Albert Einstein’s revolutionary general theory of relativity describes gravity as a curvature in the fabric of spacetime. Mathematicians at University of California, Davis have come up with a new way to crinkle that fabric while pondering shockwaves.

“We show that spacetime cannot be locally flat at a point where two shockwaves collide,” says Blake Temple, professor of mathematics at UC Davis. “This is a new kind of singularity in general relativity.”

Temple and his collaborators study the mathematics of how shockwaves in a perfect fluid affect the curvature of spacetime. Their new models prove that singularities appear at the points where shock waves collide. Vogler’s mathematical models simulated two shockwaves colliding. Reintjes followed up with an analysis of the equations that describe what happens when the shockwaves cross. He dubbed the singularity created a “regularity singularity.”

“What is surprising,” Temple told Universe Today, “is that something as mundane as the interaction of waves could cause something as extreme as a spacetime singularity — albeit a very mild new kind of singularity. Also surprising is that they form in the most fundamental equations of Einstein’s theory of general relativity, the equations for a perfect fluid.”

The results are reported in two papers by Temple with graduate students Moritz Reintjes and Zeke Vogler in the journal Proceedings of the Royal Society A.

Einstein revolutionized modern physics with his general theory of relativity published in 1916. The theory in short describes space as a four-dimensional fabric that can be warped by energy and the flow of energy. Gravity shows itself as a curvature of this fabric. “The theory begins with the assumption that spacetime (a 4-dimensional surface, not 2 dimensional like a sphere), is also “locally flat,” Temple explains. “Reintjes’ theorem proves that at the point of shockwave interaction, it [spacetime] is too “crinkled” to be locally flat.”

We commonly think of a black hole as being a singularity which it is. But this is only part of the explanation. Inside a black hole, the curvature of spacetime becomes so steep and extreme that no energy, not even light, can escape. Temple says that a singularity can be more subtle where just a patch of spacetime cannot be made to look locally flat in any coordinate system.

“Locally flat” refers to space that appears to be flat from a certain perspective. Our view of the Earth from the surface is a good example. Earth looks flat to a sailor in the middle of the ocean. It’s only when we move far from the surface that the curvature of the Earth becomes apparent. Einstein’s theory of general relativity begins with the assumption that spacetime is also locally flat. Shockwaves create an abrupt change, or discontinuity, in the pressure and density of a fluid. This creates a jump in the curvature of spacetime but not enough to create the “crinkling” seen in the team’s models, Temple says.

The coolest part of the finding for Temple is that everything, his earlier work on shockwaves during the Big Bang and the combination of Vogler’s and Reintjes’ work, fits together.

There is so much serendipity,” says Temple. “This is really the coolest part to me.

I like that it is so subtle. And I like that the mathematical field of shockwave theory, created to address problems that had nothing to do with General Relativity, has led us to the discovery of a new kind of spacetime singularity. I think this is a very rare thing, and I’d call it a once in a generation discovery.”

While the model looks good on paper, Temple and his team wonder how the steep gradients in spacetime at a “regularity singularity” could cause larger than expected effects in the real world. General relativity predicts gravity waves might be produced by the collision of massive objects, such as black holes. “We wonder whether an exploding stellar shock wave hitting an imploding shock at the leading edge of a collapse, might stimulate stronger than expected gravity waves,” Temple says. “This cannot happen in spherical symmetry, which our theorem assumes, but in principle it could happen if the symmetry were slightly broken.”

*Image caption: Artist rendition of the unfurling of spacetime at the beginning of the Big Bang. John Williams/TerraZoom*

Here are the two relevant (PDF) papers:

A Proof of Convergence for Numerical Approximations Generated by the Locally Inertial Godunov Method in General Relativity;

Points of General Relativisitic Shock Wave Interaction are “Regularity Singularities” where Spacetime is Not Locally Flat.

The description of this is strange. A spacetime where no local patch can be made flat? This seems to go against the whole atlas-chart construction with transition functions used in differential geometry.

I will try to read the second of these papers in the next day and give some idea of what is going on here. This subject does appear to warrant some attention.

LC

That would be nice! I looked at it briefly, and their construction seems to surprise them too, they claim shock wave interaction empties the available degrees of freedom for different velocity shocks. The devil is in the details, as most always.

FWIW, I quickly googled that shock wave interactions seems to be known to break Einstein equations in this way:

“It is found that a surface tension is always associated in cases where the metrics are discontinuous. In some cases, the joined metrics satisfy Einstein equations (in the sense of distributions) while, in others, the surface tension associated with the limiting discontinuous metric depends on the interpolating functions used to produce it suggesting sensitivity to short distance effects beyond general relativity.” [Thakur, 1998]

From the present paper:

“Since the gravitational metric tensor is not locally inertial at points of shock wave interaction, it begs the question as to whether there are general relativistic gravitational effects at points of shock wave interaction that cannot be predicted from the compressible Euler equations in special relativity alone. Indeed, even if there are dissipativity terms, like those of the Navier Stokes equations,13 which regularize the gravitational metric at points of shock wave interaction, our results assert that the steep gradients in the derivative of the metric tensor at small viscosity cannot be removed uniformly while keeping the metric determinant uniformly bounded away from zero, so one would expect the general relativistic effects at points of shock wave interaction to persist. We thus wonder whether shock wave interactions might provide a physical regime where new general relativistic effects might be observed.”

It is a nice way to break the effective theory, perhaps? Shock waves wouldn’t be guaranteed diffeomorphisms, which is the way I think of these maps. While I’m sure I could follow the distribution part somewhat, I have to give up the maps at shocks. :-/

A shock wave is a wave front such that A(x + ?) – A(x) , the change in amplitude over a distance, is finite over the interval (x + ?, x). This is a discontinuity in the wave front, which distinguishes shock waves from ordinary waves that is continuous. Shock waves occur physically most often due to some process that travels through a medium faster than sound in that medium. A sonic boom is one example. If a chemical combustion propagates through the medium faster than the speed of sound the energy released is converted to a shock wave and the result is an explosion.

I have yet to read the paper in detail, however I can think of some physical arguments. A shock wave in a medium made of atoms with a diameter D can’t be more abrupt than these atoms. There is a sort of scale cut off to the mathematical definition above, where the infinitesimal ? is replaced with a very small but larger D. With spacetime physics a real spacetime shock wave would have a wave front that rises within a Planck distance or a string distance L_p = sqrt{G?/c^3} ~ 10^{-33}cm. A spacetime shock wave would then have some quantum gravity aspect to it, which makes this an interesting topic. Material shock waves can’t have wave fronts that physically rise over an interval larger than the particle length or atomic size of the material. So it is then physically difficult to understand how a shock wave in a supernova, a shock wave that passes through a plasma filled with protons of size 10^{-13}cm, can create a spacetime shock wave.

The only plausible case where this might occur is in a quark-gluon plasma at the center of a sufficiently large neutron star. Some phase change or energy release might send a shock wave through the neutron star core that has a wave front that rises on the scale of quarks and gluons. This could then conceivably be at the Planck or string length.

LC

I have only gotten part way through the second of these papers. So far it seems understandable. I will say this looks to be a formal extension of the Lanczos junction conditions. These are similar to the Gauss’ law analysis of fields across a boundary between two media.

The picture for this UT article is similar to the phase diagram of chaotic dynamics for the standard map. The more or less vertical strip of what appear to be galaxies is almost identical to the islands of regular dynamics in the chaotic KAM breaking or Kantori “dust.”

LC.

Hmmm … hypothetically, could this behavior be harnessed at a vastly smaller quantum to create one of the singularity propulsion systems envisioned by Arthur C. Clarke in his novel, 2525?

Read balloon inside balloon theory.

You mean this?

Is this just a long-winded way of saying GR can’t accommodate an extreme condition like a shock wave collision, the same way it can’t handle the extreme condition of a black hole?

This is one way of looking at it. The shock wave is not C^1 (first order differentiable) at the shock front. If this is maintained as an absolute aspect of reality then GR breaks down. General relativity requires C^n, for n > 2 where derivatives take one

Metric — > connection — > curvature — > Bianchi identity

with derivatives at each arrow. Sudden jumps in the metric means covariant derivatives are not continuous.

This is a singularity result, but one which has some curious aspects to it. It is also not clear whether there are event horizons associated with them. Physically there is the question of whether material that has an “atomic graininess” to it can result in a spacetime jump. For this to be a singularity the jump must be on a scale 10^{20} times smaller than the nucleus of an atom.

LC

I read the second paper through. It takes time because I have to read this along with other reading and work. This is an interesting paper, and I think it utility is in understanding how quantum physics works with gravity. Clearly if spacetime and physics in spacetime can’t be measured on a scale smaller than the Planck scale L_p = sqrt{G?/c^3} then these shock fronts can’t be “sharper” than this. This would then remove this curious issue of there not being any locally flat region of spacetime. The derivation here relies upon an absolute discontinuity, but quantum physics would remove that — in effect smearing things out so to speak.

I thought I would give a last post on this before this blog entry falls off the front page. The reports about the Mars “Curiosity” Science Lab are starting to come fast and will doubtless swamp everything for a while.

LC

OK dumb question time. Shockwave? what is it? I can undertand a shockwave in a medium such as interstellar gas but I wasn’t aware you could have a shockwave in spacetime or are we talking an EM radiation shockwave?

I’m guessing they mean a ripple in space-time, like gravitational waves are supposed to be.

I hope that “perfect fluid” doesn’t imply an isentropic fluid. Although constant entropy shocks can be calculated, the flow which includes them will violate conservation of energy. The flow through real shocks always increases significantly in entropy.