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Covariant BSSN formulation in bimetric relativity. (English) Zbl 1478.83213

Classical Quantum Gravity 37, No. 2, Article ID 025013, 33 p. (2020); corrigendum ibid. 37, No. 7, Article ID 079501, 4 p. (2020).
Summary: Numerical integration of the field equations in bimetric relativity is necessary to obtain solutions describing realistic systems. Thus, it is crucial to recast the equations as a well-posed problem. In general relativity, under certain assumptions, the covariant BSSN formulation is a strongly hyperbolic formulation of the Einstein equations, hence its Cauchy problem is well-posed. In this paper, we establish the covariant BSSN formulation of the bimetric field equations. It shares many features with the corresponding formulation in general relativity, but there are a few fundamental differences between them. Some of these differences depend on the gauge choice and alter the hyperbolic structure of the system of partial differential equations compared to general relativity. Accordingly, the strong hyperbolicity of the system cannot be claimed yet, under the same assumptions as in general relativity. In the paper, we stress the differences compared with general relativity and state the main issues that should be tackled next, to draw a roadmap towards numerical bimetric relativity.

MSC:

83D05 Relativistic gravitational theories other than Einstein’s, including asymmetric field theories
83C27 Lattice gravity, Regge calculus and other discrete methods in general relativity and gravitational theory
49K40 Sensitivity, stability, well-posedness
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
58J47 Propagation of singularities; initial value problems on manifolds
70S15 Yang-Mills and other gauge theories in mechanics of particles and systems
58J45 Hyperbolic equations on manifolds
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References:

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