Curvature Scalar Diagnostics and Progress on a Chern-Simons Initial Value Formulation

Open Access
Wood, Shaun Patrick
Graduate Program:
Doctor of Philosophy
Document Type:
Date of Defense:
August 20, 2009
Committee Members:
  • Abhay Vasant Ashtekar, Dissertation Advisor
  • Abhay Vasant Ashtekar, Committee Chair
  • Pablo Laguna, Committee Member
  • Martin Bojowald, Committee Member
  • Nigel David Higson, Committee Member
  • general relativity
  • numerical relativity
  • Beetle-Burko scalar
  • Chern-Simons gravity
In the recent past, many numerical relativity groups have gained the ability to perform fully relativistic simulations of black hole mergers. One of the key aims of these simulations is to compute gravitational waveforms. Such waveforms are interesting not only from a theoretical standpoint, but also because they are necessary for the ongoing effort to detect gravitational waves. Current numerical methods for computing gravitational waveforms depend on assumptions made about the background spacetime, typically via a choice of tetrad. Such methods require additional checks to insure the assumptions are correct. In this dissertation, we first present the results of a project to use scalars which can be computed using only contractions of the curvature tensor as diagnostic tools in numerical relativity simulations. The scalars have been shown to contain information about the background spacetime and gravitational radiation, and are independent of any choice of tetrad, background, or coordinates. The Baker Campanelli speciality index is used to determine when the spacetime separates into background and radiation, subject to the conditions that the radiation be weak and be purely outgoing. We find that at least the second condition is not satisfied in the simulations we perform. Then, the Beetle-Burko radiation scalar is used to check the assumption that the tetrad used to compute gravitational waveforms represents the principle null directions of the background spacetime. It is essential that two vectors of the tetrad are parallel to the two principle null directions of the background in order to interpret the Newman Penrose scalar $Psi_4$ as the outgoing gravitational wave. We find the tetrad passes this test whenever it can be applied in our simulations. While this does not permit us to conclude that the scalar $Psi_4$ is the outgoing gravitational wave, the test can be used in more generic simulations to catch errors in the $Psi_4$ calculation. Next, we present an attempt to write Chern-Simons modified gravity as an initial value problem. Chern-Simons modified gravity is an extension to general relativity in which a parity violating term is added to the action. One effect the modification has is to enhance or suppress the different polarizations of gravitational radiation. Therefore, there is great interest in simulating binary black hole mergers in the theory and computing the resulting gravitational radiation. We provide an incomplete set of constraints and evolution equations for the theory, and discuss difficulties of finding a complete set. Projections of the modified field equations are also provided. Once a complete initial value formulation is found and simulations are performed, gravitational wave detection can provide a strong test of Chern-Simons modified gravity.