José María Martín-García

Constraint damping in the Z4 formulation and harmonic gauge

We show that by adding suitable lower-order terms to the Z4 formulation of the Einstein equations, all constraint violations except constant modes are damped. This makes the Z4 formulation a particularly simple example of a ?-system as suggested by Brodbeck et al (1999 J. Math. Phys. 40 909). We also show that the Einstein equations in harmonic coordinates can be obtained from the Z4 formulation by a change of variables that leaves the implied constraint evolution system unchanged. Therefore, the same method can be used to damp all constraints in the Einstein equations in harmonic gauge.

xPerm: fast index canonicalization for tensor computer algebra ☆

Abstract We present a very fast implementation of the Butler–Portugal algorithm for index canonicalization with respect to permutation symmetries. It is called xPerm , and has been written as a combination of a Mathematica package and a C subroutine. The latter performs the most demanding parts of the computations and can be linked from any other program or computer algebra system. We demonstrate with tests and timings the effectively polynomial performance of the Butler–Portugal algorithm with respect to the number of indices, though we also show a case in which it is exponential. Our implementation handles generic tensorial expressions with several dozen indices in hundredths of a second, or one hundred indices in a few seconds, clearly outperforming all other current canonicalizers. The code has been already under intensive testing for several years and has been essential in recent investigations in large-scale tensor computer algebra. Program summary Program title: xPerm Catalogue identifier: AEBH_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEBH_v1_0.html Program obtainable from: CPC Program Library, Queens University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 93 582 No. of bytes in distributed program, including test data, etc.: 1 537 832 Distribution format: tar.gz Programming language: C and Mathematica (version 5.0 or higher) Computer: Any computer running C and Mathematica (version 5.0 or higher) Operating system: Linux, Unix, Windows XP, MacOS RAM:: 20 Mbyte Word size: 64 or 32 bits Classification: 1.5, 5 Nature of problem: Canonicalization of indexed expressions with respect to permutation symmetries. Solution method: The Butler–Portugal algorithm. Restrictions: Multiterm symmetries are not considered. Running time: A few seconds with generic expressions of up to 100 indices. The xPermDoc.nb notebook supplied with the distribution takes approximately one and a half hours to execute in full.

Hyperbolicity of second-order in space systems of evolution equations

A possible definition of strong/symmetric hyperbolicity for a second-order system of evolution equations is that it admits a reduction to first order which is strongly/symmetric hyperbolic. We investigate the general system that admits a reduction to first order and give necessary and sufficient criteria for strong/symmetric hyperbolicity of the reduction in terms of the principal part of the original second-order system. An alternative definition of strong hyperbolicity is based on the existence of a complete set of characteristic variables, and an alternative definition of symmetric hyperbolicity is based on the existence of a conserved (up to lower-order terms) energy. Both these definitions are made without any explicit reduction. Finally, strong hyperbolicity can be defined through a pseudo-differential reduction to first order. We prove that both definitions of symmetric hyperbolicity are equivalent and that all three definitions of strong hyperbolicity are equivalent (in three space dimensions). We show how to impose maximally dissipative boundary conditions on any symmetric hyperbolic second-order system. We prove that if the second-order system is strongly hyperbolic, any closed constraint evolution system associated with it is also strongly hyperbolic, and that the characteristic variables of the constraint system are derivatives of a subset of the characteristic variables of the main system, with the same speeds.

The Invar tensor package

Abstract The Invar package is introduced, a fast manipulator of generic scalar polynomial expressions formed from the Riemann tensor of a four-dimensional metric-compatible connection. The package can maximally simplify any polynomial containing tensor products of up to seven Riemann tensors within seconds. It has been implemented both in Mathematica and Maple algebraic systems. Program summary Program title: Invar Tensor Package Catalogue identifier: ADZK_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADZK_v1_0.html Program obtainable from: CPC Program Library, Queens University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 136 240 No. of bytes in distributed program, including test data, etc.: 2 711 923 Distribution format: tar.gz Programming language: Mathematica and Maple Computer: Any computer running Mathematica versions 5.0 to 5.2 or Maple versions 9 and 10 Operating system: Linux, Unix, Windows XP RAM: 30 Mb Word size: 64 or 32 bits Classification: 5 External routines: The Mathematica version requires the xTensor and xPerm packages. These are freely available at http://metric.iem.csic.es/Martin-Garcia/xAct Nature of problem: Manipulation and simplification of tensor expressions. Special attention on simplifying scalar polynomial expressions formed from the Riemann tensor on a four-dimensional metric-compatible manifold. Solution method: Algorithms of computational group theory to simplify expressions with tensors that obey permutation symmetries. Tables of syzygies of the scalar invariants of the Riemann tensor. Restrictions: The present versions do not fully address the problem of reducing differential invariants or monomials of the Riemann tensor with free indices. Running time: Less than a second to fully reduce a monomial of the Riemann tensor of degree 7 in terms of independent invariants.

Gauge-invariant and coordinate-independent perturbations of stellar collapse. II: Matching to the exterior

In Paper I in this series we constructed evolution equations for the complete gauge-invariant linear perturbations of a time-dependent spherically symmetric perfect fluid spacetime. A key application of this formalism is the interior of a collapsing star. Here we derive boundary conditions at the surface of the star, matching the interior perturbations to the well-known perturbations of the vacuum Schwarzschild spacetime outside the star.

The Invar tensor package: Differential invariants of Riemann☆

Abstract The long standing problem of the relations among the scalar invariants of the Riemann tensor is computationally solved for all 6 ⋅ 10 23 objects with up to 12 derivatives of the metric. This covers cases ranging from products of up to 6 undifferentiated Riemann tensors to cases with up to 10 covariant derivatives of a single Riemann. We extend our computer algebra system Invar to produce within seconds a canonical form for any of those objects in terms of a basis. The process is as follows: (1) an invariant is converted in real time into a canonical form with respect to the permutation symmetries of the Riemann tensor; (2) Invar reads a database of more than 6 ⋅ 10 5 relations and applies those coming from the cyclic symmetry of the Riemann tensor; (3) then applies the relations coming from the Bianchi identity, (4) the relations coming from commutations of covariant derivatives, (5) the dimensionally-dependent identities for dimension 4, and finally (6) simplifies invariants that can be expressed as product of dual invariants. Invar runs on top of the tensor computer algebra systems xTensor (for Mathematica ) and Canon (for Maple ). Program summary Program title: Invar Tensor Package v2.0 Catalogue identifier: ADZK_v2_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADZK_v2_0.html Program obtainable from: CPC Program Library, Queens University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 3 243 249 No. of bytes in distributed program, including test data, etc.: 939 Distribution format: tar.gz Programming language: Mathematica and Maple Computer: Any computer running Mathematica versions 5.0 to 6.0 or Maple versions 9 and 11 Operating system: Linux, Unix, Windows XP, MacOS RAM: 100 Mb Word size: 64 or 32 bits Supplementary material: The new database of relations is much larger than that for the previous version and therefore has not been included in the distribution. To obtain the Mathematica and Maple database files click on this link. Classification: 1.5, 5 Does the new version supersede the previous version?: Yes. The previous version (1.0) only handled algebraic invariants. The current version (2.0) has been extended to cover differential invariants as well. Nature of problem: Manipulation and simplification of scalar polynomial expressions formed from the Riemann tensor and its covariant derivatives. Solution method: Algorithms of computational group theory to simplify expressions with tensors that obey permutation symmetries. Tables of syzygies of the scalar invariants of the Riemann tensor. Reasons for new version: With this new version, the user can manipulate differential invariants of the Riemann tensor. Differential invariants are required in many physical problems in classical and quantum gravity. Summary of revisions: The database of syzygies has been expanded by a factor of 30. New commands were added in order to deal with the enlarged database and to manipulate the covariant derivative. Restrictions: The present version only handles scalars, and not expressions with free indices. Additional comments: The distribution file for this program is over 53 Mbytes and therefore is not delivered directly when download or Email is requested. Instead a html file giving details of how the program can be obtained is sent. Running time: One second to fully reduce any monomial of the Riemann tensor up to degree 7 or order 10 in terms of independent invariants. The Mathematica notebook included in the distribution takes approximately 5 minutes to run.

Second and higher-order perturbations of a spherical spacetime

The Gerlach and Sengupta (GS) formalism of coordinate-invariant, first-order, spherical and nonspherical perturbations around an arbitrary spherical spacetime is generalized to higher orders, focusing on second-order perturbation theory. The GS harmonics are generalized to an arbitrary number of indices on the unit sphere and a formula is given for their products. The formalism is optimized for its implementation in a computer-algebra system, something that becomes essential in practice given the size and complexity of the equations. All evolution equations for the second-order perturbations, as well as the conservation equations for the energy-momentum tensor at this perturbation order, are given in covariant form, in Regge-Wheeler gauge.

High-order gauge-invariant perturbations of a spherical spacetime

We construct a covariant and gauge-invariant framework to deal with arbitrary high-order perturbations of a spherical spacetime. It can be regarded as the generalization to high orders of the Gerlach and Sengupta formalism for first-order nonspherical perturbations. The Regge-Wheeler-Zerilli harmonics are generalized to an arbitrary number of indices and a closed formula is deduced for their products. An iterative procedure is given in order to construct gauge-invariant quantities up to any perturbative order. Focusing on second-order perturbation theory, we explicitly compute the sources for the gauge invariants as well as for the evolution equations.

Critical Phenomena in Gravitational Collapse: The Role of Angular Momentum

After reviewingthe basics of Critical Phenomena in Gravitational Collapse of a spherically symmetric perfect fluid system, we address the relevance of adding angular momentum to the process. We study two different examples: the same perfect fluid but now with angular momentum, and Vlasov matter (collisionless particles, each with angular momentum). Using linear perturbation theory we show that in the former case there are still critical phenomena, explicitly predictingthe associated scalingla ws. We show that, on the contrary, critical phenomena are not generic for Vlasov matter.