I would like to know if there are some open mathematical problems in General Relativity, that are important from the point of view of Physics.
Is there something that still needs to be justified mathematically in order to have solid foundations?
Let me sneak in an answer before this gets closed, because the answer to this question is likely to be very very different depending if you ask a mathematician or a physicist.
:) So here is a long list of questions that a mathematician considers to be important open problems in general relativity, and why they are physically relevant.
Let me first get the biggie out of the way. There are actually two different cosmic censorship hypotheses, which are minimally related to each other. The Weak Cosmic Censorship (WCC) hypothesis states that (a fairly modern formulation)
For generic initial data to the evolution problem in general relativity, there cannot be naked singularities.
Unfortunately, if you ask me to define the terms in that hypothesis, I can just shrug and tell you, “no can do”. This question is so open as an open problem that we don’t even know the correct formulation of the statement! If you are interested, you may want to consult Christodoulou, D. “On the global initial value problem and the issue of singularities”, Classical Quant. Grav., 1999, 16, A23. Let me give just a short digression on what the difficulty of formulating the problem is.
The first is the problem of what do you mean by a naked singularity? In certain situations, the definition of a naked singularity is very clear cut (for example, if we assume mathematically that the space-time is spherically symmetric). But in general, the problem of what a naked singularity is is very difficult to define accurately in mathematics. The main problem being this: the equations of motion given by the Einstein’s equations are hyperbolic. As such, we cannot necessarily meaningfully extend the notion of a solution to points which can “see” the singularity. (Sort of a catch 22: morally speaking, if you can see the singularity, you are at a (possibly different) singular event, so it doesn’t make much sense to use the physics definition of a naked singularity being one that can be observed by a far away observer.) (This is not to say this is completely impossible: there are singular solutions which the solution can extend continuously to the singular boundary, and so technically you can exist all the way up to the moment you see the singularity, and then poof, you are gone.) So we have to probe at the notion of a naked singularity using alternative definitions that are more stable and can be mathematically formulated. And that is hard. (Once physical intuition fails, things get hard.)
The second problem is generic. It is already known that there are examples of solutions in general relativity which contains a naked singularity for all reasonable definitions of “naked singularity”. So it is in fact impossible to prove a statement that says “naked singularity cannot exists. Period.”. Any statement you prove must be a statement that is true away from some exceptional set. The hope is that the bad set is positive codimension in the space of all initial data. But we don’t know.
The other cosmic censorship hypothesis is the Strong Cosmic Censorship (SCC). Despite the name it is independent of WCC. The SCC states that
A generic solution to the Einstein’s equation cannot be continued beyond the Cauchy horizon.
Again, this is from the view of an evolution equation. A funny thing about general relativity as an evolution equation is that you build your manifold and your solution on the manifold at the same time, which causes an interesting problem. Roughly speaking, usually when you solve an nonlinear wave equation, and you extended your solution as much as you can, then you either exhausted the entire space-time, or you ran into a singularity. But in general relativity, “running out of space-time” is not a hard obstruction to extending the solution further. (This goes back to the hyperbolicity of the equation.) Basically, you can “run out of space-time that can be completely predicted by your initial data”, while still have a possible extension of your solution “to infinity and beyond”. The classic examples of this are the Reissner-Nordstrom and the Kerr-Newman black hole solutions. If you solve it as an initial value problem in partial differential equations, you get a black hole solution. But if you start from this black hole solution, you can actually analytically continue the solution (similar to how analytic continuation is done in complex analysis) to beyond what can be defineable from the inital data. The continuation is in some sense “outside” our universe, so the existence of this type of solutions causes some, at the very least, epistemological difficulties. The SCC posits that these type of solutions are rare. Just like the case of the WCC, the correct mathematical statement for this problem is also not known, and therefore it is hard to say what techniques of mathematics will be required to resolve it.
Final state conjecture
The final state conjecture in classical relativity states that (assuming the universe is not expanding) if you let the universe run long enough, because gravity is an attractive force, and is the strongest force over cosmological distances, we expect the entire universe to either collapse into a black hole, or become “mostly” empty (things drifting away from each other). Or slightly more jargony: the end state of the universe is either a single black hole or vacuum.
Mathematically this requires deriving the asymptotic behaviour of solutions to the Einstein’s equations in general relativity. The Einstein equations are a highly nonlinear system of partial differential equations, and it is very difficult to study these things asymptoticly. There are however, some special cases that have either been worked out or are receiving a lot of specialist attention.
Stability of Minkowski space
One of the first things to consider, in terms of the final state conjecture, is that whether, if we start with a universe that is already very sparse, we can guarantee that this will end up asymptotic to Minkowski space. The answer to this, as it turns out, is “yes”. The first work was due to D. Christodoulou and S. Klainerman, with extensions of the result and improvements by H. Lindblad and I. Rodnianski, L. Bieri, N. Zipser, J. Speck, etc.
Uniqueness of black holes
Now, if we posit that the final state of the universe is either Minkowski space or a black hole, we better show that a black hole is the only possible stationary solution! (Any other stationary solution would be another steady state!) The problem of black hole uniqueness is almost, but not completely, resolved.
In the case where we assume some additional symmetry (that the universe is either axially symmetric or rotationally symmetric), the uniqueness of black holes is known. In the case where we assume that all the objects we are dealing with are real analytic, we also know that black holes are unique.
The remaining case, which is expected to be true, is when we relax the regularity assumption to just infinitely differentiable, but not real analytic. In this case we only have some partial results, recently due to A. Ionescu, S. Alexakis, S. Klainerman, plus some additional contributions by P. Yu and yours truly. So far we have two major class of results: one is that if we allow only small perturbations of a stationary black hole, then there are no other stationary solutions that are approximately a known black hole solution without being one. The other is that if you assume certain special structures on the event horizon (roughly speaking, assume that the event horizon looks like the one for a known black hole solution), you cannot have other stationary exteriors.
Stability of Kerr-Newman black holes
Now, assuming we accept that the known Kerr-Newman family of black holes form the unique stationary state of general relativity, the next problem is to prove that they are actually stable under perturbations. That is, if we start out with initial data very similar to a Kerr-Newman black hole, is the evolution going to “track” a Kerr-Newman black hole? So far, there are quite some progress made in the linearised problem, due to contributions from (in no particular order) I. Rodnianski, M. Dafermos, P. Blue, A. Soffer, W. Schlag, R. Donniger, D. Tataru, J. Luk, J. Sterbenz, G. Holzegel, and many others. (It is quite an active field at the moment.) But the way forward for the full nonlinear problem is still somewhat elusive.
Penrose inequality and friends
Recently between H. Bray and the team of Huisken and Illmanen, what is known as the Riemannian Penrose inequality was proved. The Riemannian Penrose inequality sits in a large class of mass inequalities for space-time manifolds. These types of inequalities all take the form
Mass of the space-time is bounded below by Blah.
The first, and most famous, of these type of results is the positive mass theorem of Schoen and Yau, which states that the mass is bounded below by 0. The Riemannian Penrose Inequality states that the mass is bounded below by number related to the size of the black hole (when there is one). In other words, a black hole must contribute a certain amount of mass to the universe, the amount depending on its surface area.
There are further refinements of the known statements conjectured. One is the version called the Gibbons-Penrose inequality, which gives some improved lower bounds when there are multiple black holes (the attracting force between the multiple black holes will creating some contribution to the mass at infinity). And there are also other versions which also factors in electromagnetic forces and so forth. Most of them are open questions.
A somewhat related problem is the Hoop Conjecture of Kip Thorne. The conjecture roughly states that
If you try to squeeze lots of mass into a small space-time volume, what determines whether the mass will collapse and form a black hole is the “circumference” of the space-time region.
This problem is being actively pursued by M. Khuri and others, and is still open to my knowledge.
Some other open problems:
Low regularity evolution Can we solve the equations of motions assuming very rough (in the sense of having very few derivatives) initial data? These types of question are important because (a) when comparing to physical measurements, the rougher the data the theory can accomodate, the less sensitive the solution is to small perturbations, and so the better we can figure out the “error bar” due to theory (b) the problem of shockwaves. Physical systems can exhibit shock waves, and it is important to have a theory that can deal with that.
Higher dimensional cases In some cosmological/string theories, one is lead to consider space-time manifolds of higher dimensions. On the one hand, radiative decay of solutions is stronger on higher dimensions, and it would enhance stability. On the other, there are more degrees of freedom, which will increase possibilities of instability. There are numerical evidence to suggest that certain types of higher dimensional black holes are in fact unstable, and could be poked and perturbed into other types of black holes.
Evolution of apparent horizon Under the banner of dynamical horizon is a collection of topics related to how the apparent horizon (the “boundary of a black hole”) will evolve in general relativity. It is important to note that a lot of the physics literature on the evolution of apparent horizon (such as black hole evaporation) is based on a linear analysis, which to first order assumes that the horizons do not move. (Yes, by assuming that the horizons do not move (much), they show that quantum effects will make the horizons shrink.) The true nonlinear versions of the evolution is not yet well understood.
Geodesic hypothesis One of the postulates in studying general relativity is that point particles with negligible mass will travel along geodesics of the space-time. This question has a surprisingly long history (it was first considered by Einstein in the 1920s), and is still not completely resolved. The main problem is how to make the process of “taking the negligible mass limit” rigorous. For a physical object in general relativity, when it moves, its motion will cause “ripples” in the space-time caused by gravitational backreaction of its own presence. While the three body problem is difficult in classical mechanics, even the two body problem in full generality is still unresolved in general relativity.
I hope the above provides some keywords or phrases for you to look up.
Here are some more open problems in mathematical relativity that concern Cosmology. To start with, let me first quickly say what I mean by cosmology here. To me, cosmology is the study of the large scale behaviour of the universe. In other words, it is the “blurry” picture of what the universe would look like if we ignore the small-scale structures such as stars and galaxies. As such, in the study of cosmology we often assume the Cosmological Principle holds. The cosmological principle is a strengthening of the Copernican principle, and says that
On the large scale, the universe can be modeled by a solution to Einstein’s equations which admits a foliation by Cauchy hypersurfaces each of which is homogeneous and isotropic.
In layman’s words, it says that we can pick a (possibly preferred) notion of cosmological time, such that at every instant of cosmological time, space looks identical in all directions, and you cannot distinguish between any two points. (If we factor in the small-scale structure, this is quite obviously false: we live on a planet we call Earth orbiting a star we call Sol, that quite clearly distinguish us from many other points in the universe. But remember that the cosmological principle is only supposed to apply “in the large”.) One can get into an argument about whether this principle holds in practice (cf. the cosmic microwave background anisotropy), but it is under this principle most of the computations in cosmology are made.
This leads to some immediate open problems:
Stability of Cosmological Solutions
Part of the cosmological principle assumes that the universe is well-modeled by the spatially homogeneous and isotropic cosmological solutions. In particular, this means that we are assuming the cosmological solutions are stable (that if we perturb the cosmological solutions a tiny bit, say, by now factoring in the small-scale structure, the evolution as governed by Einstein’s equations will be, in the large, not very different from the evolution of the cosmological solutions).
Since in general this requires studying a large and complicated set of PDEs around a special solution, this is generally tackled on a case-by-case basis and there are still many interesting cosmological models to be studied.
Cosmological principle as a derived consequence
It would be much more satisfying, however, if the cosmological principle can be derived as a consequence of general relativity (under suitable assumptions on the cosmological constant and what not), rather than something assumed a priori. Namely, we ask the question:
Can we find some conditions such that the evolution of arbitrary data in the cosmological setting will eventually converge to one for which the cosmological principle holds?
Mathematically, we are asking whether homogenisation and isotropisation can be derived as a consequence of Einstein’s equations.
We remark here that it is generally accepted that the sufficient conditions will involve the space-time being expanding. Heuristically speaking if the space-time is expanding then the relative strength of the small-scale structure should get weaker as time goes along (since they get relatively smaller and smaller) and so one should converge toward a cosmological space-time. (On the other hand, homogenisation and isotropisation are not expected to hold in contracting space-times [either toward the big-bang singularity in the past or possibly toward a big-crunch singularity in the future], because if space-time contracts, the relative size of the small-scale features get bigger…) Luckily for us, the recent astrophysical observations seem to suggest that our universe is in a regime of accelerated expansion, so it is okay to not consider the other alternatives.
One can conjecture that something even stronger than homogenisation and isotropisation should hold. This conjecture is backed up by all the currently known results in this direction.
From a powerful result in three dimensional topology, we expect that spatial slices of the universe (assuming that the universe as a compact topology) can be written as the sum of some graph manifolds with some hyperbolic three-manifolds.
Then the conjecture is the following:
In the expanding direction, as time goes to infinity, the limits of the spatial slices is such that we can rescale the metric in such a way that (a) the rescaled metric on the hyperbolic pieces converge to the standard complete hyperbolic metric and (b) the rescaled metric on the graph manifold parts collapse.
In special situations or under special assumptions, this conjecture has been confirmed by
- Lars Andersson and Vincent Moncrief by showing the stability of the Milne model
- Vincent Moncrief and Yvonne Choquet-Bruhat in their work studying the $U(1)$ symmetric space-times (under their assumptions the spatial slices must be Seifert fibered spaces, and they do in fact show the desired [rescaled] collapse).
- The work of Michael Anderson, Martin Reiris, and possibly others, which show the desired conclusion if one were to assume some strong a priori bounds on the geometry.
There are, of course, many, many interesting problems concerning the initial data for Einstein’s equations. Just a quick review: the strong ties to geometry of Einstein’s equation in general relativity means that the system of equations is somehow simultaneously over- and underdetermined. The latter problem can be cured by “gauge fixing”; but the former problem manifests at the level of initial data.
One is used to be able to freely specify “initial conditions” when attacking an evolution equation. In the case of Einstein’s equations (and many other geometric partial differential equations), the initial data must satisfy certain initial constraints to be compatible with the global geometry: here we take the naive view that the initial data should contain the metric tensor on the initial slice and also its first time derivative. (This review by Bartnik and Isenberg gives a pretty high-powered review if you are already familiar with Lorentzian geometry; if you are not then I suggest looking at standard textbooks such as Wald’s General Relativity.)
Several natural questions arise concerning these constraint equations.
How rigid are the constraints?
By writing down the constraint equations (which can be expressed as 4 partial differential equations), we immediately see that relative to the number of degrees of freedom we have, the constraint equations are under-determined. Therefore we expect “lots” of solutions. But natural questions like “how many solutions are there given a set of prescribed asymptotics” still need to be answered. This is especially so since as part of the proof of the Positive Mass Theorem is included the following rigidity statement:
… the ADM mass of a complete asymptotically flat initial data set under assumption of dominant energy vanishes if and only if the initial data is a space-like initial data slice of Minkowski space.
Therefore we know that certain asymptotics for the initial data are very rigid.
An interesting development is the gluing technique due to Justin Corvino and Rick Schoen from a few years back. A corollary to the gluing technique is that once the ADM mass is allowed to be positive, the asymptotics become a lot less rigid for restricting the initial data. In particular, they were able to construct many initial data sets with asymptotics virtually identical to that of the Schwarzschild and Kerr black holes, but with different topology. The full force of this gluing technique and its implication is still under investigation today.
On the other hand, the success of the positive mass theorem (which is essentially a result on scalar curvature comparison) has led to many current research topics concerning various aspects of the rigidity of asymptotically flat manifolds with non-negative scalar curvature. These questions are mostly Riemannian geometric in nature and practice, and in addition to the Riemannian version of the Penrose inequality mentioned above, just to give a taste the kind of results obtained and questions asked (since I am not really too much of an expert in this direction), one can check out this recent result of Michael Eichmair and Simon Brendle which does not admit an easy physical interpretation.
How to parametrise the initial data?
Giving that the initial data sets need to satisfy the constraint equations, one is led to ask whether there exists an alternative way of parametrising the initial data (as opposed to the naive initial metric + first time derivative splitting) such that
- The data can be freely prescribed. For this we need a method to start with some tensors and a method of transformations which guarantees us to recover geometric data that satisfy the constraint equations.
- Is exhaustive; that is, every solution of the constraint equations can be generated from free data using this method.
This dream dates back to at least Andre Lichnerowicz and is still not entirely settled. A part of the story is the immense success of his conformal method in constructing and parametrising initial data sets of general relativity; it turns out that however, this conformal method is very well suited for a certain situation (data that has constant mean curvature or almost constant mean curvature), but may (the understanding on this issue is a bit muddy at the moment; but this is a post on open questions after all) be very much less well suited for other situations (data that has strongly varying mean curvature). If you are interested, I would highly recommend the two videos of David Maxwell’s lectures from September of last year.