VandyGRAF

Vanderbilt initiative for gravity, waves, and fluids

VandyGRAF Initiative

VandyGRAF Initiative

The Vanderbilt Initiative for Gravity, Waves, and Fluids is an interdisciplinary research venture providing mathematicians, physicists, and astrophysicists with the resources and space to connect and collaboratively work on problems of outstanding scientific merit, such as:

  • General relativity: theoretical, mathematical, numerical, or experimental, including, but not restricted to, black holes, gravitational radiation, and multimessenger astrophysics.
  • Fluid mechanics: theoretical, mathematical, numerical, or experimental, including, but not restricted to, relativistic fluids far from equilibrium.
  • Evolution of partial differential equations related to fluids and gravity, including, but not restricted to, the geometric analysis of waves and fluids.
  • The physics and mathematics of neutron star mergers and high-energy nuclear collisions.

VandyGRAF Seminar Series

All VandyGRAF talks will take place in the Chapel in the 17th&Horton building, unless indicated below.

Gregory Galloway, University of Miami, September 5, 12:30 pm, Room A1013

Topology and singularities in cosmological spacetimes

A theme in general relativity of long standing interest (at least to the speaker!) concerns the relationship between the topology of spacetime and the occurrence of singularities (causal geodesic incompleteness). Many such results center around the notion of topological censorship, which has to do with black hole formation and the topology of black holes. In this talk we focus on the cosmological setting. More specifically, for 3+1 dimensional spacetimes which are spatially closed (i.e. whose spatial sections are compact without boundary), we establish a precise connection between the topology of these spatial slices and the occurrence of past singularities. This result applies to spacetimes with positive cosmological constant, where, for example, Hawking’s cosmological singularity theorem in general does not. Instead we will make use of Penrose’s singularity theorem (for which Penrose was awarded half the Nobel Prize in Physics in 2020). The proof also makes use of fundamental existence results for minimal surfaces and results in 3-manifold topology that were motivated by Thurston’s geometrization conjecture. We will begin the talk with some basic background in spacetime geometry. This is joint work with Eric Ling.

Oem Trivedi, Vanderbilt University, September 12, 12:30 pm

Marcel Disconzi, Vanderbilt University, September 19, 12:30 pm

The mathematics of general-relativistic stars

Astronomy is arguably the oldest scientific discipline. Precise measurements of the motion of celestial bodies date back to the ancient Babylonians, Chinese, Greeks, and indigenous peoples outside Eurasia. Starting in the 19th century, systematic applications of physical principles to the formation and dynamics of stars marked the birth of astrophysics as a subfield of physics. Present-day astrophysics employs an array of theoretical and observational tools to construct sophisticated and predictive models of the origin, evolution, and death of stars. While stars can be largely described within Newtonian physics, some of their most interesting properties, such as bounds on their mass-radius ratio, their potential collapse into a black hole, or effects of viscosity on gravitational waves emitted by mergers of neutron stars, can only be studied via applications of general relativity. Moreover, as a matter of principle, we ought to be able to fully understand stars as general-relativistic phenomena. The mathematical treatment of stars within general relativity, however, has lagged. Little progress has been made on this front since the discovery of the Tolman-Oppenheimer-Volkoff (TOV) equations and the Oppenheimer-Snyder solution in the late ‘30s. The former describes a static (i.e., time independent), perfectly spherically symmetric star, whilst the latter describes the collapse of a perfectly spherically symmetric star with no pressure into a black hole. Despite being landmark results in general relativity, both situations are highly idealized. Inferences about generic properties of general-relativistic stars derived from such models are, therefore, a priori unjustified.

In this talk, I will discuss the problem of formulating a sound mathematical theory of general-relativistic star evolution based on the Einstein-Euler system. After setting up the problem, I will explain its main challenges, but also how a great deal of rich physics and mathematics is involved in its study. A fundamental difficulty involves understanding the mathematics of the fluid-vacuum interface which separates the body of the star from vacuum. This interface displays a singular behavior which is not amenable to current mathematical techniques. This difficulty, however, can be circumvented if we consider stars that are spherically symmetric but not static. This corresponds to a dynamic (i.e., time-dependent) generalization of the TOV equations.