Physics of Black Holes A Guided Tour
Black Holes are still considered to be among the most mysterious and fascinating objects in our universe. Awaiting the era of gravitational astronomy, much progress in theoretical modeling and understanding of classical and quantum black holes has already
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Colliding Black Holes and Gravitational Waves U. Sperhake
Abstract This article presents a summary of numerical simulations of black-hole spacetimes in the framework of general relativity. The first part deals with the 3+1 decomposition of generic spacetimes as well as the Einstein equations which forms the basis of most work in numerical relativity. Technical aspects of the resulting numerical evolutions and the diagnostics of the resulting spacetimes are discussed. The second part presents an overview of the history of numerical simulations of black-hole spacetimes. Finally, we summarize results derived from numerical blackhole simulations obtained after the breakthrough in 2005. The relevance of these results in the context of astrophysics, gravitational wave physics, and fundamental physics is discussed.
4.1 Introduction In Einstein’s theory of general relativity gravitation is a manifestation of the curvature of the spacetime rather than a force in the traditional sense. The fundamental quantity which encapsulates all information about the spacetime curvature is the spacetime metric, a set of ten functions of space and time. This metric obeys the Einstein equations which equates the Einstein tensor, a complex combination of the metric and its first and second derivatives, with the mass–energy tensor describing the matter distribution. The Einstein equations thus represent a system of ten second-order partial differential equations, one of the most complicated systems of equations in all of physics. Einstein himself did not expect physically meaningful solutions to be found analytically and it came as a surprise when Karl Schwarzschild found his famous solution of a static, spherically symmetric vacuum spacetime just a few months after the publication of general relativity in 1916. This solution is now known as a “Schwarzschild black hole”, but the term black hole was not coined until much later by John Wheeler. The Schwarzschild solution has lead to invaluable insight into general relativity and was soon generalized to include electric charge U. Sperhake (B) Theoretisch Physikaslisches Institut, Universit¨at Jena, Max-Wien-Platz 1, 07743 Jena, Germany [email protected]
Sperhake, U.: Colliding Black Holes and Gravitational Waves. Lect. Notes Phys. 769, 125–175 (2009) c Springer-Verlag Berlin Heidelberg 2009 DOI 10.1007/978-3-540-88460-6 4
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U. Sperhake
in the form of the Reissner-Nordstr¨om solution. The key simplification leading to these analytic solutions is the high degree of symmetry of the spacetime which reduces the Einstein equations to a 1D problem with no time dependence. Relaxing the assumption of spherical symmetry to allow for a spacetime with non-vanishing angular momentum led to a much more complex system of equations even in the limit of stationarity. It took more than four decades until Roy Kerr found the analytic expressions for the metric of an axisymmetric spacetime containing a rotating black hole [168]. Again, the inclusion of electric charge resulted in a generalization, t
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