The Cauchy Problem

A simple and purely algebraic construction of super-energy (s-e) tensors for arbitrary fields is presented in any dimensions. These tensors have good mathematical and physical properties, and they can be used in any theory having as basic arena an n-dimensional manifold with a metric of Lorentzian signature. In general, the completely timelike component of these s-e tensors has the mathematical features of an energy density: they are positive definite and satisfy the dominant property, which provides s-e estimates useful for global results and helpful in other matters, such as the causal propagation of the fields. Similarly, super-momentum vectors appear with the mathematical properties of s-e flux vectors. The classical Bel and Bel-Robinson tensors for the gravitational fields are included in our general definition. The energy-momentum and super-energy tensors of physical fields are also obtained, and the procedure will be illustrated by writing down these tensors explicitly for the cases of scalar, electromagnetic, and Proca fields. Moreover, higher order (super)k-energy tensors are defined and shown to be meaningful and in agreement for the different physical fields. In flat spacetimes, they provide infinitely many conserved quantities. In non-flat spacetimes, the fundamental question of the interchange of s-e quantities between different fields is addressed, and answered affirmatively. Conserved s-e currents are found for any minimally coupled scalar field whenever there is a Killing vector. Furthermore, the exchange of gravitational and electromagnetic super-energy is also shown by studying the propagation of discontinuities. This seems to open the door for new types of conservation physical laws.

http://iopscience.iop.org/article/10.1088/0264-9381/17/14/313/pdf

Super-energy tensors

Understanding the predictions of general relativity for the dynamical interactions of two black holes has been a long-standing unsolved problem in theoretical physics. Black-hole mergers are monumental astrophysical events ' releasing tremendous amounts of energy in the form of gravitational radiation ' and are key sources for both ground- and spacebased gravitational wave detectors. The black-hole merger dynamics and the resulting gravitational waveforms can only he calculated through numerical simulations of Einstein's equations of general relativity. For many years, numerical relativists attempting to model these mergers encountered a host of problems, causing their codes to crash after just a fraction of a binary orbit cnuld be simulated. Recently ' however, a series of dramatic advances in numerical relativity has ' for the first time, allowed stable / robust black hole merger simulations. We chronicle this remarkable progress in the rapidly maturing field of numerical relativity, and the new understanding of black-hole binary dynamics that is emerging. We also discuss important applications of these fundamental physics results to astrophysics, to gravitationalwave astronomy, and in other areas.

Black-hole binaries, gravitational waves, and numerical relativity

Numerical relativity and compact binaries

We present a new many-parameter family of hyperbolic representations of Einstein’s equations, which we obtain by a straightforward generalization of previously known systems. We solve the resulting evolution equations numerically for a Schwarzschild black hole in three spatial dimensions, and find that the stability of the simulation is strongly dependent on the form of the equations (i.e. the choice of parameters of the hyperbolic system), independent of the numerics. For an appropriate range of parameters we can evolve a single three-dimensional black hole to t ≃ 600 M – 1300 M, and we are apparently limited by constraint-violating solutions of the evolution equations. We expect that our method should result in comparable times for evolutions of a binary black hole system.

/pdf/extending-the-lifetime-of-3d-black-hole-computations-with-a-4gj0nhfhzs.pdf

Extending the lifetime of 3D black hole computations with a new hyperbolic system of evolution equations

We consider the evolution part of the Cauchy problem in General Relativity [8] as the time history of the two fundamental forms of a spacelike hypersurface: its metric g and its extrinsic curvature K. On such a hypersurface, for example an “initial” one, these two quadratic forms must satisfy four initial value or constraint equations. These constraints can be posed and solved as an elliptic system by known methods that will not be discussed here. (See, for example, [8].) The Ricci tensor of the spacetime metric can be displayed in a straightforward 3 + 1 decomposition giving the time derivatives of g and K in terms of the space derivatives of these quantities. These expressions contain also the lapse and shift functions characterizing the threading of the spacelike hypersurfaces by time lines. However, proof of the existence of a causal evolution in local Sobolev spaces into an Einsteinian spacetime does not result directly from these equations, which do not form a hyperbolic system for arbitrary lapse and shift, despite the fact that their characteristics are only the light cone and the normal to the time slices [13].

/pdf/einstein-bianchi-hyperbolic-system-for-general-relativity-4o5jldh38f.pdf

Einstein-Bianchi hyperbolic system for general relativity

We establish a variant of the symmetric quasi linear first order system
given by H Friedrich for the evolution equations of gravitating fluid
bodies in General Relativity which can be important to solve realistic
problems Our version has the advantage of introducing only physical
characteristics We state explicitly the conditions under which the system
is hyperbolic and admits a well posed Cauchy problem

/pdf/on-h-friedrich-s-formulation-of-the-einstein-equations-with-2ddcap4ksz.pdf

Yvonne Choquet-Bruchat

Papers

Einstein-Bianchi hyperbolic system for general relativity

On H. Friedrich's formulation of the Einstein equations with fluid sources