… Data aequatione quotcunque fluentes quantitates involvente fluxiones invenire et vice versa …

Category: Papers

Bubbles!

Joachim Krieger and I posted a new pre-print on the critical nonlinear wave equation. After close to four years of the existence of this blog I finally have a paper out that actually relates to the title of this blog! Considering that the paper itself is only ten pages long, I will just direct readers to the arXiv instead of writing more about it here

Non-existence of multiple-black-hole solutions that look locally like subextremal Kerr-Newman

Pin Yu and I just posted a paper on arXiv based on some joint work we started doing when we were both graduate students (and parts of which appeared in his dissertation). I think I did a decent job explaining the motivation in a mathematical way in the introduction to the paper, so I’ll motivate the problem here a little differently.

Ever since the discovery of the black hole solutions, there had been interest in whether static or stationary black hole solutions with more than one hole can exists in equilibrium. One of the earliest considerations was by Bach and Weyl in 1922, fairly soon after the discovery of the Schwarzschild solutions. In general it was concluded that a static configuration is impossible due to singularities forming between two holes. Roughly speaking, a black hole in vacuum spacetime is a strongly gravitating object, and gravity attracts. If two black holes were to form and were to be kept apart, an external force will be needed to hold on to them. This manifests in a singularity between the two black holes.

On the other hand, if we were to add electrical charge to the black holes, then two black holes with charges of the same sign stands a chance of being in equilibrium: the electromagnetic interaction between two like charges is repulsive, while gravity is attractive, so maybe they can cancel out! And indeed such a configuration is possible. Under what are now called Majumdar-Papapetrou solutions are precisely these types of balanced multiple black hole solutions. But the balance has to be precise! The electric charge has to be large enough to equal the gravitational mass. And this in turn requires that the black holes represent what are called “extremal black holes”.

For the stationary, as opposed to static, case, the situation is less clear. In the static case, the black holes must remain fixed in place. In the stationary case, we can allow the black holes to orbit each other. As we know well from our own solar system, orbiting systems can happen where static systems are disallowed. This is because of the “fictitious” centrifugal force which can balance out the gravitational attraction (or, more correctly stated in accordance to American high school physics curricula, the gravitational force provides the centripetal force for the bound orbit).

In our paper we show that orbiting systems (as opposed to inspiraling ones where the multiple black holes eventually crash into one another) cannot exist provided that outside the black hole things looks more or less like the situation with only one black hole. A posteriori, knowing that these systems cannot exist, we are justified in our inability to provide examples. But even a priori it is difficult to imagine a system which our assumptions do not immediately rule out.

• If we have two black holes with large mass placed close to each other, we’d expect there to be lots of gravitational interaction and the local geometry will be distorted heavily away from the single black hole Kerr Newman solution.
• If we have two black holes with significant mass place far from each other, we’d naively expect that the gravitational field from infinity sees a single body with the combined masses of the two black hole, but near one individual black hole we only see the gravitational effects from that one black hole. So intuitively a situation like this cannot have an exterior that looks everywhere just like that with one single black hole.

The situation that intuitively we cannot rule out here is the case where we start with one gigantic black hole and one tiny black hole. The effect of the tiny black hole on the gravitational field is but a mere blip compared to the big black hole. So we can say expect that the space-time metric looks like that of when there is just the one big black hole. This is the one to which our theorem applies.

Our theorem also says nothing about the extremal case. We only consider the case where the charge is insufficient to balance out the mass. As we know from the Majumdar-Papapetrou solutions we do have a need to ruling out that class of solutions. Where this restriction enters the proof, however, is in a cute way. First, let me describe the general strategy of the proof. Our method is roughly inspired by Morse theory. We construct a real valued function on the space-time such that its level surfaces foliate the space-time with homeomorphic leaves. We then show that the boundaries of the black holes (the event horizons) necessary are almost level surfaces for this real valued function. This will give rise to a contradiction: on the one hand near infinity, the level surfaces look like spheres and is a connected surface; the boundary of the black holes, together form a level surface that has two components. This contradicts the homeomorphism between level surfaces. In the actual argument, however, we do not prove homeomorphism. Instead, we show that the real valued function has no critical points. The contradiction then is provided by a mountain pass lemma. It is here we need to use subextremality.

We can only show that the real valued function has no critical points where its values are at least as large as the values on the event horizons (there is a sign issue). So to actually get a contradiction by the mountain pass lemma, we need that as we go out from the event horizon, the value of the real-valued function increases. For subextremal black holes, the event horizon is non-degenerate, and the near horizon geometry forces this to be true. For extremal black holes, the event horizon is degenerate, and the near horizon geometry allows the value of the function to remain the same or decrease.

Dominant energy condition versus hyperbolicity

I’ve posted a new paper to arXiv over the weekend. The goal of this paper is to clarify some misconceptions in the literature about the connection between the dominant energy condition of general relativity and “hyperbolicity” and domain of dependence properties of partial differential equations. (If you don’t know what hyperbolicity is, don’t worry. To paraphrase Jacques Hadamard: What a partial differential equation is, is well known. What hyperbolicity is, will be explained.) (This paper also solves a question that has been on my Questions and Answers page for a while.)

Dominant energy condition
Take Einstein’s equation $G_{\mu\nu} = 8\pi T_{\mu\nu}$. The left hand side is the Einstein tensor, composed from the Ricci tensor, the curvature scalar, and the metric tensor; it captures exactly the geometry of the space-time. The right hand side is the Einstein-Hilbert stress-energy, and captures the matter content of the universe. This equation connects matter to its effect on gravity.

The dominant energy condition is an assumption often made about the tensor $T_{\mu\nu}$. It requires that given two future-directed, time-like vector fields $X,Y$, the scalar quantity $T_{\mu\nu}X^\mu Y^\nu$ be non-negative, and in the case where $X = Y$, vanish only when the tensor itself vanishes identically. The usual interpretation (and in some cases, used as an a priori justification of the condition) is drawn from fluids and elasticity, and $T_{\mu\nu}X^\mu Y^\nu$ is treated to mean the flux in $X$ direction of the energy measured by an observer moving with velocity $Y$. That the quantity is always non-negative is supposed to reflect the fact that “energy cannot flow faster than the speed of gravity”.

The assumption is a very powerful one when it comes to dealing with the geometry of space-time. Many important theorems in general relativity about the global structure of space-time can be proven under the assumption of dominant energy. Prime among these theorems are probably the Singularity Theorem of Roger Penrose, and the Positive Mass Theorem first proved by Rick Schoen and S-T Yau, and later by Ed Witten using different methods. On the other hand, the condition has relatively little to say about the matter side of the equation. The only well-know result in this direction being a classical theorem of Stephen Hawking, which states the following:

Let $\Sigma$ be a space-like hypersurface in space-time, and let $U\subset \Sigma$ be a region. If $T_{\mu\nu}$ satisfies the dominant energy condition and vanishes on $U$, then $T$ must also vanish on the space-time region $D$ composed of all points $p$ with the property that any time-like curve emanating from $p$ must intersect $U$ exactly once. $D$ is called the domain of dependence of $U$.

Roughly speaking, this captures the causality of classical events. If energy cannot flow faster than the speed of gravity by the dominant energy condition, the edge of “pure vacuum” cannot recede faster than the speed of gravity either. However, this is about as much as the dominant energy condition can say: contrary to some suppositions, the dominant energy condition does nto guarantee causality be preserved in an absolute sense: it leaves a certain loophole. We can illustrate this by a bit of science fiction.

Hawking’s theorem can be re-interpreted as the following: a signal cannot penetrate into vacuum at a speed faster than that of gravity. But can we circumvent the theorem by forcing a medium? Imagine an infinitely rigid stick that reaches from the earth to the moon. Then if I push on one end of the stick here on earth, I should be able to poke the moon instantaneously, due to infinite rigidity of the stick. The usual resolution to this problem is that, in real life, there does not exist an infinitely rigid stick. That the stick will be somewhat elastic, and the motion of my push on one end can only propagate at the speed of sound inside the stick, which in general is slower than the speed of gravity. Many people assume that the dominant energy condition rules out the infinitely rigid stick; in the paper I show that this is not the case.

In particular, I show that relativistic models of fluids and elastic matter are perfectly happy to deal with tachyonic particles if one only impose the dominant energy condition. Then if we were somehow able to fill a tube with tachyonic fluid, the tube can be used for super-luminal information transfer!

Hyperbolicity
In the sense of Hadamard, an evolutionary partial differential equation is said to be hyperbolic (or locally well-posed) if it satisfies three conditions, given some initial conditions.

1. Any reasonable initial condition leads to a reasonable solution of the equations. (Reasonableness is a fairly relaxed, mathematical condition; it has little to do with whether the initial conditions are reasonable physically.)
2. The reasonable solution is unique: the same initial condition cannot lead to two different futures.
3. The solution is stable with respect to small errors. Two sets of initial data that differs by a tiny bit will, for some period of time (whose length is inversely proportional to the difference), give rise to similar solutions. This of course cannot be always true in the long run (think chaos theory), but the fact that it can be done for short periods of time is what allows us to, say, predict the weather for the next 5 days: small errors in our measurements and calculations will only propagate and lead to small errors for next week’s predictions. But those errors exponentially compound upon themselves if you try to make a prediction for a month.

One may say that Hadamard’s notion of hyperbolicity is what defines hyperbolic partial differential equations as the most useful type for physics.

There are various mathematically precise way of checking whether an equation is hyperbolic. (Almost all of these give sufficient conditions for hyperbolicity, but not necessary ones. So equations failing the “test” imposed by one method may yet be hyperbolic, just that it is sufficiently degenerate that the “test” used cannot discern it.) In the above paper I use the notion of regular hyperbolicity as described by Demetrios Christodoulou in his book The Action Principle and Partial Differential Equations. (I give a self-contained summary of the framework in the paper.)

It turns out that a sufficient condition for the hyperbolicity of a system of PDEs is, roughly speaking, a hierarchy of energy estimates. For linear equations, the hierarchy only needs one level. For non-linear equations, we need higher levels to control higher derivatives, which in turn by Sobolev embedding theorem allows us to control the non-linear terms. As it turns out, the dominant energy condition is, roughly speaking, equivalent to the existence of the first-level of this hierarchy of energy estimates. For linear equations, then, the dominant energy condition is sufficient to guarantee local well-posedness. Slightly less obvious is the fact that the same is true for semi-linear systems. The key idea is that the dominant energy condition is a condition on the form of the equation: it doesn’t really matter if what you “plug into” the stress-energy tensor is actually a solution. For semi-linear equations, this becomes a strong condition on the coefficients in the equations: the freedom in plugging in arbitrary data strongly constraint the coefficients. And since those coefficients also play a role in the higher order energies, we get the control we desired. So in this case, the dominant energy condition is actually a lot stronger than one may naively expect.

For quasi-linear equations, however, there is a problem. Since the coefficients now depend strongly on the data you plug-in, there is a chance that there is a conspiratorial miraculous cancellation! This can cause problems when we consider the linearised problem to compute the higher energy estimates. Another way to think about it is to separate the input for an equation into two parts. One part is to give some data, from which we derive the coefficients. The second part is to evaluate the energy for the data set using the (derived) coefficients. The semi-linear case corresponds the first step being trivial: regardless of what data we input we always get the same coefficients back. In the quasi-linear case, however, the dominant energy condition is a condition on the diagonal part of this process, the case when the data you input for the first and second steps are the same. In general, however, to establish the hierarchy of energy estimates, we need to consider cases where the data input for the first and second steps are different (this is to guarantee stability, the third of Hadamard’s conditions). Think about a square matrix $M$. Suppose we know all the diagonal elements of $M$ are positive. Then if we know $M$ is a diagonal matrix, we have that $M$ must be positive definite (this is the semi-linear case). But in general, the off-diagonal terms can be so bad that $M$ is in fact indefinite (this is the bad quasi-linear case).

In the paper I make explicit this difference between the dominant energy condition and hyperbolicity. In particular, I showed that the Skyrme model always obeys the dominant energy condition, but there are cases where it fails to be hyperbolic. In hindsight, this is perhaps not too surprising, as one can treat the Skyrme model as a model of relativistic elasticity. The hyperbolicity break-down corresponds to a strongly tachyonic regime, which for fluids (another, different, sub-case of elasticity) is also well-known to be non-hyperbolic.

Skyrmions are narcissistic

Gary Gibbons, Claude Warnick and I just announced a new paper, in which we prove that Skyrmions are narcissistic (this colourful name is due to Gary). Basically what we have demonstrated is a version of the rule well-known to tods, that opposites attract and likes repel. (In this case Skyrmions carry parity: so its reflection in a mirror becomes its anti-particle.) A more precise version of the statement is that “finite energy solutions to the Skyrme model in Minkowski space that are symmetric or anti-symmetric across a mirror plane cannot remain in static equilibria.”

Physically the intuition is simple: when something remains in static equilibrium, the total net force across any plane must vanish. This is just Newton’s second law where the action of a force will produce an acceleration. So to show that such static equlibria cannot exist, it suffices to show that any configurations of these kinds must have a net force across the mirror surface.

For linear theories (for which the particles obey the superposition principle), the situation is simple. The linearity essentially implies that there cannot be internal structure to the particles which can provide a counter-balance to the interactions between the particles. A simple way to look at this is to hold up one’s pinky and drop a bead of water on it. The top of your finger should be slightly curved, so parts of the bead of the water is sitting on a slope. But it doesn’t flow down hill! Why? This is because water has internal structure (hydrogen bonds, van der Waals forces, surface tension, etc.) that holds the small bead together. When the bead of water is small enough, the internal energy is enough to overcome gravity on a gentle slope to prevent the drop from breaking up and flowing off. Now if you take some other liquid, say rubbing alcohol, with much less surface tension, and you take the same volume of liquid and try to bead it on your pinky, you’d find it much more difficult.

In the linear theory, without the internal structure, each infinitesimal volume acts independently of other infinitesimal volumes. So for a macroscopic configuration to remain in equilibrium, it is necessary that the potential energy everywhere is constant. And for most field theories this implies that the only finite energy solution must have zero energy, and thus the solution itself is trivial. (This is a reflection of the fact that in linear theories one typically do not expect the existence of solitons.)

In nonlinear theory, however, the fields can have internal structure. From these internal attractions come the possibility of solitary states, solutions which are concentrated spatially. A striking example is the phenomenon of tidal bores. For small amplitude surface waves on water, the equation of motion is well approximated by the linear wave equation. Hence we see the waves disperse as it propagates outwards in rings. For larger amplitude water waves in a narrow and shallow channel, however, the equation of motion is better described by the Kortewag-de Vries equation, whose nonlinearity better models the internal structure of the wave. The tidal bores observed in nature are reproducible theoretically as a soliton solution to the KdV equation.

Now, the soliton solutions to KdV are necessarily traveling waves. However, for other equations that are used to model nature, soliton solutions can be stationary or even static. Some examples are given by the focusing nonlinear Schrodinger equation, focusing semi-linear wave equation, the Yang-Mills instantons, and, in this particular case we are considering, the Skyrmions. The Skyrmions are used in nuclear physics to model baryons. Their equation of motion also has a nonlinearity that captures the presence of internal structure. The question we are interested in then is whether two such baryons can remain in static equilibrium.

Now, in the case of gravity, two astronomical bodies cannot remain in static equilibrium: this is because gravity is a purely attractive force. But the interaction of Skyrmions, like interaction of magnets, can be either attractive or repulsive. So one may try to look for situations where the attractive force exactly counter balances the repulsive force. In the case of the Maxwell theory of electromagnetism, because the theory is purely linear, the only way for two bodies to have exactly counter-balancing forces is for them to be without electromagnetic charge. (Whenever there is a potential gradient the charges will flow to even out the electromagnetic potential.) In the case of a non-linear theory like Skyrmions, it is possible for an extended body to have “positively charged” and “negatively charged” regions that are held apart by the internal structure and do not immediately cancel each other out, unlike the case for classical electromagnetism. Then it is conceivable that certain configurations can exist in which two such extended bodies have their various regions aligned just right so that the net force between them is zero.

And such configurations do exist under the name of Sphalerons.

What we prove in this paper is that in other types of arrangements unlike that of sphalerons, we can mathematically rigorously show that the two bodies cannot be kept at equilibrium.

The trick is one about symmetries. For a scalar valued function, there are basically two symmetry types you can have after reflection across a plane: even or odd. For a vector valued function, however, there are more allowed symmetries. A symmetry compatible with the reflection across a plane is just any symmetry that, when you do it twice, you recover the identity (what we call an action of $\mathbb{Z}_2$). If a function takes vector values, then besides the simple symmetries like the identity $x\to x$ and the complete negation $x\to -x$, we can also have reflections across vector-subspaces in the target. The sphaleron solution is exactly one such: it has a symmetry that is a nontrivial reflection in the target.

In our paper, we show that the two simplest symmetry types (identity and total negation) cannot lead to static equilibria. The proof essentially boils down to a statement about the internal structure of the Skyrmions, that in these types of symmetries, the complicated “region-by-region” interactions that may allow to two bodies to remain in equilibrium in fact completely cancel each other. So for these types of symmetries the interaction between two bodies is dictated only by the “total charge” of the each of them. And thus we again have that the opposites attract…

And now for the mathematics. What we exploit in the Skyrme model is a manifestation of the dominant energy condition: we consider is the internal stress of the solution. In general, the stress can take arbitrary sign, as long as they average out to zero so there is no bulk motion. But by imposing a symmetry condition, we require that on the mirror surface the solution has either a Dirichlet (negation symmetry) or Neumann (identity symmetry) boundary. The Skyrme model, along with some other Lagrangian field theories, has the property that on such boundaries the stress has a sign (positive if Dirichlet; negative in Neumann). Now, since there is no bulk motion in a static solution, the total stress across the mirror surface cannot be anything but zero. This in fact forces the solution to have both a Dirichlet and a Neumann boundary condition. Using certain properties associated to the ellipticity of the static solution (either the maximum principle or strong unique continuation), we can then conclude that the solution must then vanish everywhere.