New mathematical formulation combines viscosity and relativity

In a new paper published in Physical Review X, Illinois Physics Professor Jorge Noronha and his co-authors provide a formulation of the equations of fluid dynamics that both accounts for viscosity and is compatible with Einstein’s theory of general relativity. This contribution moves scientists one step closer to solving a problem that has lingered in physics and mathematics for decades.

Since the 1940’s physicists and mathematicians have tried to consistently model viscous effects in relativity, especially in the case where strong gravitational fields are present and spacetime becomes a malleable entity described by Einstein’s equations. Previous efforts have either failed entirely—matter appears to travel faster than the speed of light—or made assumptions that could not be mathematically proven correct. 

The new work by Noronha and collaborators establishes a general framework that describes how viscous fluids behave in Einstein’s theory of general relativity. That is, the approach holds true in cases where the velocity of a viscous fluid is relativistic (i.e., close to the speed of light) or the fluid is in the presence of strong gravitational fields. In addition, this framework helps explain how the curvature of spacetime is sensitive to changes of energy, density, and velocity and how these changes lead to dissipation (e.g. heat). 

Noronha, a nuclear theorist, says, “Physics has to make sense. There is nothing in nature that prevents a relativistic fluid from having viscosity, but we didn’t have the appropriate equations to describe it. Previous attempts always broke down at some point, and they often led to the breaking of causality or simply predicted unstable behavior incompatible with observations. So, it was clear that those theories had to be wrong, but we didn’t know how to systematically fix them. This was fascinating to me.” 

This research has important applications for nuclear physics, especially for neutron star mergers. So far, scientists have largely assumed that viscous effects are too small to be relevant in mergers. For years, systematic investigations of this problem were severely hampered by the lack of a consistent formulation of viscous fluids in general relativity. The new formalism will allow theorists to incorporate viscosity into their models in a way that preserves causality. 

This ability to include viscosity in neutron star merger models may have implications for interpreting gravitational wave data from the Laser Interferometer Gravitational Wave Observatory (LIGO). 

Professor Nicolás Yunes, a theoretical physicist specializing in gravitational waves, explains, “In the next LIGO run, the instrument’s sensitivity will be higher, and we will likely be able to see the merger of two neutron stars more clearly. [Noronha et al.’s] work will allow us to model these mergers more precisely, which in turn will allow us to interpret the LIGO data more accurately.”

Professor Noronha adds: “This work provides a clear example of the benefits of establishing multi-disciplinary collaborations. For example, Prof. Disconzi, one of my collaborators, is a mathematician. The results of this work are proven using rigorous mathematical arguments, which may be of interest to mathematicians working on fluid dynamics.

“This advance in our understanding of fluid dynamics in general relativistic regimes paves the way for new simulations of neutron star mergers and relativistic heavy-ion collisions. I’m very excited to see how that plays out. I look forward also to seeing future applications of our approach in these and other fields.”

This work was done in collaboration with Marcelo Disconzi (Vanderbilt University) and Fabio Bemfica (Federal University of Rio Grande do Norte, Brazil). The full text of the paper can be found here.