Philippos Papadopoulos

Towards a stable numerical evolution of strongly gravitating systems in general relativity: The conformal treatments

We study the stability of three-dimensional numerical evolutions of the Einstein equations, comparing the standard ADM formulation to variations on a family of formulations that separate out the conformal and traceless parts of the system. We develop an implementation of the conformal-traceless ~CT! approach that has improved stability properties in evolving weak and strong gravitational fields, and for both vacuum and spacetimes with active coupling to matter sources. Cases studied include weak and strong gravitational wave packets, black holes, boson stars and neutron stars. We show under what conditions the CT approach gives better results in 3D numerical evolutions compared to the ADM formulation. In particular, we show that our implementation of the CT approach gives more long term stable evolutions than ADM in all the cases studied, but is less accurate in the short term for the range of resolutions used in our 3D simulations.

Dynamics of perturbations of rotating black holes

We present a numerical study of the time evolution of perturbations of rotating black holes. The solutions are obtained by integrating the Teukolsky equation written as a first order in time, coupled system of equations. We address the numerical difficulties of solving the equation in its original form. We follow the propagation of generic initial data through the burst, quasinormal ringing and power-law tail phases. In particular, we calculate the effects due to the rotation of the black hole on the scattering of incident gravitational wave pulses. These effects include the amplitude enhancement due to so-called super-radiance. The results may help explain how the angular momentum of the black hole affects the gravitational waves that are generated during the final stages of black hole coalescence.

Non-axisymmetric relativistic Bondi-Hoyle accretion on to a Kerr black hole

In our programme of studying numerically the so-called Bondi–Hoyle accretion in the fully relativistic regime, we present here the first results concerning the evolution of matter accreting supersonically on to a rotating (Kerr) black hole. These computations generalize previous results where the non-rotating (Schwarzschild) case was extensively considered. We parametrize our initial data by the asymptotic conditions for the fluid and explore the dependence of the solution on the angular momentum of the black hole. Towards quantifying the robustness of our numerical results, we use two different geometrical foliations of the black hole space–time, the standard form of the Kerr metric in Boyer–Lindquist coordinates as well as its Kerr–Schild form, which is free of coordinate singularities at the black hole horizon. We demonstrate some important advantages of using such horizon-adapted coordinate systems. Our numerical study indicates that regardless of the value of the black hole spin the final accretion pattern is always stable, leading to constant accretion rates of mass and momentum. The flow is characterized by a strong tail shock, which, unlike the Schwarzschild case, is increasingly wrapped around the central black hole as the hole angular momentum increases. The rotation-induced asymmetry in the pressure field implies that, besides the well-known drag, the black hole will experience also a lift normal to the flow direction. This situation exhibits some analogies with the Magnus effect of classical fluid dynamics.

Relativistic hydrodynamics around black holes and horizon adapted coordinate systems

Despite the fact that the Schwarzschild and Kerr solutions for the Einstein equations, when written in standard Schwarzschild and Boyer-Lindquist coordinates, present coordinate singularities, all numerical studies of accretion flows onto collapsed objects have been widely using them over the years. This approach introduces conceptual and practical complications in places where a smooth solution should be guaranteed, i.e., at the gravitational radius. In the present paper, we propose an alternative way of solving the general relativistic hydrodynamic equations in background (fixed) black hole spacetimes. We identify classes of coordinates in which the (possibly rotating) black hole metric is free of coordinate singularities at the horizon, independent of time, and admits a spacelike decomposition. In the spherically symmetric, non-rotating case, we re-derive exact solutions for dust and perfect fluid accretion in Eddington-Finkelstein coordinates, and compare with numerical hydrodynamic integrations. We perform representative axisymmetric computations. These demonstrations suggest that the use of those coordinate systems carries significant improvements over the standard approach, especially for higher dimensional studies.

Relativistic hydrodynamics on space - like and null surfaces: Formalism and computations of spherically symmetric space-times

We introduce a formulation of Eulerian general relativistic hydrodynamics which is applicable for (perfect) fluid data prescribed on either spacelike or null hypersurfaces. Simple explicit expressions for the characteristic speeds and fields are derived in the general case. A complete implementation of the formalism is developed in the case of spherical symmetry. The algorithm is tested in a number of different situations, predisposing for a range of possible applications. We consider the Riemann problem for a polytropic gas, with initial data given on a retarded/advanced time slice of Minkowski spacetime. We compute perfect fluid accretion onto a Schwarzschild black hole spacetime using ingoing null Eddington-Finkelstein coordinates. Tests of fluid evolution on dynamic background include constant density and TOV stars sliced along the radial null cones. Finally, we consider the accretion of self-gravitating matter onto a central black hole and the ensuing increase in the mass of the black hole horizon.

A "horizon-adapted" approach to the study of relativistic accretion flows onto rotating black holes

We present a new geometrical approach to the study of accretion flows onto rotating black holes. Instead of Boyer-Lindquist coordinates, the standard choice in all existing numerical simulations in the literature, we employ the simplest example of a horizon-adapted coordinate system, the Kerr-Schild coordinates. This choice eliminates unphysical divergent behavior at the event horizon. Computations of Bondi-Hoyle accretion onto extreme Kerr black holes, performed here for the first time, demonstrate the key advantages of this procedure. We argue that it offers the best approach to the numerical study of the observationally increasingly, more accessible, relativistic inner region around black holes.

Spherical collapse of supermassive stars: Neutrino emission and gamma-ray bursts

We present the results of numerical simulations of the spherically symmetric gravitational collapse of supermassive stars (SMS). The collapse is studied using a general relativistic hydrodynamics code. The coupled system of Einstein and fluid equations is solved employing coordinates adapted to a foliation of the spacetime by means of outgoing null hypersurfaces. The code contains an equation of state which includes eects due to radiation, electrons and baryons, and detailed microphysics to account for electron-positron pairs. In addition energy losses by thermal neutrino emission are included. We are able to follow the collapse of SMS from the onset of instability up to the point of black hole formation. Several SMS with masses in the range 5 10 5 M 10 9 M are simulated. In all models an apparent horizon forms initially, enclosing the innermost 25% of the stellar mass. From the computed neutrino luminosities, estimates of the energy deposition by -annihilation are obtained. Only a small fraction of this energy is deposited near the surface of the star, where, as proposed recently by Fuller & Shi (1998), it could cause the ultrarelativistic flow believed to be responsible for -ray bursts. Our simulations show that for collapsing SMS with masses larger than 5 10 5 M the energy deposition is at least two orders of magnitude too small to explain the energetics of observed long-duration bursts at cosmological redshifts. In addition, in the absence of rotational eects the energy is deposited in a region containing most of the stellar mass. Therefore relativistic ejection of matter is impossible.

Scalar field induced oscillations of relativistic stars and gravitational collapse

We study the interaction of massless scalar fields with relativistic stars by means of fully dynamic numerical simulations of the Einstein-Klein-Gordon perfect fluid system. Our investigation is restricted to spherical symmetry and the stars are approximated by relativistic polytropes. Studying the nonlinear dynamics of isolated compact objects is very effectively performed within the characteristic formulation of general relativity, in which the spacetime is foliated by a family of outgoing light cones. We are able to compactify the entire spacetime on a computational grid and simultaneously impose natural radiative boundary conditions and extract accurate radiative signals. We study the transfer of energy from the scalar field to the fluid star. We find, in particular, that depending on the compactness of the stellar model, the scalar wave forces the star either to oscillate in its radial modes of pulsation or to undergo gravitational collapse to a black hole on a dynamical time scale. The radiative signal, read off at future null infinity, shows quasinormal oscillations before the setting of a late time power-law tail.

Dynamics of scalar fields in the background of rotating black holes

A numerical study of the evolution of a massless scalar field in the background of rotating black holes is presented. First, solutions to the wave equation are obtained for slowly rotating black holes. In this approximation, the background geometry is treated as a perturbed Schwarzschild spacetime with the angular momentum per unit mass playing the role of a perturbative parameter. To first order in the angular momentum of the black hole, the scalar wave equation yields two coupled one-dimensional evolution equations for a function representing the scalar field in the Schwarzschild background and a second field that accounts for the rotation. Solutions to the wave equation are also obtained for rapidly rotating black holes. In this case, the wave equation does not admit complete separation of variables and yields a two-dimensional evolution equation. The study shows that, for rotating black holes, the late time dynamics of a massless scalar field exhibit the same power-law behavior as in the case of a Schwarzschild background independently of the angular momentum of the black hole.

Simulating the dynamics of relativistic stars via a light-cone approach

We present new numerical algorithms for the coupled Einstein--perfect-fluid system in axisymmetry. Our framework uses a foliation based on a family of light cones, emanating from a regular center, and terminating at future null infinity. This coordinate system is well adapted to the study of the dynamical spacetimes associated with isolated relativistic compact objects such as neutron stars. In particular, the approach allows the unambiguous extraction of gravitational waves at future null infinity and avoids spurious outer boundary reflections. The code can accurately maintain long-term stability of polytropic equilibrium models of relativistic stars. We demonstrate global energy conservation in a strongly perturbed neutron star spacetime, for which the total energy radiated away by gravitational waves corresponds to a significant fraction of the Bondi mass. As a first application we present results in the study of pulsations of axisymmetric relativistic stars, extracting the frequencies of the different fluid modes in fully relativistic evolutions of the Einstein--perfect-fluid system and making a first comparison between the gravitational news function and the predicted wave using the approximations of the quadrupole formula.