Thread by @getjonwithit

Shock waves, solitons and spacetime singularities: what do sonic booms and water ripples have to do with cosmic censorship and the determinism (or otherwise) of general relativity?
Brace for a brief adventure in hyperbolic partial differential equations (PDEs). (1/20)

PDEs may be broadly classified into three types: elliptic, which are time-independent and where information propagation is instantaneous; parabolic, which are time-dependent and diffusive; and hyperbolic, which are time-dependent and non-diffusive (i.e. wavelike). (2/20)

Hyperbolic PDEs are quite special, and can be analyzed using the method of “characteristics”, in which the PDE is reduced to a family of ordinary differential equations (ODEs), each of which is associated to a path through spacetime called a "characteristic line/curve". (3/20)

These characteristic curves determine the flow of information through the domain of the PDE - when they get closer together, this indicates a compression wave; when they spread out, this indicates a rarefaction wave. But when they *intersect*, that's where the fun starts. (4/20)

When two characteristic curves intersect, the solution becomes multi-valued at that point. The PDE no longer tells us uniquely how to evolve the solution from that point onwards, so we say that it has become "ill-posed" (i.e. it is no longer deterministic/well-posed). (5/20)

In some cases (e.g. for waves in granular materials), this non-determinism/multi-valuedness is tolerable. But in others (e.g. in electromagnetism, or inviscid fluid mechanics), it's not. But here there's an old trick: just integrate the PDE over a finite spacetime volume. (6/20)

This gives rise to the "weak" or "integral" form of the PDE, and the behaviour of the PDE at ill-posed points can be described in terms of solutions to this form (so-called "weak" or "nonlinear" solutions). Weak/nonlinear solutions form a superset of standard solutions. (7/20)

These weak solutions typically take the form of distributions, like delta functions: objects which are not themselves classical functions (hence not solutions to the original PDE), but which can be integrated to yield functions (hence solutions to the weak/integral form). (8/20)

In inviscid fluid mechanics, nonlinear solutions tend to be shock waves - discontinuous pressure waves propagating faster than the local speed of sound. For other PDE systems, like in electromagnetism, they may form solitons - stable, self-sustaining wave packets. (9/20)

In general, whether a nonlinear solution becomes a shock wave or a soliton depends upon whether dispersive or dissipative effects dominate, and hence whether the PDE acts more like the inviscid Burgers' equation (shocks) or the Korteweg-de Vries equation (solitons). (10/20)

What does any of this have to do with black holes? The Einstein field equations are a mixed hyperbolic-elliptic system, but using the so-called ADM formalism we can split these 10 mixed equations into 6 hyperbolic "evolution" equations and 4 elliptic "gauge" equations. (11/20)

In so doing, we "foliate" our spacetime into a time-ordered sequence of spacelike hypersurfaces (Cauchy surfaces). The characteristic curves of the hyperbolic evolution equations are then just the normal (perpendicular) curves to these hypersurfaces. (12/20)

Under gravitational collapse, these characteristics get closer together. If the collapse is strong enough, they intersect (as do the corresponding hypersurfaces), and so the evolution equations become ill-posed, and our spacetime ceases to be "globally hyperbolic". (13/20)

Physically, this corresponds to the formation of a singularity - the relativistic analogue of a shock wave or soliton. In fact, we can say that many black hole metrics (such as Schwarzschild for a static black hole or Kerr for a spinning one) are "gravitational solitons". (14/20)

Formally, this means that they can be generated by the Belinski-Zakharov (inverse) transform. Informally, it means that they are self-sustaining objects of "pure gravity" (i.e. they are vacuum solutions, sustained solely by their own gravitational field). (15/20)

Incidentally, this explains why such black hole metrics are, paradoxically, vacuum solutions: a Schwarzschild black hole has "mass", yet its Ricci tensor (and hence its energy-momentum) is zero everywhere, except at the r=0 singularity, where it is undefined. (16/20)

But now we see that this is because it's acting just like a distribution or delta function: it may be undefined at r=0 and zero everywhere else, but it can still be integrated over a spacetime volume to yield a finite result (e.g. the ADM or Bondi mass of the black hole). (17/20)

The question of whether the Einstein field equations remain well-posed (and hence deterministic) for "physically reasonable" initial conditions remains open: that's the Strong Cosmic Censorship conjecture. Yes, we know that singularities are a robust prediction of GR... (18/20)

...as per the Penrose-Hawking singularity theorems, yet, for as long as they remain "cloaked" behind event horizons, they can't affect the hyperbolicity/well-posedness of the external spacetime: that hope of "cloaking" is the Weak Cosmic Censorship conjecture. (19/20)

Yet these lofty questions about the formation of black holes and the determinism of GR are ultimately grounded in exactly the same mathematics that governs the formation of shock waves around fighter jets, or the formation of ripples around canal boats. (20/20)