Assuming only a basic knowledge of classical general relativity, the book develops the mathematical formalism from first principles, and then highlights some of the pioneering simulations involving black holes and neutron stars, gravitational collapse and gravitational waves. The book contains exercises to help readers master new material as it is presented. Numerous illustrations, many in color, assist in visualizing new geometric concepts and highlighting the results of computer simulations. Summary boxes encapsulate some of the most important results for quick reference. Applications covered include calculations of coalescing binary black holes and binary neutron stars, rotating stars, colliding star clusters, gravitational and magnetorotational collapse, critical phenomena, the generation of gravitational waves, and other topics of current physical and astrophysical significance.

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Overview[ edit ] A primary goal of numerical relativity is to study spacetimes whose exact form is not known. The spacetimes so found computationally can either be fully dynamical , stationary or static and may contain matter fields or vacuum. In the case of stationary and static solutions, numerical methods may also be used to study the stability of the equilibrium spacetimes. In the case of dynamical spacetimes, the problem may be divided into the initial value problem and the evolution, each requiring different methods.

Numerical relativity is applied to many areas, such as cosmological models , critical phenomena , perturbed black holes and neutron stars , and the coalescence of black holes and neutron stars, for example. While Cauchy methods have received a majority of the attention, characteristic and Regge calculus based methods have also been used.

All of these methods begin with a snapshot of the gravitational fields on some hypersurface , the initial data, and evolve these data to neighboring hypersurfaces. In this line, much attention is paid to the gauge conditions , coordinates, and various formulations of the Einstein equations and the effect they have on the ability to produce accurate numerical solutions. Numerical relativity research is distinct from work on classical field theories as many techniques implemented in these areas are inapplicable in relativity.

Many facets are however shared with large scale problems in other computational sciences like computational fluid dynamics , electromagnetics, and solid mechanics. Numerical relativists often work with applied mathematicians and draw insight from numerical analysis , scientific computation , partial differential equations , and geometry among other mathematical areas of specialization.

History[ edit ] Foundations in theory[ edit ] Albert Einstein published his theory of general relativity in These form a set of coupled nonlinear partial differential equations PDEs. After more than years since the first publication of the theory, relatively few closed-form solutions are known for the field equations, and, of those, most are cosmological solutions that assume special symmetry to reduce the complexity of the equations.

The field of numerical relativity emerged from the desire to construct and study more general solutions to the field equations by approximately solving the Einstein equations numerically.

A necessary precursor to such attempts was a decomposition of spacetime back into separated space and time. Misner in the late s in what has become known as the ADM formalism. At the time that ADM published their original paper, computer technology would not have supported numerical solution to their equations on any problem of any substantial size.

The first documented attempt to solve the Einstein field equations numerically appears to be Hahn and Lindquist in , [4] followed soon thereafter by Smarr [5] [6] and by Eppley. At around the same time Tsvi Piran wrote the first code that evolved a system with gravitational radiation using a cylindrical symmetry. Applying symmetry reduced the computational and memory requirements associated with the problem, allowing the researchers to obtain results on the supercomputers available at the time.

Early results[ edit ] The first realistic calculations of rotating collapse were carried out in the early eighties by Richard Stark and Tsvi Piran [9] in which the gravitational wave forms resulting from formation of a rotating black hole were calculated for the first time. For nearly 20 years following the initial results, there were fairly few other published results in numerical relativity, probably due to the lack of sufficiently powerful computers to address the problem.

In the late s, the Binary Black Hole Grand Challenge Alliance successfully simulated a head-on binary black hole collision. As a post-processing step the group computed the event horizon for the spacetime. This result still required imposing and exploiting axisymmetry in the calculations. This provides an excellent test case in numerical relativity because it does have a closed-form solution so that numerical results can be compared to an exact solution, because it is static, and because it contains one of the most numerically challenging features of relativity theory, a physical singularity.

One of the earliest groups to attempt to simulate this solution was Anninos et al. With respect to black hole simulations specifically, two techniques were devised to avoid problems associated with the existence of physical singularities in the solutions to the equations: 1 Excision, and 2 the "puncture" method. In addition the Lazarus group developed techniques for using early results from a short-lived simulation solving the nonlinear ADM equations, in order to provide initial data for a more stable code based on linearized equations derived from perturbation theory.

More generally, adaptive mesh refinement techniques, already used in computational fluid dynamics were introduced to the field of numerical relativity. Excision[ edit ] In the excision technique, which was first proposed in the late s, [12] a portion of a spacetime inside of the event horizon surrounding the singularity of a black hole is simply not evolved.

In theory this should not affect the solution to the equations outside of the event horizon because of the principle of causality and properties of the event horizon i. Thus if one simply does not solve the equations inside the horizon one should still be able to obtain valid solutions outside. One "excises" the interior by imposing ingoing boundary conditions on a boundary surrounding the singularity but inside the horizon.

While the implementation of excision has been very successful, the technique has two minor problems. The first is that one has to be careful about the coordinate conditions. While physical effects cannot propagate from inside to outside, coordinate effects could.

For example, if the coordinate conditions were elliptical, coordinate changes inside could instantly propagate out through the horizon. This then means that one needs hyperbolic type coordinate conditions with characteristic velocities less than that of light for the propagation of coordinate effects e.

The second problem is that as the black holes move, one must continually adjust the location of the excision region to move with the black hole. The excision technique was developed over several years including the development of new gauge conditions that increased stability and work that demonstrated the ability of the excision regions to move through the computational grid. This is a generalization of the Brill-Lindquist [21] prescription for initial data of black holes at rest and can be generalized to the Bowen-York [22] prescription for spinning and moving black hole initial data.

Until , all published usage of the puncture method required that the coordinate position of all punctures remain fixed during the course of the simulation. Of course black holes in proximity to each other will tend to move under the force of gravity, so the fact that the coordinate position of the puncture remained fixed meant that the coordinate systems themselves became "stretched" or "twisted," and this typically led to numerical instabilities at some stage of the simulation.

Breakthrough[ edit ] In researchers demonstrated for the first time the ability to allow punctures to move through the coordinate system, thus eliminating some of the earlier problems with the method. This allowed accurate long-term evolutions of black holes.

Lazarus project[ edit ] The Lazarus project — was developed as a post-Grand Challenge technique to extract astrophysical results from short lived full numerical simulations of binary black holes. It combined approximation techniques before post-Newtonian trajectories and after perturbations of single black holes with full numerical simulations attempting to solve General Relativity field equations. The Lazarus approach, in the meantime, gave the best insight into the binary black hole problem and produced numerous and relatively accurate results, such as the radiated energy and angular momentum emitted in the latest merging state, [26] [27] the linear momentum radiated by unequal mass holes, [28] and the final mass and spin of the remnant black hole.

Adaptive mesh refinement[ edit ] Adaptive mesh refinement AMR as a numerical method has roots that go well beyond its first application in the field of numerical relativity. Mesh refinement first appears in the numerical relativity literature in the s, through the work of Choptuik in his studies of critical collapse of scalar fields. This technique extended to astrophysical binary systems involving neutron stars and black holes, [38] and multiple black holes.

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## BSSN formalism

Overview[ edit ] A primary goal of numerical relativity is to study spacetimes whose exact form is not known. The spacetimes so found computationally can either be fully dynamical , stationary or static and may contain matter fields or vacuum. In the case of stationary and static solutions, numerical methods may also be used to study the stability of the equilibrium spacetimes. In the case of dynamical spacetimes, the problem may be divided into the initial value problem and the evolution, each requiring different methods. Numerical relativity is applied to many areas, such as cosmological models , critical phenomena , perturbed black holes and neutron stars , and the coalescence of black holes and neutron stars, for example. While Cauchy methods have received a majority of the attention, characteristic and Regge calculus based methods have also been used. All of these methods begin with a snapshot of the gravitational fields on some hypersurface , the initial data, and evolve these data to neighboring hypersurfaces.

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## Numerical relativity

Mezishura Read more Read less. Numerous illustrations, many in color, assist in visualizing new geometric concepts and highlighting the results of computer simulations. Assuming only a basic knowledge of classical general relativity, the book develops the mathematical formalism from first principles, and then highlights some of the pioneering simulations involving black holes and neutron stars, gravitational collapse and gravitational waves. Amazon Restaurants Food delivery from local restaurants. Binary neutron star evolution.

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## Numerical Relativity

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