Gravitational Waves General Relativity predicts emission of gravitational waves during the formation and evolution of black holes and neutron stars and the inspiral and coalescence of systems of these stars.

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Black Holes We study the gravitational collapse and formation of black holes and the two-body problem in General Relativity. Neutron Stars Neutron stars are invaluable laboratories for probing the state of matter under extreme conditions, and observations of neutron stars provide complementary information to that obtained from particle accelerators such as the Large Hadron Collider. Name Position. Professor of Applied Mathematics. Professor of Theoretical Physics. Fabian Gittins. Postgraduate research student in Theoretical Astrophysics. Professor of Mathematical Physics, Senior Tutor.

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Recommended for you. PAGE 1. Results for the shock heating test of a cold, relativistically inflowing gas against a wall using the explicit Eulerian techniques of Centrella and Wilson [ 54 ]. The relative error is measured with respect to the average value of the density over a region in the shocked material. The data are from Centrella and Wilson [ 54 ] and the plot reproduces a similar one from Norman and Winkler [ ].

These terms, commonly called Q in the literature see Section 3. However, [ ] proposed to add the Q terms in a relativistically consistent way, in order to consider the artificial viscosity as a real viscosity. Hence, the hydrodynamic equations should be rewritten for a modified stress-energy tensor of the following form:.

In general, Q is a nonlinear function of the velocity and, hence, the quantity QW 2 V in the momentum density of Equation 26 is a highly nonlinear function of the velocity and its derivatives. This fact, together with the explicit presence of the Lorentz factor in the convective terms of the hydrodynamic equations, as well as the pressure in the specific enthalpy, make the relativistic equations much more coupled than their Newtonian counterparts.

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As a result, Norman and Winkler proposed the use of implicit schemes as a way to describe more accurately such coupling. Their code, which in addition incorporates an adaptive grid, reproduces very accurate results even for ultrarelativistic flows with Lorentz factors of about 10 in one-dimensional, flat spacetime simulations. Very recently, Anninos and Fragile [ 13 ] have compared state-of-the-art artificial viscosity schemes and high-order non-oscillatory central schemes see Section 3.

Using a three-dimensional Cartesian code, these authors found that earlier results for artificial viscosity schemes in shock tube tests or shock reflection tests are not improved, i. On the other hand, results for the shock reflection problem with a second-order finite difference central scheme show the suitability of such a scheme to handle ultrarelativistic flows, the underlying reason being, most likely, the use of a conservative formulation of the hydrodynamic equations rather than the particular scheme employed see Section 3.

This formulation was aimed to take fundamental advantage of the hyperbolic and conservative character of the equations, contrary to the one discussed in the previous section. From the numerical point of view, the hyperbolic and conservative nature of the relativistic Euler equations allows for the use of schemes based on the characteristic fields of the system, bringing to relativistic hydrodynamics the existing tools of classical fluid dynamics.

This procedure departs from earlier approaches, most notably in avoiding the need for either artificial dissipation terms to handle discontinuous solutions [ , ], or implicit schemes as proposed in [ ]. If a numerical scheme written in conservation form converges, it automatically guarantees the correct Rankine-Hugoniot jump conditions across discontinuities — the shock-capturing property see, e.

Writing the relativistic hydrodynamic equations as a system of conservation laws, identifying the suitable vector of unknowns, and building up an approximate Riemann solver permitted the extension of state-of-the-art high-resolution shock-capturing schemes HRSC in the following from classical fluid dynamics into the realm of relativity [ ]. Theoretical advances on the mathematical character of the relativistic hydrodynamic equations were first achieved studying the special relativistic limit.

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## General Relativity And Relativistic Astrophysics - Proceedings Of The 5th Canadian Conference

Besides verifying the hyperbolic character of the relativistic hydrodynamic equations, Anile and collaborators [ 10 ] obtained the explicit expressions for the eigenvalues and eigenvectors in the local rest frame, characterized by. In Font et al. The approach followed in [ 93 ] for the equations of special relativistic hydrodynamics was extended to general relativity in [ 21 ].

It comprises the rest-mass density D , the momentum density in the j -direction S j , and the total energy density E , measured by a family of observers which are the natural extension for a generic spacetime of the Eulerian observers in classical fluid dynamics. Interested readers are directed to [ 21 ] for more complete definitions and geometrical foundations. With this choice of variables the equations can be written in conservation form.

## [] General relativity and relativistic astrophysics

Strict conservation is only possible in flat spacetime. For curved spacetimes there exist source terms, arising from the spacetime geometry. However, these terms do not contain derivatives of stress-energy tensor components. More precisely, the first-order flux-conservative hyperbolic system, well suited for numerical applications, reads. The state vector is given by. The local characteristic structure of the previous system of equations was presented in [ 21 ]. The eigenvalues characteristic speeds of the corresponding Jacobian matrices are all real but not distinct, one showing a threefold degeneracy as a result of the assumed directional splitting approach , and a complete set of right-eigenvectors exists.

System 30 satisfies, hence, the definition of hyperbolicity. As it will become apparent in Section 3. This information can be found in [ 21 ] see also [ 96 ]. The range of applications considered so far in general relativity employing the above formulation of the hydrodynamic equations, Equation 30 , 31 , 32 , 33 , is still small and mostly devoted to the study of stellar core collapse and accretion flows onto black holes see Sections 4.

In the special relativistic limit this formulation is being successfully applied to simulate the evolution of ultra- relativistic extragalactic jets, using numerical models of increasing complexity see, e. The first applications in general relativity were performed, in one spatial dimension, in [ ], using a slightly different form of the equations.

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Preliminary investigations of gravitational stellar collapse were attempted by coupling the above formulation of the hydrodynamic equations to a hyperbolic formulation of the Einstein equations developed by [ 39 ]. These results are discussed in [ , 38 ]. More recently, successful evolutions of fully dynamical spacetimes in the context of adiabatic stellar core collapse, both in spherical symmetry and in axisymmetry, have been achieved [ , , 67 ]. These investigations are considered in Section 4.

An ambitious three-dimensional, Eulerian code which evolves the coupled system of Einstein and hydrodynamics equations was developed by Font et al. The formulation of the hydrodynamic equations in this code follows the conservative Eulerian approach discussed in this section.

The code is constructed for a completely general spacetime metric based on a Cartesian coordinate system, with arbitrarily specifiable lapse and shift conditions. In [ 96 ] the spectral decomposition eigenvalues and right-eigenvectors of the general relativistic hydrodynamic equations, valid for general spatial metrics, was derived, extending earlier results of [ 21 ] for non-diagonal metrics.

Due to the paramount importance of the characteristic structure of the equations in the design of upwind HRSC schemes based upon Riemann solvers, we summarize all necessary information in Section 5. General covariant conservative formulations of the general relativistic hydrodynamic equations for ideal fluids, i. The form invariance of these approaches with respect to the nature of the spacetime foliation implies that existing work on highly specialized techniques for fluid dynamics i. In the next two sections we describe the existing covariant formulations in some detail.

Eulderink and Mellema [ 76 , 78 ] first derived a covariant formulation of the general relativistic hydrodynamic equations. As in the formulation discussed in Section 2.

## Numerical Hydrodynamics in General Relativity

Such a solver was first applied to the non-relativistic Euler equations in [ ] and has been widely employed since in simulating compressible flows in computational fluid dynamics. After the appropriate choice of the state vector variables, the conservation laws, Equations 7 and 8 , are re-written in flux-conservative form. The vector F 0 represents the state vector the unknowns , and each vector F i is the corresponding flux in the coordinate direction x i.

Further technical details can be found in [ 76 , 78 ].