrose, Some Unsolved Problems in Classical General Relativity. – Chris Gerig Apr 4 '15 at 0:04
• The Smarandache volume is quite the eye-opener. From the preface, here's the conclusion of the first paper: "Einstein’s explanation [of the perihelion precession of Mercury], based on wrong integral calculus and arbitrary approximations, is a complete failure." It keeps going like that! – user5117 Apr 4 '15 at 0:34
• I just ran across this recent (Jan 2015) workshop: Mathematical Problems in General Relativity. – Joseph O'Rourke Apr 4 '15 at 16:24
• Smarandache? Seriously? – Emil Jeřábek Apr 4 '15 at 21:06

Fix a manifold $M$ and consider the set of globally hyperbolic solutions $\mathcal{S}(M)$ of Einstein's equations, possibly with appropriate asymptotic conditions. (a) Give $\mathcal{S}(M)$ the structure or an infinite dimensional manifold (or more general kind of smooth space). (b) Pick a subalgebra $\mathcal{A} \subseteq C^\infty(\mathcal{S}(M))$ that separates points, contains at least the local and multi-local functionals. (c) Show that the choice in (b) can be made such that the diffeomorphism-invariant subalgebra $\mathcal{A}^\mathrm{Diff} \subset \mathcal{A}$ separates the orbits of diffeomorphisms and is closed under Poisson brackets (naturally defined by the variational nature of the equations). (d) Explicitly construct the formal deformation quantization of the Poisson algebra $\mathcal{A}^\mathrm{Diff}$ (say using an infinite dimensional version of Fedosov's method).