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Recently, I bumped into this interesting comment by Valter Moretti which made me wonder about the following, more general question (to which I suspect the answer is affirmative):

Can we easily tell, just from the operators appearing in a differential equation, whether the solutions to this differential equation will turn out to violate causality?

This question goes in much the same direction, although it is restricted to the heat equation. In the comments to joshphysics' answer to that question, there is also some mention of the notion of hyperbolicity of a partial differential equation. I've heard things like hyperbolicity, ellipticity, etc. before in the context of causality, and the Wikipedia page on hyperbolic PDE's mentions something along these lines as well:

If a disturbance is made in the initial data of a hyperbolic differential equation, then not every point of space feels the disturbance at once. Relative to a fixed time coordinate, disturbances have a finite propagation speed.

So perhaps the question can be reframed as follows:

Can we classify (a reasonably broad class of) PDE's in terms of the causality properties of their solutions? Does this classification involve "hyperbolicity" and related notions, and why (perhaps there is an intuitive, or physical, picture associated with them)? If yes, is there an easy way to see whether any given PDE belongs in one of those categories?

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  • $\begingroup$ In order for this question to make any sense, the differential equation has to be defined in terms of the coordinates of a space where some notion of causality can be defined. So I'd point out you're already dealing with only a subset of diffeq's, dependent on the underlying space. Still, interesting question. $\endgroup$ – David Z Oct 22 '15 at 12:53
  • $\begingroup$ @DavidZ Of course, I envisioned all of this as taking place in the context of spacetime. I might edit the question later to make this more explicit. $\endgroup$ – Danu Oct 22 '15 at 13:13
  • $\begingroup$ here is a nice article analizing the causality and predictive power of dynamical equations space.mit.edu/home/tegmark/dimensions.pdf $\endgroup$ – user83548 Oct 22 '15 at 16:03
  • $\begingroup$ I believe that the case of linear and quasi-linear PDEs is trivially categorized in this sense because you obtain the local propagation speed in terms of the "speed" of the characteristic at the given point. The non-linear case would be complicated even though you can read in Wald or Hawking and Ellis how this is handled in relativity (by transforming into a quasilinear system, tadah!). $\endgroup$ – Void Oct 22 '15 at 17:11
  • $\begingroup$ @Void Can you elaborate? That method only works for hyperbolic PDE's, or not? $\endgroup$ – Thibaut Demaerel Nov 1 '16 at 15:56
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I would say first that the question that is being asked here is a very deep one, and considerable care is warranted when trying to answer it. In that vein, we have to define the meaning of the terms we are using carefully. Category errors abound in this area. So, let's first be clear that the term "causality" refers to properties of physical processes, that take place in space and (crucially) time. Solutions of mathematical equations, on the other hand, are not physical processes so we need to make sure we are clear about what we mean by "causality properties" of solutions to (differential equations). This may sound like I'm needlessly splitting hairs, but here are some examples:

Example I: Projectile Motion:

Let's say we consider the motion of a projectile, which we could describe by a certain ordinary differential operator. Following my earlier comment, it is now important to note that we are going to solve an initial/boundary value problem, not an operator. As an analogy from linear algebra, the latter would be like saying "we are solving a coefficient matrix" rather than "we are solving a linear system of equations". Obviously, the former makes no sense. Now, going back to our projectile motion, a standard way to pose such a problem would be to provide initial values for the initial position and velocity of the projectile, and then solve the second-order ODE. Doing so seems to correspond closely to a causal process, where the progress of the projectile is causally determined from its prior state.

However, notice that we could pose a mathematically different problem, by providing the positions of the projectile at two points in time. Mathematically we would then be solving a boundary value problem, and the trajectory of the projectile would follow from a process that does not appear "causal" in the above sense at all, in that its details depend on boundary data at two end points. Yes, one might argue that this way to phrase the problem is somewhat "artificial", but the point is that we were dealing with the exact same differential operator, and may obtain the exact same solution.

Example II: Planetary Motion:

My next example is cribbed from the second Messenger Lecture Feyman gave at Cornell University in 1964, on "The Relation of Mathematics to Physics". In the lecture, Feynman discusses the problem of planetary motion. Using Newtonian Mechanics, we end up with an initial value problem to define the planet's trajectory. We could see the motion of the planet again as a process where the gravitational forces cause the planet to move in a certain way. At each step, the central force will "bend" the path of the planet a little so that eventually we end up with an ellipse. However, Feynman points out that we could use a radically different mathematical formulation of the exact same physics based on a variational principle. Such a formulation does not contain the concept of causality in an obvious way (I would argue, in any way at all). In this case the equations are different, but the solution is exactly the same.

And of course, we could instead move to the more advanced physical model of general relativity, in which case causality might again disappear when we speak of the planet simply following geodesics in a curved space-time.

Partial differential equations:

So let's talk about partial differential equations now. Just as for ODE's, and actually much more so, it is possible to formulate equivalent problems in very different ways. I could solve a problem described by an elliptic operator (which would typically be considered an "a-causal operator", I think) by providing boundary conditions that look like the ones for a hyperbolic operator, and my solution process could look like a "causal process". It is also worth pointing out that much more complex types of boundary conditions can and do appear e.g. in certain formulations of the fundamental equations of fluid flow (for example, a vorticity transport formulation of the Navier-Stokes equations for incompressible flow naturally requires boundary conditions in an integral form).

To summarize, I would say that I am quite doubtful that it is possible to classify differential operators as being "causal" or "having causal solutions". Based on what I said above one might be tempted to augment this notion by classifying, say, initial/boundary value problems (so, operator plus initial/boundary conditions) as having "causal solutions". However, I am very skeptical as to what such a classification would, or even could accomplish.

Finally, I have come to feel that the concept of causality in general may be problematic in principle, see John Norton's paper on "Causation as Folk Science". As a matter of fact, after all these years I am now less certain of my understanding of the concept of causality than I may have ever been. I want to emphasize that I offer the above more as food for thought and further discussion than a formal answer. The way I have come to know this forum, people may well complain that the above isn't really an answer at all. And they may be right...

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You already got it, I guess. Hyperbolicity is the property you are looking for.

The wikipedia article is very good: https://en.wikipedia.org/wiki/Hyperbolic_partial_differential_equation

As Void said in an short answer below your question, for linear PDEs it is trivially categorized, since every disturbance travels along characteristics. So if you can write down your PDE in first oder state form, find the eigenvalues of the Jacobean matrix A to be real, then characteristics exist. I think it makes no sense to write down the equations here again, since wikipedia has it very accurately. You also asked for an intuitive picture: Wave fronts propagate along these characteristics with finite speed. A smooth coverage of the x,t-plane with characteristics is possible and every disturbance is carried along these characteristics. Therefore two points are only causal if they are within cones of characteristics. In addition the solution can be written down using only ODEs on the characteristics (see method of characteristics). As already said it becomes more complicated if dealing with nonlinear PDEs. Then characteristics cannot be used anymore in the above sense. But you are asking for causlity and if disturbances can capture the whole space in finite time, which is non-causal. In nonlinear systems even the fastest wave speed is not fixed, but depends on something, for example the amplitude. So if the amplitude is limited, then the fastest wave is limited in speed also.

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