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?

  • $\begingroup$ Does "mathematical" rule out "search for unified field theory"? $\endgroup$ – The Masked Avenger Apr 3 '15 at 23:54
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    $\begingroup$ 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. $\endgroup$ – Chris Gerig Apr 4 '15 at 0:04
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    $\begingroup$ 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! $\endgroup$ – user5117 Apr 4 '15 at 0:34
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    $\begingroup$ I just ran across this recent (Jan 2015) workshop: Mathematical Problems in General Relativity. $\endgroup$ – Joseph O'Rourke Apr 4 '15 at 16:24
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    $\begingroup$ Smarandache? Seriously? $\endgroup$ – Emil Jeřábek Apr 4 '15 at 21:06

Perhaps the cosmic censorship conjecture (the absence of singularities outside event horizons) is the most compelling, at least that is what is argued by Klainerman in Cosmic censorship and other great mathematical challenges of general relativity, with reference to Hilbert's requirement that a great problem in mathematics "should be clear and easy to comprehend, difficult yet not completely inaccessible lest it mocks at our efforts. It should provide a landmark on our way through the confusing maze and thus guide us towards hidden truth."

There is no doubt that the cosmic censorship conjecture verifies this last criterion, its solution will be a great advance in our understanding of general solutions to the Einstein field equations. There is also no doubt that it is very difficult. Though young in comparison to the other big challenges in mathematics, such as the magnificent seven millennium problems, it has resisted so far all our efforts and it is obvious to all concerned that a solution is nowhere in sight. The conjecture is also clear and easy to comprehend even though a completely tight formulation could only be given once a solution will be found. It thus only remains to argue, as I will attempt here, that the cosmic censorship conjecture is not completely inaccessible as it has generated, and will continue to generate, new mathematical techniques which allow us to see a glimmer of light at the end of the tunnel.

  • $\begingroup$ Thanks, especially for the citation of Sergui Klainerman's paper. $\endgroup$ – Joseph O'Rourke Apr 4 '15 at 16:22
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    $\begingroup$ A recent paper: James Isenberg, "On Strong Cosmic Censorship." May 2015. arXiv abs link. $\endgroup$ – Joseph O'Rourke May 26 '15 at 12:12

Since questions on the interface between GR and QFT are admissible, here's what I consider the open problem in that direction.

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).

The end point of the above construction would constitute a (formal) quantization of GR as a QFT. As stated, there are various choices that can be made in the construction, but if a minimal set of requirements is satisfied, then any such choice is fine. Hence, I consider the above as a "clear mathematical question". In principle, one would also like to replace "formal deformation quantization" in the final step by "strict deformation quantization", but there is not yet a precise consensus on what that means for QFTs, nor is there a constructive method of obtaining strict deformations (unlike formal ones).


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