Simulations Numerical of Extreme-Mass-Ratio Binaries
Carlos F. Sopuerta Numerical Relativity Center for Gravitational Wave Physics Penn State University
5/30/2004 7th CAPRA Meeting 1
Approaches to the Extreme-Mass-Ratio Binary System. Main Issues for the Numerical Implementation: 3+1 formulation, Numerical schemes, Gauges, etc.
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7th CAPRA Meeting
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Brute Force Approach: Requires strong AMR.
The small object induces a small deviation on the background spacetime of the SMBH:
Simplest way of solving for the problem:
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7th CAPRA Meeting
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Problems:
Contracted Second Bianchi Identities imply that a particle follows a geodesic of the background. Motion is not determined by the field equations.
Alternative: Consider second-order terms in the metric deviation
Assuming the metric deviations can be approximated by the linear perturbations we have:
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7th CAPRA Meeting
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Algorithm using time-domain integration:
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7th CAPRA Meeting
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Alternative Approach: To Mimic Matched Asymptotic Expansions Techniques. A small black hole moving in the
background of a SMBH.
Advantage: Instead of AMR we would need Moving Grid tecniques
M
m
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7th CAPRA Meeting
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Exterior zone Interior zone Perturbations.
SMBH spacetime : Kerr + Linear Perturbations. Small BH spacetime: Schwarzschild + Linear
Hypersurface joining the SMBH and the Small BH Buffer zone spacetimes: Continuity of the first and second fundamental forms at Exchange of information and motion of the the hypersurface small BH (hypersurface equation) Requires knowledge on the structure of the perturbations.
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7th CAPRA Meeting
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Formulation the of dynamical equations: We will split our equations by using the well-known 3+1 splitting:
We will consider systems of equations that are both first-order in time (ADM, BSSN, etc. Expertise in Full Numerical Relativity) and secondorder in time (Systems of wave-type equations Expertise in the Finite Element community)
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7th CAPRA Meeting
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Gauge: We have to fix two different gauges.
Background Gauge: It is convenient to choose horizon-penetrating slicings in order to excise the SBMH singularity from the computational domain.
Perturbations Gauge: May depend on the formulation.
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7th CAPRA Meeting
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In order to try different formulations, different gauges, and different numerical schemes, we need to build scripts that generate the equations (and perhaps even the code) in order to change in an efficient way the model setup. The best way to generate these scripts is to use advance computer algebra systems like MathTensor (Mathematica), GRTensor (...