Get PDF General Relativity, Cosmology and Astrophysics: Perspectives 100 years after Einsteins stay in Prague

Free download. Book file PDF easily for everyone and every device. You can download and read online General Relativity, Cosmology and Astrophysics: Perspectives 100 years after Einsteins stay in Prague file PDF Book only if you are registered here. And also you can download or read online all Book PDF file that related with General Relativity, Cosmology and Astrophysics: Perspectives 100 years after Einsteins stay in Prague book. Happy reading General Relativity, Cosmology and Astrophysics: Perspectives 100 years after Einsteins stay in Prague Bookeveryone. Download file Free Book PDF General Relativity, Cosmology and Astrophysics: Perspectives 100 years after Einsteins stay in Prague at Complete PDF Library. This Book have some digital formats such us :paperbook, ebook, kindle, epub, fb2 and another formats. Here is The CompletePDF Book Library. It's free to register here to get Book file PDF General Relativity, Cosmology and Astrophysics: Perspectives 100 years after Einsteins stay in Prague Pocket Guide.

We're sorry, something went wrong. Please try again. About this product. Make an offer:. Stock photo.

Account Options

Brand new: lowest price The lowest-priced brand-new, unused, unopened, undamaged item in its original packaging where packaging is applicable. Buy It Now. Add to cart. Make Offer. New condition. See photos. See details.

Einstein and the Monks - Chris Impey at TEDx Tucson 2013

Mars, A. Pasar al contenido principal.

Noticias Todas las noticias. Marc Mars Lloret. Nombre: Marc. Apellidos: Mars Lloret. Despacho: T Regge and Wheeler worked out the equations for axial perturbations of Schwarzschild BHs; later on, Zerilli derived the equations for polar perturbations [ ]. It turns out to be possible to define a specific combination of the axial perturbation functions , and a combination of the polar perturbation functions , K lm which describe completely the propagation of GWs.

This approach was soon extended to general spherically symmetric BH backgrounds and a gauge-invariant formulation in terms of specific combinations of the perturbation functions that remain unchanged under perturbative coordinate transformations [ , ]. We further note that the Weyl and Riemann tensors are identical in vacuum. In this framework, the perturbation equations reduce to a wave equation for the perturbation of which is called the Teukolsky equation [ ]. The main advantage of the Bardeen-Press-Teukolsky approach is that it is possible to separate the angular dependence of perturbations of the Kerr background, even though such background is not spherically symmetric.

Its main drawback is that it is very difficult to extend it beyond its original setup, i. The tensor harmonic approach is much more flexible. In particular, spacetime perturbation theory with tensor harmonic decomposition has been extended to spherically symmetric stars [ , , , ] the extension to rotating stars is much more problematic [ ]. As we discuss in Section 5. It is not clear whether such generalizations are possible with the Bardeen-Press-Teukolsky approach. The sources describe the objects that excite the spacetime perturbations, and can arise either directly from a non-vanishing stress-energy tensor or by imposing suitable initial conditions on the spacetime.

In the point particle limit the source term is a nontrivial perturbing stress-tensor, which describes for instance the infall of a small object along generic geodesics. Thus, the spacetime perturbation approach is in principle able to describe qualitatively, if not quantitatively, highly dynamic BHs under general conditions. The original approach treats the small test particle moving along a geodesic of the background spacetime.

Gravitational back-reaction can be included by taking into account the energy and angular momentum loss of the particle due to GW emission [ , , ]. For a general account on the self-force problem, we refer the interested reader to the Living Reviews article on the subject [ ]. In the close limit approximation the source term can be traced back to nontrivial initial conditions. In particular, the original approach tackles the problem of two colliding, equal-mass BHs, from an initial separation small enough that they are initially surrounded by a common horizon.

Thus, this problem can be looked at as a single perturbed BH, for which some initial conditions are known [ , ]. Because at late times the forcing caused by the source term has died away, it is natural to describe this phase as the free oscillations of a BH, or in other words as solutions of the homogeneous version of Eq. Together with the corresponding boundary conditions, the Regge-Wheeler and Zerilli equations then describe a freely oscillating BH. Due to GW emission, these oscillations are damped, i.

Such intuitive picture of BH ringdown can be given a formally rigorous meaning through contour integration techniques [ , 95 ]. The extension of the Regge-Wheeler-Zerilli approach to asymptotically dS or AdS spacetimes follows with the procedure outlined above and decomposition 16 ; see also Ref. It turns out that the Teukolsky procedure can also be generalized to these spacetimes [ , , ]. The Regge-Wheeler-Zerilli approach has proved fruitful also in other contexts including alternative theories of gravity.

Generically, the decomposition works by using the same metric ansatz as in Eq. Important examples where this formalism has been applied include scalar-tensor theories [ , , ], Dynamical Chern-Simons theory [ , , ], Einstein-Dilaton-Gauss-Bonnet [ ], Horndeski gravity [ , ], and massive theories of gravity [ ].

Spacetime perturbation theory is a powerful tool to study BHs in higher-dimensional spacetimes. The tensor harmonic approach has been successfully extended by Kodama and Ishibashi [ , ] to GR in higher-dimensional spacetimes, with or without cosmological constant. Since many dynamical processes involving higher-dimensional BHs in particular, the collisions of BHs starting from finite distance can be described in the far field limit by a perturbed spherically symmetric BH spacetime, the Kodama and Ishibashi approach can be useful to study the GW emission in these processes.

  • Advances in Protective Structures Research.
  • 14th IEEE International Conference on Embedded and Real-time Computing Systems and Applications (Rtcsa 2008).
  • Lassar-Cohn-Application of some general reactions to investigation of organic chemistry.
  • All AEI publications | Max Planck Institute for Gravitational Physics (Albert Einstein Institute).
  • Literature and Evil?
  • About the conference.
  • Linking Constructions into Functional Linguistics: The role of constructions in grammar.

The relevance of this approach therefore extends well beyond the study of spherically symmetric solutions. For applications of this tool to the wave extraction of NR simulations see for instance [ ]. The GW amplitude and its energy and momentum fluxes can be expressed in terms of these master functions. For illustration of this procedure, we consider here the special case of scalar perturbations. We define the gauge-invariant quantities. The total radiated energy is obtained from integration in time and summation over all multipoles.

In summary, this approach can be used, in analogy with the Regge-Wheeler-Zerilli formalism in four dimensions, to determine the quasi-normal mode spectrum see, e. The generalization of this setup to higher-dimensional rotating Myers-Perry [ ] BHs is still an open issue, since the decoupling of the perturbation equations has so far only been obtained in specific cases and for a subset of the perturbations [ , , ].

Spacetime perturbation theory has also been used to study other types of higher-dimensional objects as for example black strings. Gregory and Laflamme [ , ] considered a very specific sector of the possible gravitational perturbations of these objects, whereas Kudoh [ ] performed a complete analysis that builds on the Kodama-Ishibashi approach. Astrophysical systems.

Relativity and Gravitation

Perturbation theory has been applied extensively to the modelling of BHs and compact stars, either without source terms, including in particular quasi-normal modes [ , , 95 ], or with point particle sources. Note that wave emission from extended matter distributions can be understood as interference of waves from point particles [ , , ]. Equations for perturbations of stars have been derived for spherically symmetric [ , , ] and slowly rotating stars [ , ]. Equations of BH perturbations with a point particle source have been studied as a tool to understand BH dynamics.

This is a decades old topic, historically divided into investigations of circular and quasi-circular motion, and head-ons or scatters. Circular and quasi-circular motion. Gravitational radiation from point particles in circular geodesics was studied in Refs. This problem was reconsidered and thoroughly analyzed by Poisson, Cutler and collaborators, and by Tagoshi, Sasaki and Nakamura in a series of elegant works, where contact was also made with the PN expansion see the Living Reviews article [ ] and references therein.

The emission of radiation, together with the self-gravity of the objects implies that particles do not follow geodesics of the background spacetime. Inclusion of dissipative effects is usually done by balance-type arguments [ , , , ] but it can also be properly accounted for by computing the self-force effects of the particle motion see the Living Reviews article [ ] and references therein.

EM waves from particles in circular motion around BHs were studied in Refs. Head-on or finite impact parameter collisions: non-rotating BHs. Seminal work by Davis et al. This work has been generalized to include head-on collisions at non-relativistic velocities [ , , , 93 ], at exactly the speed of light [ , 93 ], and to non-head-on collisions at non-relativistic velocities [ , 93 ].

Holdings information at the University of Liverpool Library

The infall of multiple point particles has been explored in Ref. Shapiro and collaborators have investigated the infall or collapse of extended matter distributions through superpositions of point particle waveforms [ , , ]. Electromagnetic radiation from high-energy collisions of charged particles with uncharged BHs was studied in Ref. Head-on or finite impact parameter collisions: rotating BHs. Gravitational radiation from point particle collisions with Kerr BHs has been studied in Refs.

Suggestions that cosmic censorship might fail in high-energy collisions with near-extremal Kerr BHs, have recently inspired further scrutiny of these scenarios [ 71 , 72 ] as well as the investigation of enhanced absorption effects in the ultra-relativistic regime [ ]. Close Limit approximation. The close limit approximation was first compared against nonlinear simulations of equal-mass, non-rotating BHs starting from rest [ ]. It has since been generalized to unequal-mass [ 35 ] or even the point particle limit [ ], rotating BHs [ ] and boosted BHs at second-order in perturbation theory [ ].

Recently the close limit approximation has also been applied to initial configurations constructed with PN methods [ ]. Beyond electrovacuum GR. The resurgence of scalar-tensor theories as a viable and important prototype of alternative theories of gravity, as well as the conjectured existence of a multitude of fundamental bosonic degrees of freedom, has revived interest in BH dynamics in the presence of fundamental fields. Radiation from collisions of scalar-charged particles with BHs was studied in Ref. Radiation from massive scalar fields around rotating BHs was studied in Ref.

Similar effects do not occur for massless gravitons [ ]. Beyond four-dimensions and asymptotic flatness. The formalism to handle gravitational perturbations of four-dimensional, spherically symmetric asymptotically A dS BHs has been developed in Ref. Gravitational perturbations of higher-dimensional BHs can be handled through the elegant approach by Kodama and Ishibashi [ , ], generalized in Ref.

Perturbations of higher-dimensional, rotating BHs can be expressed in terms of a single master variable only in few special cases [ ]. The generic case has been handled by numerical methods in the linear regime [ , ]. We are not aware of any studies on gravitational or electromagnetic radiation emitted by particles in orbit about BHs in spacetimes with a cosmological constant. The quadrupole formula was generalized to higher-dimensional spacetimes in Ref. The first fully relativistic calculation of GWs generated by point particles falling from rest into a higher-dimensional asymptotically flat non-rotating BH was done in Ref.

The mass multipoles induced by an external gravitational field i. The close limit approximation was extended to higher-dimensional, asymptotically flat, space-times in Refs. While conceptually simple, the spacetime perturbation approach does involve solving one or more second-order, non-homogeneous differential equations. A very simple and useful estimate of the energy spectrum and total radiated gravitational energy can be obtained by using what is known as the ZFL or instantaneous collision approach. The technique was derived by Weinberg in [ , ] from quantum arguments, but it is equivalent to a purely classical calculation [ ].

The approach is a consequence of the identity. Thus, the low-frequency spectrum depends exclusively on the asymptotic state of the colliding particles which can be readily computed from their Coulomb gravitational fields. Because the energy spectrum is related to Open in a separate window via.

Furthermore, if the asymptotic states are an accurate description of the collision at all times, as for instance if the colliding particles are point-like, then one expects the ZFL to be an accurate description of the problem. The particles collide head-on along the z -axis and we use standard spherical coordinates. The spectrum is flat, i. The approach neglects the details of the interaction and the internal structure of the colliding and final objects, and the price to pay is the absence of a lengthscale, and therefore the appearance of this divergence.

The divergence can be cured by introducing a phenomenological cutoff in frequency. These results have been generalized to include collisions with finite impact parameter and to a computation of the radiated momentum as well [ , 93 ]. Finally, recent nonlinear simulations of high-energy BH or star collisions yield impressive agreement with ZFL predictions [ , 93 , , ]. The zero-frequency limit for head-on collisions of particles was used by Smarr [ ] to understand gravitational radiation from BH collisions and in Ref. It was later generalized to the nontrivial finite impact parameter case [ ], and compared extensively with fully nonlinear numerical simulations [ 93 ].

Beyond four-dimensional, electrovacuum GR. Recent work has started applying the ZFL to other spacetimes and theories. Brito [ ] used the ZFL to understand head-on collisions of scalar charges with four-dimensional BHs.

Mathematical General Relativity - INSPIRE-HEP

The ZFL has been extended to higher dimensions in Refs. An alternative technique to model the dynamics of collisons of two particles or two BHs at high energies describes the particles as gravitational shock waves. This method yields a bound on the emitted gravitational radiation using an exact solution, and provides an estimate of the radiation using a perturbative method.

In the following we shall review both. In this limit, the geometry becomes that of an impulsive or shock gravitational pp -wave, i. This is the Aichelburg-Sexl geometry [ 16 ] for which the curvature has support only on a null plane. In Brinkmann coordinates, the line element is:.

The usefulness of shock waves in modelling collisions of particles or BHs at very high energies relies on the following fact. Since the geometry of a single shock wave is flat outside a null plane, one can superimpose two shock wave solutions traveling in opposite directions and still obtain an exact solution of the Einstein equations, valid up to the moment when the two shock waves collide.

But it is more convenient to write down the geometry in coordinates for which test particle trajectories vary continuously as they cross the shock. These are called Rosen coordinates, ; their relation with Brinkmann coordinates can be found in [ ] and the line element for the superposition becomes. This metric is a valid description of the spacetime with the two shock waves except in the future light-cone of the collision, which occurs at.

Remarkably, and despite not knowing anything about the future development of the collision, an AH can be found for this geometry within its region of validity, as first pointed out by Penrose. Its existence indicates that a BH forms and moreover its area provides a lower bound for the mass of the BH [ ]. This AH is the union of two surfaces,. The relevant null normals to S 1 and S 2 are, respectively,.

One must then guarantee that these normals have zero expansion and are continuous at the intersection. This yields the solution. In particular, at the intersection, the AH has a polar radius. Instead of providing a bound on the inelasticity, a more ambitious program is to determine the exact inelasticity by solving the Einstein equations in the future of the collision. Whereas an analytic exact solution seems out of reach, a numerical solution of the fully nonlinear field equations might be achievable, but none has been reported. The approach that has produced the most interesting results, so far, is to solve the Einstein equations perturbatively in the future of the collision.

In a boosted frame, say in the negative z direction, one of the shock waves will become blueshifted whereas the other will become redshifted. It so happens that expressing the exact solution in such perturbative fashion only has terms up to second order:. This perturbative expansion is performed in dimensionless coordinates of Brinkmann type, as in Eq. The geometry to the future of the strong shock, on the other hand, will be of the form.

For instance, to obtain one solves the linearized Einstein equations. In the de Donder gauge these yield a set of decoupled wave equations of the form , where the is the trace reversed metric perturbation. The wave equation must then be subjected to the boundary conditions At higher orders, the problem can also be reduced to solving wave equations for but now with sources provided by the perturbations of lower order [ ]. After obtaining the metric perturbations to a given order, one must still compute the emitted gravitational radiation, in order to obtain the inelasticity.

The first-order results can equivalently be obtained using the Landau-Lifshitz pseudo-tensor for GW extraction [ ]. The results in first and second order are, respectively:. Let us close this subsection with three remarks on these results. Firstly, the results 38 are below the AH bound 35 , as they should.

Secondly, and as we shall see in Section 7. Finally, as we comment in the next subsection, the generalisation to higher dimensions of the first-order result reveals a remarkably simple pattern. The technique of superimposing two Aichelburg-Sexl shock waves [ 16 ] was first used by Penrose in unpublished work but quoted, for instance, in Ref. Penrose showed the existence of an AH for the case of a head-on collision, thus suggesting BH formation.

Computing the area of the AH yields an upper bound on the fraction of the overall energy radiated away in GWs, i. They computed the metric in the future of the collision perturbatively to second order in the head-on case. A formalism for higher order and the caveats of the method in the presence of electric charge were exhibited in [ ]. AH formation in shock wave collisions with generalized profiles and asymptotics has been studied in [ 19 , , 31 , ].

Generating time-dependent solutions to the Einstein equations using numerical methods involves an extended list of ingredients which can be loosely summarized as follows. Choose a specific formulation that admits a well-posed IBVP, i. Handle singularities such that they do not result in the generation of non-assigned numbers which rapidly swamp the computational domain. Construct initial data that solve the Einstein constraint equations and represent a realistic snapshot of the physical system under consideration.

Apply diagnostic tools that measure GWs, BH horizons, momenta and masses, and other fields. Elegant though this tensorial form of the equations is from a mathematical point of view, it is not immediately suitable for a numerical implementation. For one thing, the character of the equations as a hyperbolic, parabolic or elliptic system is not evident. In other words, are we dealing with an initial-value or a boundary-value problem? Well-posedness of the IBVP then requires a suitable formulation of the evolution equations, boundary conditions and initial data. We shall discuss this particular aspect in more detail further below, but first consider the general structure of the equations.

A key ingredient for the spacetime split of the equations is the projection of tensors onto time and space directions. For this purpose, the space projection operator is defined as. For a generic tensor , its spatial projection is given by projecting each index speparately. A particularly important tensor is obtained from the spatial projection of the spacetime metric.

As we see from Eq. The final ingredient required for the spacetime split of the Einstein equations is the extrinsic curvature or second fundamental form defined as. A straightforward calculation shows that the extrinsic curvature can be expressed in terms of the Lie derivative of the spatial metric along either n or m according to. We have now assembled all tools to calculate the spacetime projections of the Riemann tensor. In the following order, these are known as the Gauss, the contracted Gauss, the scalar Gauss, the Codazzi, the contracted Codazzi equation, as well as the final projection of the Riemann tensor and its contractions:.

For simplicity, we have kept all spacetime indices here even for spatial tensors. As mentioned above, the time components can and will be discarded eventually. By using Eq. It turns out helpful for this purpose to introduce the corresponding projections of the energy-momentum tensor which are given by.

Using the explicit expressions for the Lie derivatives. By virtue of the Bianchi identities, the constraints 54 and 55 are preserved under the evolution equations. The suitability of a given system of differential equations for a numerical time evolution critically depends on a continuous dependency of the solution on the initial data. This aspect is referred to as well posedness of the IBVP and is discussed in great detail in Living Reviews articles and other works [ , , , ].

Here, we merely list the basic concepts and refer the interested reader to these articles. Consider for simplicity an initial-value problem in one space and one time dimension for a single variable u t, x on an unbounded domain. We note that F t may be a rapidly growing function, for example an exponential, so that well posedness represents a necessary but not sufficient criterion for suitability of a numerical scheme.

Well posedness of formulations of the Einstein equations is typically studied in terms of the hyperbolicity properties of the system in question. Hyperbolicity of a system of PDEs is often defined in terms of the principal part , that is, the terms of the PDE which contain the highest-order derivatives. We consider for simplicity a quasilinear first-order system for a set of variables u t, x. The system is called strongly hyperbolic if P is a smooth differential operator and its associated principal symbol is symmetrizeable [ ]. For the special case of constant coefficient systems this definition simplifies to the requirement that the principal symbol has only imaginary eigenvalues and a complete set of linearly independent eigenvectors.

If linear independence of the eigenvectors is not satisfied, the system is called weakly hyperbolic. For more complex systems of equations, strong and weak hyperbolicity can be defined in a more general fashion [ , , , ]. In our context, it is of particular importance that strong hyperbolicity is a necessary condition for a well posed IBVP [ , ]. The ADM equations 52 — 53 , in contrast, have been shown to be weakly but not strongly hyperbolic for fixed gauge [ ]; likewise, a first-order reduction of the ADM equations has been shown to be weakly hyperbolic [ ]. These results strongly indicate that the ADM formulation is not suitable for numerical evolutions of generic spacetimes.

It is interesting to note that the BSSN formulation had been developed in the s before a comprehensive understanding of the hyperbolicity properties of the Einstein equations had been obtained; it was only about a decade after its first numerical application that strong hyperbolicity of the BSSN system [ ] was demonstrated.

Publications of AEI Potsdam and AEI Hannover

We see here an example of how powerful a largely empirical approach can be in the derivation of successful numerical methods. And yet, our understanding of the mathematical properties is of more than academic interest as we shall see in Section 6. The modification of the ADM equations which results in the BSSN formulation consists of a trace split of the extrinsic curvature, a conformal decomposition of the spatial metric and of the traceless part of the extrinsic curvature and the introduction of the contracted Christoffel symbols as independent variables.

For generality, we will again write the definitions of the variables and the equations for the case of an arbitrary number D of spacetime dimensions. We define. Note that the definition 58 implies two algebraic and one differential constraints. Inserting the definition 58 into the ADM equations 52 — 53 and using the Hamiltonian and momentum constraints respectively in the evolution equations for K and results in the BSSN evolution system. In practical applications, it turns out necessary for numerical stability to enforce the algebraic constraint whereas enforcement of the unit determinant appears to be optional.

A further subtlety is concerned with the presence of the conformal connection functions on the right-hand side of the BSSN equations. Two recipes have been identified that provide long-term stable numerical evolutions. An overview of the specific choices of variables and treatment of the BSSN constraints for the present generation of codes is given in Section 4 of [ ].

Abstracts and Presentations

It has been realized a long time ago that the Einstein equations have a mathematically appealing form if one imposes the harmonic gauge condition [ ]. Taking the derivative of this condition eliminates a specific combination of second derivatives from the Ricci tensor such that its principal part is that of the scalar wave operator.

In consequence of this simplification of the principal part, the Einstein equations in harmonic gauge can straightforwardly be written as a strongly hyperbolic system. This formulation even satisfies the stronger condition of symmetric hyperbolicity which is defined in terms of the existence of a conserved, positive energy [ ], and harmonic coordinates have played a key part in establishing local uniqueness of the solution to the Cauchy problem in GR [ , , ]. This particularly appealing property of the Ricci tensor can be maintained for arbitrary coordinates by introducing the functions [ , ].

This is called the Generalized Harmonic Gauge formulation. With this definition, it turns out convenient to consider the generalized class of equations. The addition of the term replaces the contribution of to the Ricci tensor in terms of and thus changes the principal part to that of the scalar wave operator.

A solution to the Einstein equations is now obtained by solving Eq. It can then be shown [ ] that the ADM constraints 54 , 55 imply. By virtue of the contracted Bianchi identities, the evolution of the constraint system obeys the equation. A key addition to the GHG formalism has been devised by Gundlach et al. Including these damping terms and using the definition 71 to substitute higher derivatives in the Ricci tensor, the generalized Einstein equations 72 can be written as.

An alternative first-order system of the GHG formulation has been presented in Ref. In particular, the constraint subsystem of the BSSN equations contains a zero-speed mode [ , , ] which may lead to large Hamiltonian constraint violations. The GHG system does not contain such modes and furthermore admits a simple way of controlling constraint violations in the form of damping terms [ ].

Finally, the wave-equation-type principal part of the GHG system allows for the straightforward construction of constraint-preserving boundary conditions [ , , ]. On the other hand, the BSSN formulation is remarkably robust and allows for the simulation of BH binaries over a wide range of the parameter space with little if any modifications of the gauge conditions; cf. Section 6. Combination of these advantages in a single system has motivated the exploration of improvements to the BSSN system and in recent years resulted in the identification of a conformal version of the Z 4 system, originally developed in Refs.

The key idea behind the Z 4 system is to replace the Einstein equations with a generalized class of equations given by. The resulting evolution equations given in the literature vary in details, but clearly represent relatively minor modifications for existing BSSN codes [ 28 , , ]. Investigations have shown that the conformal Z 4 system is indeed suitable for implementation of constraint preserving boundary conditions [ ] and that constraint violations in simulations of gauge waves and BH and NS spacetimes are indeed smaller than those obtained for the BSSN system, in particular when constraint damping is actively enforced [ 28 , ].

This behaviour also manifests itself in more accurate results for the gravitational radiation in binary inspirals [ ]. In summary, the conformal Z 4 formulation is a very promising candidate for future numerical studies of BH spacetimes, including in particular the asymptotically AdS case where a rigorous control of the outer boundary is of utmost importance; cf. Another modification of the BSSN equations is based on the use of densitized versions of the trace of the extrinsic curvature and the lapse function as well as the traceless part of the extrinsic curvature with mixed indices [ , ].

Some improvements in simulations of colliding BHs in higher-dimensional spacetimes have been found by careful exploration of the densitization parameter space [ ]. The formulations discussed in the previous subsections are based on a spacetime split of the Einstein equations. A natural alternative to such a split is given by the characteristic approach pioneered by Bondi et al.

Here, at least one coordinate is null and thus adapted to the characteristics of the vacuum Einstein equations. For generic four-dimensional spacetimes with no symmetry assumptions, the characteristic formalism results in a natural hierarchy of two evolution equations, four hypersurface equations relating variables on hypersurfaces of constant retarded or advanced time, as well as three supplementary and one trivial equations.

A comprehensive overview of characteristic methods in NR is given in the Living Reviews article [ ]. Although characteristic codes have been developed with great success in spacetimes with additional symmetry assumptions, evolutions of generic BH spacetimes face the problem of formation of caustics, resulting in a breakdown of the coordinate system; see [ 59 ] for a recent investigation. In the form of Cauchy-characteristic extraction, i. All the Cauchy and characteristic or combined approaches we have discussed so far, evolve the physical spacetime, i.

The additional cost resulting from the larger set of variables, however, is mitigated by the fact that these include projections of the Weyl tensor that directly encode the GW content. Even though the conformal field equations have as yet not resulted in simulations of BH systems analogous to those achieved in BSSN or GHG, their elegance in handling the entire spacetime without truncation merits further investigation. A brief historic overview of many formulations of the Einstein equations including systems not discussed in this work is given in Ref.

Image reproduced with permission from [ ], copyright by APS. Illustration of mesh refinement for a BH binary with one spatial dimension suppressed. Around each BH marked by the spherical AH , two nested boxes are visible. These are immersed within one large, common grid or refinement level. We finally note that for simulations of spacetimes with high degrees of symmetry, it often turns out convenient to directly impose the symmetries on the shape of the line element rather than use one of the general formalisms discussed so far.

As an example, we consider the classic study by May and White [ , ] of the dynamics of spherically symmetric perfect fluid stars. A four-dimensional spherically symmetric spacetime can be described in terms of the simple line element. May and White employ Lagrangian coordinates co-moving with the fluid shells which is imposed through the form of the energy-momentum tensor. Section II in Ref. The addition of matter to the spacetime can, in principle, be done using the formalism just laid down Vector fields can be handled in similar fashion, we refer the reader to Ref. In summary, a great deal of progress has been made in recent years concerning the well-posedness of the numerical methods used for the construction of spacetimes.

We note, however, that the well-posedness of many problems beyond electrovacuum GR remains unknown at present. This includes, in particular, a wide class of alternative theories of gravity where it is not clear whether they admit well-posed IBVPs. Performing numerical simulations in generic higher-dimensional spacetimes represents a major challenge for simple computational reasons. Contemporary simulations of compact objects in four spacetime dimensions require cores and Gb of memory for storage of the fields on the computational domain.