Tomáš Ledvinka

General covariant evolution formalism for numerical relativity

A general-covariant extension of Einsteins field equations is considered with a view to numerical relativity applications. The basic variables are taken to be the metric tensor and an additional four-vector

Relativistic closed-form Hamiltonian for many-body gravitating systems in the post-Minkowskian approximation.

{Z}_{\ensuremath{\mu}}.

A Symmetry breaking mechanism for the Z4 general covariant evolution system

Einsteins solutions are recovered when the additional four-vector vanishes, so that the energy and momentum constraints amount to the covariant algebraic condition

Static fluid cylinders and their fields: global solutions

{Z}_{\ensuremath{\mu}}=0.

Constraint-preserving boundary conditions in the Z4 numerical relativity formalism

The extended field equations can be supplemented by suitable coordinate conditions in order to provide symmetric hyperbolic evolution systems: this is actually the case for either harmonic coordinates or normal coordinates with harmonic slicing.

A 3+1 covariant suite of numerical relativity evolution systems

The Hamiltonian for a system of relativistic bodies interacting by their gravitational field is found in the post-Minkowskian approximation, including all terms linear in the gravitational constant. It is given in a surprisingly simple closed form as a function of canonical variables describing the bodies only. The field is eliminated by solving inhomogeneous wave equations, applying transverse-traceless projections, and using the Routh functional. By including all special relativistic effects our Hamiltonian extends the results described in classical textbooks of theoretical physics. As an application, the scattering of relativistic objects is considered.

Black holes and magnetic fields

The general-covariant Z4 formalism is further analyzed. The gauge conditions are generalized with a view to Numerical Relativity applications and the conditions for obtaining strongly hyperbolic evolution systems are given both at the first and the second order levels. A symmetry-breaking mechanism is proposed that allows one, when applied in a partial way, to recover previously proposed strongly hyperbolic formalisms, like the BSSN and the Bona-Masso ones. When applied in its full form, the symmetry breaking mechanism allows one to recover the full five-parameter family of first order KST systems. Numerical codes based in the proposed formalisms are tested. A robust stability test is provided by evolving random noise data around Minkowski space-time. A strong field test is provided by the collapse of a periodic background of plane gravitational waves, as described by the Gowdy metric.

Comparing Hamiltonians of a spinning test particle for different tetrad fields

The global properties of static perfect-fluid cylinders and their external Levi-Civita fields are studied both analytically and numerically. The existence and uniqueness of global solutions is demonstrated for a fairly general equation of state of the fluid. In the case of a fluid admitting a non-vanishing density for zero pressure, it is shown that the cylinders radius has to be finite. For incompressible fluid, the field equations are solved analytically for nearly Newtonian cylinders and numerically in fully relativistic situations. Various physical quantities such as proper and circumferential radii, external conicity parameter and masses per unit proper/coordinate length are exhibited graphically.

POST-MINKOWSKIAN CLOSED-FORM HAMILTONIAN FOR GRAVITATING N-BODY SYSTEMS

The constraint-preserving approach is discussed in parallel with other recent developments with the goal of providing consistent boundary conditions for numerical relativity simulations. The case of the first-order version of the Z4 system is considered, and constraint-preserving boundary conditions of the Sommerfeld type are provided. The stability of the proposed boundary conditions is related to the choice of the ordering parameter. This relationship is explored numerically and some values of the ordering parameter are shown to provide stable boundary conditions in the absence of corners and edges. Maximally dissipative boundary conditions are also implemented. In this case, a wider range of values of the ordering parameter is allowed, which is shown numerically to provide stable boundary conditions even in the presence of corners and edges.

Bronislaw Malinowski and the Anthropology of Law

A suite of three evolution systems is presented in the framework of the 3+1 formalism. The first one is of second order in space derivatives and has the same causal structure of the Baumgarte-Shapiro-Shibata-Nakamura (BSSN) system for a suitable choice of parameters. The second one is the standard first order version of the first one and has the same causal structure of the Bona-Masso system for a given parameter choice. The third one is obtained from the second one by reducing the space of variables in such a way that the only modes that propagate with zero characteristic speed are the trivial ones. This last system has the same structure of the ones recently presented by Kidder, Scheel and Teukolski: the correspondence between both sets of parameters is explicitly given. The fact that the suite started with a system in which all the dynamical variables behave as tensors (contrary to what happens with BSSN system) allows one to keep the same parametrization when passing from one system to the next in the suite. The direct relationship between each parameter and a particular characteristic speed, which is quite evident in the second and the third systems, is a direct consequence of the manifest 3+1 covariance of the approach.