Centered in Mathematics

A far-reaching conjecture in Einstein’s theory of general relativity, called the Final State Conjecture, states (approximately) that the final state of gravitational processes is one or more black holes plus some gravitational radiation. This conjecture is tightly linked to our understanding of the LIGO discovery of colliding black holes: Indeed, the observed LIGO events are best explained in terms of a final state consisting of just one large, spinning black hole (called a Kerr black hole) plus a burst of gravitational radiation which propagated through the universe for over a billion years before reaching us. Surprisingly, for all our sophisticated numerical modeling of black hole collisions, it remains out of reach to definitely prove that a single spinning black hole is what remains after the in-spiral of the two colliding black holes is complete. And LIGO-type mergers are just the beginning of the Final State Conjecture: It is clearly possible, given generic initial conditions, for the final state to be several spinning black holes moving apart from one another. The Final State Conjecture is regarded as the holy grail of mathematical general relativity and is on a par in depth and beauty with any other major open problem in Mathematics, such as the millennium Clay problems.

While still out of reach in full generality, the Final State Conjecture is related to a myriad of deep and challenging problems which are the focus of active research. For example, if true, the conjecture implies either that the initial data is too small to concentrate and thus must disperse to zero, or that it is large enough to produce bound states, i.e. black holes. The first possibility comes under the heading of the stability of Minkowski space; the second is the problem of collapse. The fact that final states must asymptote locally to Kerr black hole solutions implies, in particular, that any stationary solution of the equations must be a Kerr solution, and that Kerr is stable under perturbations. These two claims are referred to as the problem of rigidity and the stability of Kerr. Moreover, singularities are expected not to appear, at least from generic initial conditions; a statement postulating this is known as the weak cosmic censorship conjecture. The Final State Conjecture also requires us to understand the theoretical underpinnings of interactions of black holes such as the in-spiral typical of colliding black holes. In short, the Final State Conjecture provides a concise and intuitive guide to many of the main problems in mathematical general relativity.

Led by the senior faculty members Mihalis Dafermos, Alex Ionescu, Sergiu Klainerman, Igor Rodnianski and junior faculty Yakov Shlapentokh-Rothman, the Princeton Mathematics Department is at the forefront of research on all the above mentioned problems. Our faculty members have also made fundamental contributions to problems concerning the interior of black holes (in particular relating to the strong cosmic censorship conjecture), and cosmological spacetimes.

Our faculty’s interest in these problems goes beyond their obvious physical interest. As mathematicians, they also see them as challenges to develop new mathematical ideas and techniques relevant in a broader setting. In that sense it helps to remember the essay of Eugene Wigner (former Nobel Prize Princeton Physics professor) “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” according to which mathematical methods and ideas developed in the effort of understanding a specific problem can have an unexpected impact on seemingly unrelated problems in other fields.