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 . 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 . It requires that given two future-directed, time-like vector fields , the scalar quantity be non-negative, and in the case where , 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 is treated to mean the flux in direction of the energy measured by an observer moving with velocity . 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 be a space-like hypersurface in space-time, and let be a region. If satisfies the dominant energy condition and vanishes on , then must also vanish on the space-time region composed of all points with the property that any time-like curve emanating from must intersect exactly once. is called the domain of dependence of .
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!
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.
- 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.)
- The reasonable solution is unique: the same initial condition cannot lead to two different futures.
- 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 . Suppose we know all the diagonal elements of are positive. Then if we know is a diagonal matrix, we have that must be positive definite (this is the semi-linear case). But in general, the off-diagonal terms can be so bad that 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.