1. Geometric Integrators in the Study of Gravitational Collapse
Jonathan D. Kommemi
Advisor: Michael Holst, Department of Mathematics
University of California, San Diego
Introduction
In 1915, Einstein proposed the revolutionary idea that gravity was a mere perception of geometry rather than a force exerted by massive objects. More precisely, Einstein argued that what we perceive as gravity is really the manifestation of the bending, or curvature, of both space and time due to the presence of matter and energy. The Einstein field equations:
are ten coupled, non-linear, partial differential equations which precisely describe this relationship between matter and curvature. Unfortunately, except in very special cases, obtaining explicit expressions for solutions to the equations is not possible due to their complexity; therefore, modern techniques for the numerical solution on computers is crucial to this area of science.
Toy Model -- Pendulum
As a means of showcasing the power of these geometric integrators, we apply them to a fairly simple mechanical system: the motion of a pendulum. We begin by writing the Hamiltonian which gives rise to the equations of motion of the pendulum (where we make use of geometric units, i.e. we take constants to be unit)
where q is the position of the bob and p the momentum conjugate to q.
One can show that solutions to the equations of motion:
are constant along the level curves.
Toy Model -- Pendulum (cont’d)
As one can contrast, the symplectic methods on the right are far more accurate (at even greater time stepping!) than, say, forward or backward Euler, which are not.
Application to Relativity (cont’d)
In the problem we intend to study, we consider a scalar field minimally coupled to our gravitational field. The scalar field must satisfy the Klein-Gordon wave equation and the stress-energy tensor must look as follows:
Klein-Gordon Wave Equation
Stress-Energy Tensor
While it is quite messy to work out the equations which model this problem, we quite succinctly state that the following two equations, in addition to Klein-Gordon, fully model the Einstein equations.
The latter being the constraints (of which only 2 of the 4 are non-trivial) and the former being the slicing condition imposed on the evolution of a slice of space-time.
Numerical Relativity
Numerical relativity is a research field at the intersection of physics, numerical analysis and computer science. Its goal is to develop the capability to approximate solutions to the Einstein equations in important complex physical scenarios such as the collision of black holes.
A major difficulty in building reliable numerical solutions to the Einstein equations is dealing with constraint violation. These four equations are known to “propagate” if the system is solved exactly (i.e. analytically) , but the constraints are naturally violated once numerical methods are introduced. In the end, even the slightest error in the constraints leads to the loss of all basic physical properties such as conservation of momentum and energy.
However, there are types of numerical methods, known as geometric or symplectic integrators which are designed to give highly accurate and reliable approximations to dynamics while enforcing various constraints and at the same time preserving various physical properties such as time-reversibility.
Future Directions
While these methods have been used extensively in the area of computational solid mechanics, it appears that its use in the numerical relativity community is non-existent and we are optimistic that controlling constraint violation is extremely beneficial to obtaining solutions which have some sort of validity attesting to their physical properties.
Once we are able to attest to the practicality of symplectic integrators, generalization to full 3+1 situations will be desirable.
Application to Relativity
The Einstein equations are complicated to solve in its entirety. As a result, many assumptions are made, via symmetries of the space-time geometry to “simplify” the equations; but, nevertheless, even the simplified equations are still difficult. One common assumption is that of spherical symmetry. In essence, this reduces the full 3+1 (3 spatial and 1 temporal co-ordinates) geometry to a 1+1 system. The end result is that the 10 equations reduce down to 4.
Acknowledgements
Along with Mike Holst, I’d like to thank Gabriel Nagy and Ryan Szypowski for all the time and encouragement they have given me, not only on this project, but all, past, present, and future. And also, to Calit2 for their generous funding and support.