Jason Tabachnik with Andrew Tolley
Exploring the dualities between fluids, strings, and gravity
Recently, high-energy physicists have discovered and developed powerful new tools known as “dualities.” Dualities are deep correspondences between naively different theories, allowing one to examine the same phenomena in two seemingly divergent lights. Essentially, they are like two sides to the same coin; dualities help to elucidate aspects of one theory that would otherwise be difficult to describe in the other (and vice versa). Physicist Roman Jackiw and his collaborators have pointed out a remarkable connection between the Navier-Stokes equations – the fundamental equations describing classical fluids – and the equations for a “membrane” moving in a higher-dimensional geometry. Membranes are fundamental units of matter with dimensionality greater than two, whereas a point particle is a zero-dimensional object and a string is a one-dimensional object. The membrane idea is now a standard one within the context of string theory and this connection between the Navier-Stokes equations and membrane motion is well known within the high-energy community.
Applying our knowledge of membrane dynamics, Jackiw’s duality implies the existence of symmetries in the fluid equations that haven’t been explored or described in any current model. In recent work by Tolley and de Rham, the role of symmetries in higher-dimensional membrane equations has been clarified and the authors have developed unique extensions to membrane equations which preserve the consistency of the dynamics. In other words, their work has opened the possibility that extensions to membrane models exist which keep all of the physics intact. Given the power of Jackiw’s duality, it is natural to see how corrections in the membrane equations would translate in the framework of fluid dynamics. Starting this past summer, we have already begun such endeavors. Our initial efforts focused upon extensions of the membrane equations that incorporate higher-order curvature corrections, particularly via the Ricci scalar. This is a natural avenue to pursue because membranes are almost uniquely identified by their curvature. Having obtained these corrections, we are currently in the process of exploiting Jackiw’s duality as we translate back to the frame of fluid dynamics. After exploring this avenue, we will explore another type of curvature quantity, the extrinsic curvature, using the same methods outlined above. It is expected that a more accurate description of fluids, specifically their nonlinear behavior, will be illuminated through the translation back into the fluid frame.
After examining the latter issues, we plan to investigate a number of deeper issues intimately related to this subject matter. In closely related work, physicist Andrew Strominger and his collaborators have recently pointed out a duality of their own: Einstein’s Equations for the gravitational field near a black hole horizon degenerate into the incompressible Navier-Stokes equations. In other words, gravitation seems to mirror fluid behavior near extreme boundaries and spacetime horizons. The latter is exciting since corrections to the gravitational field yield higher-order viscosity terms and hence deeply ties into the concepts surrounding turbulence. Later on, we hope to probe this second duality and consider what happens to membranes as they near a black hole horizon. The interaction of gravity with membrane dynamics will likely have enormous implications in the frame of fluid dynamics. Overall, our intention is to see if there is an over-arching connection between Jackiw’s duality and Strominger’s duality, potentially yielding tremendous insight into nonlinear fluid phenomena.