Numerical Relativity

Snapshots for the head-on collision of two black holes. The top panel shows the initial data for the lapse function α, based on Brill-Lindquist data; the middle panel at the instant when the smaller black hole moves through the origin of the coordinate system; the bottom panel at a late time after merger.

Many current numerical relativity codes (in three spatial dimensions) share several features: they adopt the Baumgarte-Shapiro-Shibata-Nakamura (BSSN) formulation of Einstein's equations, use finite-difference methods in Cartesian coordinates, and adopt moving puncture coordinates, i.e. ~a combination of 1+log slicing and the "Gamma-driver" condition. While Cartesian coordinates are well suited for many applications, in particular simulations of binaries, spherical polar coordinates have some desirable properties for simulations of single objects, for example gravitational collapse and supernova explosions. We have therefore developed and implemented a new approach that applies in spherical polar coordinates the numerical methods that have previously proven to be extremely successful in Cartesian coordinates. This approach relies on three key ingredients: a reference-metric formulation of the BSSN equations, factoring out appropriate geometrical factors from tensor components, and using a ''partially implicit" Runge-Kutta (PIRK) method. The resulting equations are still singular at the origin of the coordinate system and on the polar axis, but all singular terms can be handled analytically, and the PIRK method is stable even in the presence of these singular terms. Our approach therefore does not rely on a regularization of the equations, and can be used even in the absence of spherical or axi-symmetry. We also applied a reference-metric approach to the formulation of relativistic hydrodynamics, and implemented the resulting equations to perform what we believe are the first self-consistent and stable simulations of general relativistic hydrodynamics in dynamical spacetimes in spherical polar coordinates without the need for regularization or symmetry assumptions.


BSSN equations in spherical coordinates without regularization: Vacuum and nonvacuum spherically symmetric spacetimes
Montero, Pedro J.; Cordero-Carrión, Isabel, Physical Review D, vol. 85, Issue 12, id. 124037, 2012 [link]

Numerical relativity in spherical polar coordinates: Evolution calculations with the BSSN formulation
Baumgarte, Thomas W.; Montero, Pedro J.; Cordero-Carrión, Isabel; Müller, Ewald, Physical Review D, vol. 87, Issue 4, id. 044026, 2013 [link]

General relativistic hydrodynamics in curvilinear coordinates
Montero, Pedro J.; Baumgarte, Thomas W.; Müller, Ewald, Physical Review D, Volume 89, Issue 8, id.084043, 2014 [link]

Fully covariant and conformal formulation of the Z4 system in a reference-metric approach: Comparison with the BSSN formulation in spherical symmetry
Sanchis-Gual, Nicolas; Montero, Pedro J.; Font, José A.; Müller, Ewald; Baumgarte, Thomas W., Physical Review D, Volume 89, Issue 10, id.104033, 2014 [link]

Numerical relativity in spherical polar coordinates: Off-center simulations
Baumgarte, Thomas W.; Montero, Pedro J.; Müller, Ewald, Physical Review D, Volume 91, Issue 6, id.064035, 2015 [link]

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