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*

Big Bang, Blake Temple, Davis, general theory of relativity, Moritz Reintjes, Proceedings of the Royal Society A, regularity singularity, spacetime, University of California, Zeke Vogler

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

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?

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.