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Register a new userCauchy-Characteristic Matching In General Relativity
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Authors
Bela Szilagyi
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From Einsteins theory we know that besides the electromagnetic spectrum, objects like quasars, active galactic nuclei, pulsars and black holes also generate a physical signal of purely gravitational nature. The actual form of the signal is impossible to determine analytically, which lead to use of numerical methods. Two major approaches emerged. The first one formulates the gravitational radiation problem as a standard Cauchy initial value problem, while the other approach uses a Characteristic Initial value formulation. In the strong field region, where caustics in the wavefronts are likely to form, the Cauchy formulation is more advantageous. On the other side, the Characteristic formulation is uniquely suited to study radiation problems because it describes space-time in terms of radiation wavefronts. The fact that the advantages and disadvantages of these two systems are complementary suggests that one may want to use the two of them together. In a full nonlinear problem it would be advantageous to evolve the inner (strong field) region using Cauchy evolution and the outer (radiation) region with the Characteristic approach. Cauchy Characteristic Matching enables one to evolve the whole space-time matching the boundaries of Cauchy and Characteristic evolution. The methodology of Cauchy Characteristic Matching has been successful in numerical evolution of the spherically symmetric Klein-Gordon-Einstein field equations as well as for 3-D non-linear wave equations. In this thesis the same methodology is studied in the context of the Einstein equations.
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This paper gives a detailed pedagogic presentation of the central concepts underlying a new algorithm for the numerical solution of Einsteins equations for gravitation. This approach incorporates the best features of the two leading approaches to computational gravitation, carving up spacetime via Cauchy hypersurfaces within a central worldtube, and using characteristic hypersurfaces in its exterior to connect this region with null infinity and study gravitational radiation. It has worked well in simplified test problems, and is currently being used to build computer codes to simulate black hole collisions in 3-D.
We present several improvements to the Cauchy-characteristic evolution procedure that generates high-fidelity gravitational waveforms at $mathcal{I}^+$ from numerical relativity simulations. Cauchy-characteristic evolution combines an interior solution of the Einstein field equations based on Cauchy slices with an exterior solution based on null slices that extend to $mathcal{I}^+$. The foundation of our improved algorithm is a comprehensive method of handling the gauge transformations between the arbitrarily specified coordinates of the interior Cauchy evolution and the unique (up to BMS transformations) Bondi-Sachs coordinate system of the exterior characteristic evolution. We present a reformulated set of characteristic evolution equations better adapted to numerical implementation. In addition, we develop a method to ensure that the angular coordinates used in the volume during the characteristic evolution are asymptotically inertial. This provides a direct route to an expanded set of waveform outputs and is guaranteed to avoid pure-gauge logarithmic dependence that has caused trouble for previous spectral implementations of the characteristic evolution equations. We construct a set of Weyl scalars compatible with the Bondi-like coordinate systems used in characteristic evolution, and determine simple, easily implemented forms for the asymptotic Weyl scalars in our suggested set of coordinates.
We implement a spatially fixed mesh refinement under spherical symmetry for the characteristic formulation of General Relativity. The Courant-Friedrich-Levy (CFL) condition lets us deploy an adaptive resolution in (retarded-like) time, even for the nonlinear regime. As test cases, we replicate the main features of the gravitational critical behavior and the spacetime structure at null infinity using the Bondi mass and the News function. Additionally, we obtain the global energy conservation for an extreme situation, i.e. in the threshold of the black hole formation. In principle, the calibrated code can be used in conjunction with an ADM 3+1 code to confirm the critical behavior recently reported in the gravitational collapse of a massless scalar field in an asymptotic anti-de Sitter spacetime. For the scenarios studied, the fixed mesh refinement offers improved runtime and results comparable to code without mesh refinement.
Characteristic classes in space-time manifolds are discussed for both even- and odd-dimensional spacetimes. In particular, it is shown that the Einstein--Hilbert action is equivalent to a second Chern-class on a modified Poincare bundle in four dimensions. Consequently, the cosmological constant and the trace of an energy-momentum tensor become divisible modulo R/Z.
This lecture will present a review of the past and present tests of the General Relativity theory. The essentials of the theory will be recalled and the measurable effects will be listed and analyzed. The main historical confirmations of General Relativity will be described. Then, the present situation will be reviewed presenting a number of examples. The opportunities given by astrophysical and astrometric observations will be shortly discussed. Coming to terrestrial experiments the attention will be specially focused on ringlasers and a dedicated experiment for the Gran Sasso Laboratories, named by the acronym GINGER, will be presented. Mention will also be made of alternatives to the use of light, such as particle beams and superfluid rings.
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