# On Boundary Conditions for the Einstein Equations

- 1.1k Downloads

## **Abstract**

The use of the projection of the Einstein tensor normally to a timelike boundary as a set of boundary conditions for the initial-value problem of the vacuum Einstein equations is investigated within the setting of a particular first-order strongly hyperbolic formulation. It is found that the components of such a projection give rise to boundary conditions that are appropriate, in a certain sense, for the initial-value problem of the evolution equations and for the initial-value problem of the auxiliary system of propagation of the constraints, at the same time. It can be concluded that imposing such boundary conditions on the values of the fundamental variables of the initial-value problem guarantees the propagation of the constraints. This contribution presents a unified account of results that have recently appeared separately in the literature. The presentation is meant to be accessible to a broader readership.

## Keywords

Black Hole Evolution Equation Einstein Equation Extrinsic Curvature Cauchy Surface## Preview

Unable to display preview. Download preview PDF.

## References

- 1.Arlen Anderson, James W. York: Fixing Einstein's equations. Phys. Rev. Lett.
**82**, 4384 (1999)zbMATHMathSciNetCrossRefADSGoogle Scholar - 2.Thomas W. Baumgarte, Stuart L. Shapiro: On the numerical integration of Einstein's field equations. Phys. Rev. D
**59**, 024007 (1999)MathSciNetCrossRefADSGoogle Scholar - 3.Gioel Calabrese, Jorge Pullin, Oscar Reula, Olivier Sarbach, Manuel Tiglio: Well posed constraint preserving boundary conditions for the linearized Einstein equations. Commun. Math. Phys.
**240**, 377 (2003)zbMATHMathSciNetCrossRefADSGoogle Scholar - 4.Simonetta Frittelli, Roberto Gómez: Boundary conditions for hyperbolic formulations of the Einstein equations. Class. Quantum Grav.
**20**, 2379 (2003)zbMATHCrossRefADSGoogle Scholar - 5.Simonetta Frittelli, Roberto Gómez: Einstein boundary conditions for the 3+1 Einstein equations. Phys. Rev. D
**68**, 044014 (2003)MathSciNetCrossRefADSGoogle Scholar - 6.Simonetta Frittelli, Roberto Gómez: Einstein boundary conditions for the Einstein equations in the conformal-traceless decomposition. Phys. Rev. D
**70**, 064008 (2004)MathSciNetCrossRefADSGoogle Scholar - 7.Simonetta Frittelli, Roberto Gómez: Einstein boundary conditions in relation to constraint propagation for the initial-boundary value problem of the Einstein equations. Phys. Rev. D
**69**, 124020 (2004)MathSciNetCrossRefADSGoogle Scholar - 8.Simonetta Frittelli: Potential for ill-posedness in several second-order formulations of the Einstein equations. Phys. Rev. D
**70**, 044029 (2004)MathSciNetCrossRefADSGoogle Scholar - 9.Bertil Gustaffson, Heinz-Otto Kreiss, Joseph Oliger:
*Time-Dependent Problems and Difference Methods*(Wiley, New York 1995)Google Scholar - 10.Lawrence E. Kidder, Mark A. Scheel, Saul A. Teukolsky, Eric D. Carlson, Gregory B. Cook: Black hole evolution by spectral methods. Phys. Rev. D
**62**, 084032 (2000)MathSciNetCrossRefADSGoogle Scholar - 11.Lawrence E. Kidder, Mark A. Scheel, Saul A. Teukolsky: Extending the lifetime of 3d black hole computations with a new hyperbolic system of evolution equations. Phys. Rev. D
**64**, 064017 (2001)MathSciNetCrossRefADSGoogle Scholar - 12.Gabriel Nagy, Omar Ortiz, Oscar A. Reula: Strongly hyperbolic second order Einstein's evolution equations. Phys. Rev. D
**70**, 044102 (2004)MathSciNetADSGoogle Scholar - 13.Olivier Sarbach, Gioel Calabrese: Detecting ill-posed boundary conditions in general relativity. J. Math. Phys.
**44**, 3888 (2003)zbMATHMathSciNetCrossRefADSGoogle Scholar - 14.J. M. Stewart: The Cauchy problem and the initial boundary value problem in numerical relativity. Class. Quantum Grav.
**15**, 2865 (1998)zbMATHCrossRefADSGoogle Scholar - 15.Robert M. Wald:
*General Relativity*(University of Chicago Press, Chicago 1984)zbMATHGoogle Scholar - 16.Steven Weinberg:
*Gravitation and Cosmoloy*.*Principles and Applications of the General Theory of Relativity*(John Wiley & Sons, New York 1971)Google Scholar - 17.James W. York: Kinematics and dynamics of general relativity. In:
*Sources of Gravitational Radiation*, ed by Larry Smarr (Cambridge University Press, Cambridge 1979)Google Scholar