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Register a new userCharacteristic classes in general relativity on a modified Poincare curvature bundle
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Authors
Yoshimasa Kurihara
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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.
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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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