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Introduction to 3+1 Numerical Relativity$
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Miguel Alcubierre

Print publication date: 2008

Print ISBN-13: 9780199205677

Published to Oxford Scholarship Online: September 2008

DOI: 10.1093/acprof:oso/9780199205677.001.0001

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(p.64) 2 THE 3+1 FORMALISM
Introduction to 3+1 Numerical Relativity

Miguel Alcubierre

Oxford University Press

There are several different approaches to the problem of separating the Einstein field equations in a way that allows us to give certain initial data, and from there obtain the subsequent evolution of the gravitational field. Specific formalisms differ in the way in which this separation is carried out. This chapter focuses on the 3+1 formalism, where spacetime is split into three-dimensional space on the one hand, and time on the other. The 3+1 formalism is the most commonly used in numerical relativity, but it is certainly not the only one. Topics discussed include the 3+1 split of spacetime, extrinsic curvature, the Einstein constraints, the ADM evolution equations, free versus constrained evolution, Hamiltonian formulation, the BSSNOK formulation, alternative formalisms, the characteristic approach, and the conformal approach.

Keywords:   Einstein field equations, spacetime, extrinsic curvature, ADM evolution equations, free evolution, constrained evolution, Hamiltonian formulation

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