Visualizing Spacetime Curvature in Binary Black Hole Mergers using Tendex and Vortex Lines Público
Vu, Andrew (2013)
Published
Abstract
With the growth of supercomputing clusters and advanced instrumentation, we have seen a tremendous advancement in the study of Einstein's theory of general relativity. In particular, the two-body problem in general relativity has been solved, and we now are exploring black hole mergers and observing peculiar phenomena such as spin-flip, black hole recoil, and orbital hang-up. With the construction of LIGO and eLISA for gravitational wave detection, there now leaves an extensive task for data analysis. As of now, the best method for finding a gravitational wave is to match it against a library of waveform templates. These templates are created using numerical relativity which solves the Einstein field equations using the most complex algorithms on large supercomputer clusters. Only through this, can we then be able to identify the physical phenomena that generated the gravitational radiation. As of now, the most promising source for gravitational wave detection is the radiation given off by the collisions and mergers of binary black holes. Currently, we are in need of new tools that will aid us in the generation of more accurate waveform templates. What I will try to do is implement a code in the numerical simulations to produce lines that are analogous to the electric and magnetic field lines in Maxwell's theory. In general relativity, the Weyl curvature tensor (traceless component of the Riemann curvature tensor), which details all the information about the curvature of the local geometry, can be covariantly split into two parts, which are spatial, symmetric, and trace-free. The two tensors are the "electric" part which is the tidal field Eij, which details the stretching and compressing of an observer, and the "magnetic" part which is the frame-drag field Bij, which details the precession of an observer. Each Eij and Bij tensor has three orthogonal eigenvector fields that can be visualized by their integral curves. These lines prove to be a useful tool, as we will show in this paper, for understanding the nonlinear dynamics around regions of strong-gravity such as a stationary black hole, a rotating black hole, and binary black hole merger.
Table of Contents
1 Introduction 1
1.1 Space, Time and Special Relativity 3
1.2 The Metric Tensor and Geometry 4
1.2.1 Vectors 6
1.2.2 Covariant Vectors and Tensors 7
1.2.3 Covariant Derivatives and Christoffel Symbols 9
1.2.4 Geodesics and Parallel Transport 10
1.3 Riemann Tensor, Ricci Tensor, and the Ricci Scalar 12
1.4 Einstein Equations 15
1.5 Schwarzschild Metric 16
1.6 Linearized Gravity and Gravitational Radiation 17
2 Gravitational Waves and their detection 18
2.1 Sources 19
2.2 Numerical Relativity 19
3 Tendex and Vortex Lines 20
3.1 Newman-Penrose Formalism 20
3.2 Tendex and Vortex Lines 21
4 Implementation 24
4.1 Equations 24
4.2 Numerical Implementation 31
4.3 Visualization 32
4.3.1 Schwarzschild Black Hole 32
4.3.2 Kerr Black Hole 34
4.3.3 Binary Black Hole Merger 38
5 Conclusion 46
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