Three Kepler Problems

Depending on the chosen scale, the mathematics of the Kepler problem involves different parts of mathematics. In a celestial meso-scale setup, the Kepler problem is part of Newtonian mechanics. After changing to the center of mass, this problem leads to the ordinary differential equation, x”=-x/|x|³, a Hamiltonian system describing a mass point in the V(x)=1/|x| potential. Newton’s derivation of the Kepler laws is simple and often taught in multivariable calculus classes (it can be done on 2 pages [PDF]). In a quantum frame work, the micro scale, the problem is the Hydrogen atom, a spectral problem in the theory of partial differential equations; the problem is to find the eigenvalues and eigenvectors of H=-Δ +V, where Δ is the Laplacian and V is a multiplication operator. Similarly as the celestial Newton problem, the mathematics is still well accessible: it just requires to find an orthonormal eigensystem in which the operator H is diagonal. Spherical harmonics does the job. In physics, the microscopic Kepler problem is not used for gravitational interaction but for the electromagnetic interaction (we currently don’t know what happens with gravity on an atomic scale as the measurements do not allow to get so small values). The third Kepler problem, the general relativistic 2-body problem is the macro scale version of the three Kepler problems and is not solved at all. One of the openly available papers about this is here, where a numerical scheme is described. Even stating the problem provides difficulty: it is the problem to find a singular measure located on two curves $K$ in a 4-dimensional pseudo Riemannian manifold $(M,g)$ such that there is a metric $g$ satisfies the Einstein equations (in a weak sense of course as we deal with distributions). The mass tensor defined by the two curves which each is a geodesic in $g$ . While in the classical Kepler problem, the typical example is the earth-sun system, and in the quantum Kepler problem, we deal with a hydrogen atom consisting of an electron moving in the electric field, the key example in the relativistic case is a black body binary system. In each of the three cases, idealizations are done. In the classical Kepler problem, the bodies are assumed to be point like and tidal effects or other bodies are neglected. In the Hydrogen atom, one neglects the influence of the electron on the proton, nor does include the magnetic moments. In the black hole problem, the two black holes do radiate gravitational waves and the two bodies will eventually merge. This collision is a special challenge. The mathematics deals with the partial differential equations for the metric $g$ which involves distributions in the mass tensor as the curves are singular. Smoothing them out to get a smooth tensor already would deviate from the theory and require to look at Vlasov type dynamics to describe a mass field located in the support of the now smooth energy momentum tensor. Smearing out the curve serves as a numerical scheme and when modelling particles inside that could would assume that the geodesic flow in the metric is integrable. That is very unlikely as generic geodesic flows are chaotic. There is not even an existence theorem available for that type of system and even a numerical simulation requires sophistication. It is fair to say that from a mathematical point of view, we are completely in the dark. [I had once taught a geometric analysis course during a trimester at Caltech: writeup here [PDF], where GR appeared as an illustration to some diff geometry.]

Is there a common frame work for all these three systems? We don’t know. The existence of a concept in which all three problems are special cases could be the ultimate test for a quantum gravity theory. There is no question that all the current contenders like loop quantum gravity and string theory do not do answer that “Kepler question”, despite the many candidate theories (the Wikipedia article mentions more than 2 dozen. But these are probably only the “well known” ones as undoutably, every physisist has his or her own theory). The subject is heavily fragmented. As pointed out in this blog, there is a quote of Jorge Pullin: “Loopy people go to loopy conferences, stringy people go to stringy conferences.”, but this covers only the star theories and is mostly based on the herd instinct which is always very strong if jobs are sparse. But we can see the communities happily united as both theories fail miserably providing a unified view onto the two body problems even in the solved classical and quantum cases. We mean with `solving”, that the computation should give quantitative results using a machinery which a comparable complexity than the classical solutions. Of course, a new solution is allowed to pack the machinery into a general theory. In the case of the quantum Kepler problem it is achieved by harmonic analysis in three dimensional space. In the binary black hole problem a solution should in particular give an existence result. This does not even touch on the problem of what happens at the collision point or what happens if a black hole evaporates and disappears. The later is usually given as a test scenario for a theory combining quantum and general relativistic physics. Now one can ask why this “Kepler test” is reasonable at all. Does not physics deal with models which allow experimental verification in some range of scales and that these models patch larger parts? But this seems really the question when asking whether there is a unifying theory. Maybe it is fair to say that one can not call a theory a “theory of quantum gravity” if it does not cover even the two body interaction problem; both in the small as well as in the large. But the problem illustrates how gigantic the task is: the theories can not even solve the Kepler problem which Newton dealt with. And in the macro case, there is no mathematical existence theorem nor clarity what happens when singularities merge. We also have little insight what happens with very small masses as in the small, other forces are much more important and dwarf the gravitational force. On the Macro scale, even the “one body problem” is in the mud, as one does not know what happens when a singularity evaporates. t’Hooft tells in this book: “A healthy theory should be able to tell us how to calculate here”. This is obviously an area of physics, where experimental results are missing. Maybe more results of LIGO and the future LISA will enlighten us.

QG=CM+QM+GR ≤ UT

The commonly given definition of quantum gravity (QG) is vague: it aims to unify relativity (GR) classical mechanics (CM) and quantum mechanics (QM), building part of a unified theory (UT). It has to bridge the macro scale with gravity working on cosmological scale, with the meso scale known to engineering and include the micro scale, where quantum effects matters on an atomic scale. It has to be dynamical, allowing to compute the past and future from a given system. It also should be a UT without the need of tuning a list of parameters. Particle forces and masses should be derivable within the theory. If possible, the simplest cases should be integrable situations, giving explicit solutions. It has to overcome the difficulty that time is global in CM and QM while in GR, it is tied to geometry and mass.

As a disclaimer, I’m not working in quantum gravity but it is necessary for a mathematician (I believe) to think regularly about the nature of space and time. Geometry is a theory of “space” and dynamical systems theory is a theory of “time”. Also, as history has shown, our understanding “mass” is very much related to calculus and partial differential equations. Gravitational physics is a nonlinear geometric theory while quantum mechanics is linear functional analysis, and both are heavy on PDE’s so that all this is also not strange for mathematicians to think about it. As Dijckgraaf has pointed out at various places, the benefits of thinking about fundamental questions is often beneficial for mathematics (e.g in i.e. in this Nautilus interview). What you read here is a blog entry written on a Sunday afternoon and should not taken too seriously. To learn about quantum gravity, one better starts with books like the one by Carlo Rovelli (for which a free draft is available). Or then this draft for Thiemann’s book. For general audience is Lee Smolin’s “Three Roads to Quantum Gravity”. I must admit however that even after trying again and again to get into this subject especially through lighter literature, I have the impression that the topic of quantum gravity is far from accessible. Especially the popular books feel like: “because I told you so” and try to explain non-existent theories with analogies. It is still an entertaining subject.

As a unified theory UT, a contender for QG also needs to be elegant and not a patch of given theories. To put it with a picture: it has to be a chimera, possibly with many heads and a common body. It can even be patched with different theories like a hydra, but it should have a common body. It has to cover space, time, forces and matter in a unified manner. Such a theory does not exist yet, it might even be too ambitious. Candidates are string theory, loop quantum gravity, or then other roads among them refinements of existing theories or then a completely different approach. Also not excluded is that there is no such theory: who has proven after all that different theories can always be combined? Like the following picture of a chimera, it might remain fantasy (or then a result of a cruel surgery).

Image source: I don’t know who is the artist. There are hundreds of versions of this picture on the web, like here.

Occam’s razor

Like the antropic principle, also Occam’s razor is a meta guide in science which can be used as a guide but not be taken too seriously. It can also be used when searching for a unified theory. Like any dogma or rule of thumb. the razor can lead to poor or wrong conclusions (you can also cut your self with a razor). Albert Einstein once said that a theory can be right or wrong but that a model has a third possibility: it can be irrelevant. Meta principles like Occam’s razor are based on intuition and not based on reason. Lisa Randall once pointed out that there is no theorem which assures us that the basic laws of nature are simple. Maybe there is no simple theory. More likely is that our mathematics is not ready to appreciate the simplicity. A Greek mathematician like Archimedes would probably find our modern mathematics very strange (especially modern algebraic geometry).

Still, there are some indications from history that simplicity is a good guide. It is a healthy attitude to at least pretend there is always a simple solution). Of course, in general, the principle fails because simple solutions often have handicaps and discard one or the other aspect. Simple models often neglect important parts. Simple algorithms do not always lead to the most efficient solution for example. But the principle “simplicity, generality, clarity” is entrenched in the computer science philosophy and well alive even decades after it has been coined. Even so we know that optimal computer code is not always elegant by nature, the principle is still helpful as elegant simple code is maybe not the most efficient one, but easier to understand and proven to be error free. Finally, what is “simple” is relative. The Greeks thought that integers and circles are simple and irrational numbers or ellipses are complicated. We today know that complex numbers are much more natural than integers as experience has taught us that the natural numbers are the most complex ones.

[Added June 12, 2017: just an episode from my undergraduate years which had been traumatic but eventually very helpful. All mathematics students had to take a course analogue to CS 50 at Harvard which involved a lot of programming in Pascal. It was time consuming (for me even to such an extent that I neglected other courses as I was almost day and night programming on those old apple computers). Anyway, the course featured like all other courses problem sessions in which programming techniques were discussed. In one of those seminars, my own code was presented as an example “NOT TO PROGRAM!”. Indeed, I had optimized the code and dealt with special cases separately, which made the program ugly, inelegant and hard to read. But I believe it was very fast. It illustrates that there is always a fight between speed and beauty. I think now that the course assistant was right. Its better to be more clear than to be more optimal. One reason is that it often is even hard to read your own code, especially if it was optimized. Simplicity helps. And this is definitely also true in physics. End of addition.]

But lets look at one of the simplest equations in physics: $L v = \lambda v$ . This is the eigenvalue equation. Even so it looks simple enough, it explains the hydrogen atom and so the periodic system of elements. Eigenvalue equations are everywhere. Quantum mechanics is full of eigenvalue problems. The Wheeler deWitt equation is an eigenvalue equation. It was an early attempt to quantum gravity by bending everything to an eigenvalue problem.

Knowing that the Hydrogen atom is just an eigenvalue problem, one can try to do this for Newtonian gravity. The differential equations are nonlinear, but there is an underlying equation which is linear. It is the Gauss equation $\Delta V = \rho, F=dV$ , where $\Delta$ is the Laplacian. It gives the force $F$ from the mass distribution $\rho$ . The nonlinear Hamiltonian dynamics of a particle moving in the potential $V$ gives then the motion of particles. But now since every mass point moves, also $\rho$ moves. If $\rho$ is a smooth mass distribution, the n-body dynamics needs to be generalized to a Vlasov dynamics, where masses are not located on Dirac measures but on smooth measures. (A An overview over Vlasov dynamics [PDF]). It is a natural generalization of the n-body problem to a larger class of systems.

A bit more risky and radical approach of Wheeler-de Witt type is to assume that the potential $V$ is proportional to the mass density $\rho$ so that we just look for an eigenvector of the operator $L$ like in the Wheeler de Witt equation. But what operator L should we take? Since the density $\rho$ is part of the system, there is a gravitational self-interaction. We can model this interaction with the potential $V = L^{-1} \rho$ . We see that in such an eigenvalue problem, the distribution of mass and the Potential defined by the mass are linked.

Also in general relativity, we have to look at similar equilibrium problems. The energy momentum tensor must be invariant under the geodesic flow dynamics given in the metric defined by $\rho$ . Can this nonlinear complicated PDE problem be written as a linear equation? This is not an entirely stupid question as quantum mechanics, in a semiclassical limit produces classical nonlinear dynamics. Nature seems to like linearity. Now again, it is easy to speculate and much harder to come up with a linear system which models GR. It probably does not exist, as the statement “Nature likes linearity” could be wrong.

The Kepler test

A concrete goal is that QG needs to compute the motion of n-body system of black holes as effectively as RG, compute problems in engineering like satellite trajectories as effectively as CM and reveal the features of a quantum mechanical n-body system as effectively as QM. Most importantly, as pointed out in this exposition, QG needs to match experiments and predict outcomes which can not be obtained by other theories. A complicated theory which only can get results which other theories can do will end up on the garbage hill of science. Quantitative measurements on Kepler problems have lead to spectacular successes in the past: Tycho Brahe and Kepler experimentally got to laws which led to Newtonian classical celestial mechanics, general relativity was confirmed by perihelon advancement effects, quantum mechanics by explaining the periodic system of elements from the Hydrogen 2 body problem. Gravitational waves shed light on binary systems in the macro scale. Hossenfelder is a Baconian voice argues against a wave of Descartian assaults on QG: String theory, loop quantum gravity, causal dynamical triangulation, asymptotically safe gravity or causal sets etc. But Hossenfelder rightly points out: “We neither have a theory nor data!” Indeed: “This is not a plan. It is barely a concept” (the last sentence is a quote from the Guardians).
See the clip.

The Kepler problem on the micro cosmos scale is the Hydrogen atom $\Delta + 1/r$ . It is the “planetary Kepler problem” even so the force is not the gravitational force. We see it as part of the Meso scale as it uses standard classical mechanical frame work. The analogue problem in the Macro cosmos is the motion of two black holes. The hydrogen atom and the Kepler problem are both perfectly well understood, the 2-body problem in general relativity on the other hand is not understood at all. We don’t even know whether solutions exist. A binary system will eventually collapse leaving a single black hole. Independently from that (this is not part of the Kepler problem), black holes radiate and eventually disappear, a process definitely needing quantum gravity to be understood. The bodies move on geodesics of a metric which is determined by the path. There are gravitational waves. It is a complicated infinite dimensional Hamiltonian system coupled to finite dimensional geodesic motion. The coupling is both ways. I’m not even aware of an existence theorem in GR for a 2 body problem even if the pseudo Riemannian manifold M is $R^4$ . Mathematical existence theorems in GR are very difficult and usually assume very smooth and small initial data. We look for a pseudo Riemannian metric $g$ satisfying the Einstein equations $E=T$ for a mass distribution located on two curves such that each of the curves is a geodesic $r_i(t)$ in the metric g. If at some time t, one of the curves is within the Schwarzschild radius of the other, then the bodies have merged. The energy-momentum tensor T in the solution of the two body problem is a Schwarz distribution located on some curves which collapse. The metric $g$ is complicated and includes the propagation of gravitational waves. Unlike the planetary classical Kepler or the Hydrogen atom, the system is linked to the rest of the space-time manifold.

Hydrogen Atom	Kepler Problem	Binary Blackhole
$(L + 1/r) u = \lambda u$	$Lu=\rho$ $r''=-r/\|r\|^3$	$r''^k=\Gamma_{ij}^k r'^i r'^j$ $R_{ij} - R g_{ij}/2 = T$
PDE eigenvalue problem	PDE, ODE for point particle	Singular integro PDE
Electron does not affect nucleus, no virtual particles, no Zitterbewegung.	No tidal forces, no relativistic effects	Post Newtonian numerical schemes only

Scales and Discreteness

Whoever learns about atoms and the planetary systems in elementary school is intrigued by the similarity of the systems and probably, as many kids do, makes the obvious association and plays with the idea that the atom itself is a planetary system. Similarly, our planetary system can be imagined to be an atom in the Meso scale of some larger world. Such associations (or better short circuits triggered by naivity) are very old. They can be traced back go Greek mythology or even earlier when celestial objects had god like status. Kids can up with of perpetual motion machines and reamed about mechanisms to prevent potential energy to pass to kinetic energy allowing super man (or wonder women) like flying. These fantasies of course die quickly with more knowledge. Still, the problem of QR is exactly to understand such comic-book musings. They bridge physics from our Meso scale both to the very small and to the very large. Also quite many SciFi stories deal with this. There are stories (like Egan’s Schild’s ladder in which the “universe” is a discrete graph. There is the more recent movie “Ant man” in which the author of the comic has imagined allowing a person to enter the nano scale. The fantasies of this kind are very old, just think of the story of Aladdin, one of the 1001 stories. In parallel, many folk tales in the west deal with giants (also a concept in which scales can be bent).

Discrete geometries are tempting as the mathematics is much simpler and technical difficulties disappear. Many have proposed discrete structures. Even the Greeks starting with Democritus and put into literature by Lukretius. (Lukrez and especially the work “De Rerum Natura” was my final Matura (high school diploma) project after learning 6 years of Latin). Fascinating is also the idea to replace the target space of field theories with discrete spaces. With translational invariant local rules, a theorem of Hedlund assures that one deals then with cellular automata. Polymath Steven Wolfram wrote ones an entire opus about the idea of having cellular automata as objects of fundamental importance (also here, a concrete challenge of that theory is “describe the Kepler problem in that frame work!) Interesting musings come from serious science fiction writers like Greg Egan. Discrete structures have long been used as regularizations of quantum field theories. Examples are lattice gauge theories. In mathematical physics, discretization of space and time is entrenched. Lattice gas theories in statistical mechanics for example approach the continuum using thermodynamic limits. In solid state physics, one has the tight binding approximations, Hueckel theory in mathematical chemistry, random walk approximations of continuous time stochastic processes. As also often pointed out and also mentioned in this Hossenfelder blog, a lattice is hardly a good model for fundamental physics. It breaks too much symmetry. (I myself am currently intrigued by the possibility of Barycentric refinement). What happens there is that in the limit, isotropy and Euclidean symmetries are is somehow restored as the vertex degrees of the discrete structure become more and more refined too allowing for finer and finer rotational motions).

Continuous symmetries which are crucial to solve classical problems are not available. Solving the Kepler problem or Hydrogen atom or binary Black hole problem using discrete structures has not been done using discrete structures. Exceptions of course are numerical schemes done on a computer, which by definition is a finite system. But such simulations just try to compute numerically the continuum model. The continuum limit can always be taken as a Barycentric limit or a sequence of triangulations of space. But the Barycentric limit is an almost periodic space unfamiliar to the language of the Kepler problems. In one dimensions, the limiting space is the dyadic group of integers, a wonderful compact metric space with a translation group having a smallest unit. (See the last part of these slides). While the role of discrete structures in the continuum are not clear yet, one should look at them more carefully.

Connections between the discrete and continuum

There are strong formal connection between the discrete and continuum. This is what quantum calculus is about. Examples of fundamental connections between the discrete and continuum are: numerical approximations, discrete ultraviolet regularizations, the Barycentric limit.
An ultraviolet problem is to understand the energy $\int_{R^3} E^2/2 dV$ of an electron. In this paragraph, I follow this entry: if we integrate from $\rho=r_e$ to $\infty$ , we get with $E=q/(4\pi \rho^2)$ the mass-energy $q^2/(8\pi r_e)$ , which diverges. The classical electron radius is then defined so that the mass is the electron mass $m_e$ . This gives $r_3 = e^2/(4 \pi \epsilon_0 m_e c^2) = \alpha \hbar/(m_e c)$ which is $2.8 10^{-15} cm =2.8 Fermi$ , where $\alpha=1/137$ is the fine structure constant and $\hbar/(m_e c)$ is the Compton wave length of the electron. t Hooft tells in his book “In Search of the Ultimate Building Blocks” that if we hit upon an obstacle, even if it looks like a pure formality or just a technical complication, it should be carefully scrutinized. Nature might be telling us something, and we should find out what it is.

In this context, the Energy theorem is an interesting metaphor: the total energy of a finite abstract simplicial complex is the Euler characteristic. This result bridges some topology with a notion of energy (and so mass as energy is always tied to mass). The energy theorem means that geometry itself is energy and so mass. Mathematically, energy Euler characteristic and spectra are all multiplicative. Lets keep time an independent entity and consider the Helmholtz Hamiltonian system $\psi' = i H_{\psi}$ , where $H=(\psi, g \psi) - T S(|\psi|^2)$ . This nonlinear system is expected to feature solitons and describes gravity waves near equilibria. Additionally, we can look at an isospectral deformation of the Dirac operator leading to additional particles and forces. This evolution also deforms the connection Laplacian and so produces a deformation of the interaction geometry. One can then try to postulate that space is given as a probability space on the unit sphere of the Hilbert space defined by a simplicial complex and that this measure minimizes the free energy. We can now look at a concrete Hamiltonian system and hope to get some physics. Motivated from the quantum n-body problem, we can also look at the functional $H=\sum_{x,y} L(x,y) \psi(x) \psi(y) - \sum_{x,y} L^{-1}(x,y) \psi(x) \psi(y)$ , where the first part is the kinetic energy and where the second part is the potential energy. This is a quadratic functional on functions in finite dimensions. When restricting to a Lagrange constraint $\sum_x \psi(x) \psi(y) = 1$ , we are led to the Lagrange equations $( L - L^{-1} )\psi = \lambda \psi$ meaning that we look for eigenvalues of the operator $L-L^{-1}$ . We like that operator because it has mathematical properties: its trace is a functional which is the sum of the Euler characteristics of the unit spheres.

Pushing the Bohr model

The hydrogen atom is the quantum Kepler problem with Hamiltonian $\Delta + V$ . It is only a Kepler problem by analogy, as the force is not the gravitational force, so that there could be just mathematical affinity. This operator is defined by geometry alone as the Green function has the kernel $V(y) = 1/|x-y|= g(x,y)$ . We can write $H$ as $H(f) = -L f + V(y) f(y)$ . We have $L = \frac{\hbar^2}{2m} \Delta$ and $V(y) = -\frac{e^2}{4\pi \epsilon_0 |y|}$ . In spherical coordinates (r,t,s), the eigenfunctions factor as $R_{nl}(r) Y_{lm}(t,s) = R(r) T(t) S(s)$ with $T(t) = exp(i m t)$ . The functions $Y_{l,m}(t,s)$ are spherical harmonics, where m=-l, … ,l. The eigenvalues of H are $-h c R_{\infty}/n^2$ , where $R_{\infty} = m_e e^4/(8 \epsilon_0 h^3 c)$ is the Rydberg constant. This is a situation, where we have a single charge fixed at a point. It is assumed not to be influenced by the electrons around it nor do the electrons influence each other. A refined picture leads to refinements like the Lamb shift due to the quantization of the electric field or the hyperfine structure due to dipole moment interactions.
Now, if we have a Laplacian and a point, the analogue operator is $H(u)(y) = L u(y) + g(x,y) u(y)$ . It describes the situation, where we have a single particle at a point. If we have particles everywhere, we get $V(x) = \sum_y g(x,y)$ . In the case of the connection Laplacian, this is the curvature at a point. What is the ground state for such an operator? What is the spectrum? It is defined by the geometry alone.

One can look at the Hydrogen atom in other spaces. On the real line, it is $(1/x^2) f'' + 1/x = E f$ . This is a problem for cookbook: the Bohr quantization condition (exmple). The actual Hydrogen atom in one dimension is however $f''(x)+|x| = E f$ as the natural Laplacian in one dimensions has the Green function |x-y|.

The gravitational field also can be described by $L V = \rho$ , where $\rho$ is the mass distribution. As already mentioned, a radical approach in the Wheeler de Witt spirit is to assume that the mass is proportional to the potential. This gives an eigenvalue problem $L V = \lambda V$ . In other words, the gravitational potential is an eigenfunction. This suggests to look for solutions of the Newtonian problem in the potential of an eigenfunction. Can one look at Kepler type problems in the discrete? The ordinary differential equation of the Kepler problem might have to be replaced with a discrete time dynamics or then a deformation in a more general space of geometries. What we need is a rule telling that a pair of edges $(e,f) = (a,x,b)$ is part of a trajectory. Somehow, we need this to depend on g(x,x) the potential. What about minimizing a path for the functional $|\gamma| + \sum_x g(x,x)$ or more simply $\sum g(x_i,x_{i+1}) + \sum L(x_i,x_{i+1}) +g(x_i,x_i) + L(x_i,x_i)$ . Already unclear is whether there is a unique solution. For the binary black hole problem, the only thing which appears reasonable is solving a global variational problem, where both matter (the energy momentum tensor and geometry (the metric) are subject to perturbations.

Basic physics problems

Finally, lets look at some of the fundamental problems of physics. The following list is hardly original as the problems are widely known. But I include connections to my own interests (which are constantly changing). Basic physics questions are a strong motivator for mathematics even if the mathematics turns out to be irrelevant for physics. Also, one has always to be reminded that speculation is easy and ideas are very cheap (we have too many ideas in physics which do not work, the reason being that coming up with concepts IS easy. It is the “following through” and the matching with experimental data (and throwing away of garbage) which is difficult.

Is there an explicit solution to the Kepler problem in general relativity, maybe as a limiting case of solutions to classical variational problems? My own take is to compare that problem with the Navier stokes equations which is a model for fluid dynamics. That problem might not have a global solution (a Millenium problem), but this might just show that the model is pushed to situations, where it does not apply any more as a model to physics. In reality, a fluid is not a smooth function after all as a fluid is a finite particle dynamical system. Similarly, the Kepler problem in general relativity might not be relevant any more in extreme cases, especially if we deal with colliding black holes or evaporating black holes. These are situations, where the classical theory of general relativity most likely reaches its limits and becomes fundamentally wrong. Still, the mathematical question can be asked, similarly as one can ask about the almost everywhere existence theorem in the Newtonian n-body problem (also a notorious difficult open problem [and one of the 15 open problems of Barry Simon] as one does not know about the size of the non-collision singularities. Also these non-collision singularities, particles going to infinity in finite time are of course not realistic as relativistic considerations will kick in before). The problem of global existence of the GR 2-body problem is probably one of the most concrete problems and also one of the more accessible ones as LIGO experiments allow to test ideas and more importantly remove irrelevant models.
Can the general Kepler problem be approached using pure geometry? The idea would be that geometry alone defines mass (this is not the take of general relativity, where an energy momentum tensor is taken). Still, one can turn things around and take a manifold and assume that the Einstein tensor defines (via the Einstein equations) an energy momentum tensor. What does that mean in the Kepler case: the two mass points are implemented as a concrete geometry which can be modeled by simplicial complex and a wave function on the complete graph which is zero on elements which are not evolved. Now let the geometry evolve, for example using the isospectral deformation of the exterior derivative. How do the two bodies move? This could be studied experimentally on a computer.
Explain dark matter. There is overwhelming indirect observational evidence. After some speculative appearances of the word “dark, matter”, it was Fritz Zwicky who first suggested the term after basing it on experimental evidence. The importance of Zwicky is sometimes overlooked. It was Zwicky who discovered dark matter. Previous guesses were lucky punches not based on evidence and should not be given so much weight. (I could randomly generate a few billion theories with the help of a computer and probably get something right when checking back in 100 years). Related to dark matter is phantom energy, dark energy with negative kinetic energy. See Image credit. My take is a mathematical one: dark matter could be related to isospectral deformation of the exterior derivative as we have no physical explanation for the diagonal blocks of that Laplacian.
Explain inflation an exponential expansion in the early universe in the first 10^-36 seconds. Inflation explains the large-scale structure of the cosmos. A particle physics explanation is not known. My take is again mathematical: isospectral deformation of the exterior derivative produces inflation. This is an analogy as in mathematics we can prove that any Riemannian manifold or discrete graph expands, if we let the exterior derivative evolve freely on the isospectral set, then this expansion happens with an exponential fast bump (inflation). Whether this is physically relevant is unclear but the mathematics is there and run on the computer for small networks.
Is there are grand unified theory GUT, in which electromagnetic, weak and strong interactions merge into a single force? Candidates are the Georgi-Lashow model based on the fact that SU(5) contains the gauge groups U(1),SU(2) and SU(3) or the Pati-Salam model based on SU(4) x SU(2) x SU(2). My own take is that there are some mathematical theorems on division algebras which show that the gauge groups of the Standard model could be seen as natural. See my particles and primes paper. Not that I think that the analogies of “primes in associative division algebras” and particles are physically relevant (physics is about experiments after all and not about affinities or speculation), but the affinity shows that the structure of the gauge groups is not that arbitrary. The 3D-sphere, (which is the space of unit quaternions), as well as the unit circle are the only Euclidean spheres which carry a Lie group (this ia a very beautiful explnation why U(1) and SU(2) are natural). Now, the SO(3) symmetry is of an other kind and can be implemented as a symmetry related to the permutations of the three basic non-real coordinate axes of the quaternions. Much more difficult than the existence of GUT is the existence of TOE, a theory of everything in which also gravity is included. Related is the problem of magnetic monopoles or the problem of explaining neutrino oscillations or proton decay for which there is no evidence. My own guess is that this requires new mathematics and more importantly more work in areas where experiments can be done. For example in the case of gravity with very small masses (which is kind of the opposite of LIGO, but the technologies could be related as both require extremely sensitive detectors).
The Baryon problem asks why there is less anti-matter than matter in the universe. Are there antimatter clouds in the galactic center or other parts of the universe? Are there anti-matter comets. My take: one can ponder also the question why the orientation of the DNA double helix or why most people have the heart on the left side. But there could be a fundamental asymmetry which could be explained. We just don’t know how. The connection Laplacian for example has no symmetry $\sigma(L) = \sigma(-L)$ .
Are the super partners of the known particles? It could explain the hierarchy problem, the large discrepancy between the forces. The weak force is $10^{24}$ times stronger than gravity. My take: supersymmetry is hidden by isospectral deformation. The sectors are so close that they appear to be the same particle. Additionally, there is a natural Laplacian, the connection Laplacian which does not feature super symmetry. (Why is it interesting? Because it is always invertible.) If the connection Laplacian is related to gravitational energy, then it could be related to a lack of super symmetry as well as the broken matter-antimatter symmetry.