Numerical relativity

# Numerical relativity

Discussion

Encyclopedia
Numerical relativity is one of the branches of general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

that uses numerical methods and algorithms to solve and analyze problems. To this end, supercomputers are often employed to study black holes, gravitational waves, neutron stars and many other phenomena governed by Einstein's
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...

Theory of General Relativity. A currently active field of research in numerical relativity is the simulation of relativistic binaries and their associated gravitational waves. Other branches are also quite active.

## Overview

A primary goal of numerical relativity is to study spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...

s whose exact form is not known. The spacetimes so found computational can either be fully dynamical, stationary
Stationary spacetime
In general relativity, specifically in the Einstein field equations, a spacetime is said to be stationary if it admits a Killing vector that is asymptotically timelike....

or static
Static spacetime
In general relativity, a spacetime is said to be static if it admits a global, non-vanishing, timelike Killing vector field K which is irrotational, i.e., whose orthogonal distribution is involutive...

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
Physical cosmology
Physical cosmology, as a branch of astronomy, is the study of the largest-scale structures and dynamics of the universe and is concerned with fundamental questions about its formation and evolution. For most of human history, it was a branch of metaphysics and religion...

models, critical phenomena
Critical phenomena
In physics, critical phenomena is the collective name associated with thephysics of critical points. Most of them stem from the divergence of thecorrelation length, but also the dynamics slows down...

, perturbed
Perturbation (astronomy)
Perturbation is a term used in astronomy in connection with descriptions of the complex motion of a massive body which is subject to appreciable gravitational effects from more than one other massive body....

black hole
Black hole
A black hole is a region of spacetime from which nothing, not even light, can escape. The theory of general relativity predicts that a sufficiently compact mass will deform spacetime to form a black hole. Around a black hole there is a mathematically defined surface called an event horizon that...

s and neutron star
Neutron star
A neutron star is a type of stellar remnant that can result from the gravitational collapse of a massive star during a Type II, Type Ib or Type Ic supernova event. Such stars are composed almost entirely of neutrons, which are subatomic particles without electrical charge and with a slightly larger...

s, and the coalescence of black hole
Coalescence (meteorology)
Coalescence is the process by which two or more droplets, bubbles or particles merge during contact to form a single daughter droplet, bubble or particle. It can take place in many processes, ranging from meteorology to astrophysics. For example, it is both inve formation of raindrops as well as...

s and neutron stars, for example. In any of these cases, Einstein's equations can be formulated in several ways that allow us to evolve the dynamics. While Cauchy methods have received a majority of the attention, characteristic and Regge calculus
Regge calculus
In general relativity, Regge calculus is a formalism for producing simplicial approximations of spacetimes that are solutions to the Einstein field equation. The calculus was introduced by the Italian theoretician Tullio Regge in the early 1960s....

based methods have also been used. All of these methods begin with a snapshot of the gravitational field
Gravitational field
The gravitational field is a model used in physics to explain the existence of gravity. In its original concept, gravity was a force between point masses...

s on some hypersurface
Hypersurface
In geometry, a hypersurface is a generalization of the concept of hyperplane. Suppose an enveloping manifold M has n dimensions; then any submanifold of M of n − 1 dimensions is a hypersurface...

, the initial data, and evolve these data to neighboring hypersurfaces. http://relativity.livingreviews.org/Articles/lrr-2000-5/

Like all problems in numerical analysis, careful attention is paid to the stability
Numerical stability
In the mathematical subfield of numerical analysis, numerical stability is a desirable property of numerical algorithms. The precise definition of stability depends on the context, but it is related to the accuracy of the algorithm....

and convergence of the numerical solutions. In this line, much attention is paid to the gauge conditions
Gauge fixing
In the physics of gauge theories, gauge fixing denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. By definition, a gauge theory represents each physically distinct configuration of the system as an equivalence class of detailed local field...

, 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
Classical field theory
A classical field theory is a physical theory that describes the study of how one or more physical fields interact with matter. The word 'classical' is used in contrast to those field theories that incorporate quantum mechanics ....

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
Computational fluid dynamics
Computational fluid dynamics, usually abbreviated as CFD, is a branch of fluid mechanics that uses numerical methods and algorithms to solve and analyze problems that involve fluid flows. Computers are used to perform the calculations required to simulate the interaction of liquids and gases with...

, electromagnetics, and solid mechanics. Numerical relativists often work with applied mathematicians and draw insight from numerical analysis
Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....

, scientific computation, partial differential equation
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...

s, and geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

among other mathematical areas of specialization.

### Foundations in theory

Albert Einstein
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...

published his theory of general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

in 1915. It, like his earlier theory of special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

, described space and time as a unified spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...

subject to what are now known as the Einstein field equations
Einstein field equations
The Einstein field equations or Einstein's equations are a set of ten equations in Albert Einstein's general theory of relativity which describe the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy...

. These form a set of coupled nonlinear partial differential equations (PDEs). In the nearly 100 years since the first publication of the theory, relatively few closed-form
Closed-form expression
In mathematics, an expression is said to be a closed-form expression if it can be expressed analytically in terms of a bounded number of certain "well-known" functions...

solutions are known for the field equations, and, of those, most are cosmological
Cosmology
Cosmology is the discipline that deals with the nature of the Universe as a whole. Cosmologists seek to understand the origin, evolution, structure, and ultimate fate of the Universe at large, as well as the natural laws that keep it in order...

solutions that assume special symmetry
Symmetry
Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...

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. This was first published by Richard Arnowitt
Richard Arnowitt
Richard Lewis Arnowitt is an American physicist known for his contributions to theoretical particle physics and to general relativity.Arnowitt is a Distinguished Professor at Texas A&M University, where he is a member of the Department of Physics....

, Stanley Deser
Stanley Deser
Stanley Deser is an American physicist known for his contributions to general relativity. Currently, he is the Ancell Professor of Physics at Brandeis University in Waltham, Massachusetts....

, and Charles W. Misner
Charles W. Misner
Charles W. Misner is an American physicist and one of the authors of Gravitation. His specialties include general relativity and cosmology. His work has also provided early foundations for studies of quantum gravity and numerical relativity....

in the late 1950s in what has become known as the ADM formalism
The ADM Formalism developed in 1959 by Richard Arnowitt, Stanley Deser and Charles W. Misner is a Hamiltonian formulation of general relativity...

. Although for technical reasons the precise equations formulated in the original ADM paper are rarely used in numerical simulations, most or all practical approaches to numerical relativity use a "3+1 decomposition" of spacetime into three-dimensional space and one-dimensional time that is closely related to the ADM formulation, because the ADM procedure reformulates the Einstein field equations into a constrained
Constraint (mathematics)
In mathematics, a constraint is a condition that a solution to an optimization problem must satisfy. There are two types of constraints: equality constraints and inequality constraints...

initial value problem
Initial value problem
In mathematics, in the field of differential equations, an initial value problem is an ordinary differential equation together with a specified value, called the initial condition, of the unknown function at a given point in the domain of the solution...

that can be addressed encoded on a computer for solution.

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 1964, followed soon thereafter by Smarr
Larry Smarr
Larry Smarr is a physicist and leader in scientific computing, supercomputer applications, and Internet infrastructure.He received both his BA and MS at the University of Missouri in Columbia, Missouri and received a Ph.D...

and by Eppley. These early attempts were focused on evolving Misner data in axisymmetry (also known as "2+1 dimensions"). At around the same time Tsvi Piran wrote the first code that evolved a system with gravitational radiation using a cylindrical symmetry. In this calculations Piran has set the foundation to many of the concepts used today in evolving ADM equations, like "free evolution, Constrained evolusion which deal with the fundamental problem of treating the constraint equations that arise in the ADM formalism. 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

The first realistic calculations of rotating collapse were carried out in the early eighties by Richard Stark and Tsvi Piran 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 1990s when the Binary Black Hole Grand Challenge
Grand Challenge
Grand Challenges were USA policy terms set as goals in the late 1980s for funding high-performance computing and communications research in part in response to the Japanese 5th Generation 10-year project....

Alliance successfully simulated a head-on binary black hole
Black hole
A black hole is a region of spacetime from which nothing, not even light, can escape. The theory of general relativity predicts that a sufficiently compact mass will deform spacetime to form a black hole. Around a black hole there is a mathematically defined surface called an event horizon that...

collision. As a post-processing step the group computed the event horizon
Event horizon
In general relativity, an event horizon is a boundary in spacetime beyond which events cannot affect an outside observer. In layman's terms it is defined as "the point of no return" i.e. the point at which the gravitational pull becomes so great as to make escape impossible. The most common case...

for the spacetime. This result still required imposing and exploiting axisymmetry in the calculations.

Some of the first documented attempts to solve the Einstein equations in three dimensions were focused on a single Schwarzschild black hole
Schwarzschild metric
In Einstein's theory of general relativity, the Schwarzschild solution describes the gravitational field outside a spherical, uncharged, non-rotating mass such as a star, planet, or black hole. It is also a good approximation to the gravitational field of a slowly rotating body like the Earth or...

, which is described by a static and spherically symmetric solution to the Einstein field equations. 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
Gravitational singularity
A gravitational singularity or spacetime singularity is a location where the quantities that are used to measure the gravitational field become infinite in a way that does not depend on the coordinate system...

. One of the earliest groups to attempt to simulate this solution was Anninos et al. in 1995. In their paper they point out that
"Progress in three dimensional numerical relativity has been impeded in part by lack of computers with sufficient memory and computational power to perform well resolved calculations of 3D spacetimes."

### Maturation of the Field

In the years that followed, not only did computers became more powerful, but also various research groups developed alternate techniques to improve the efficiency of the calculations. 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
Excision
Excision is the alias of Jeff Abel, a dubstep DJ and music boss from British Columbia, Canada. He frequently works with fellow Canadian dubstep producers Datsik and Downlink. As one of the first dubstep producers and DJs in North America, he has played a significant role in the genre's growth in...

, 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
In numerical analysis, adaptive mesh refinement is a method of adaptive meshing. Central to any Eulerian method is the manner in which it discretizes the continuous domain of interest into a grid of many individual elements...

techniques, already used in computational fluid dynamics
Computational fluid dynamics
Computational fluid dynamics, usually abbreviated as CFD, is a branch of fluid mechanics that uses numerical methods and algorithms to solve and analyze problems that involve fluid flows. Computers are used to perform the calculations required to simulate the interaction of liquids and gases with...

were introduced to the field of numerical relativity.

#### Excision

In the excision technique, which was first proposed in the late 1990s, a portion of a spacetime inside of the event horizon
Event horizon
In general relativity, an event horizon is a boundary in spacetime beyond which events cannot affect an outside observer. In layman's terms it is defined as "the point of no return" i.e. the point at which the gravitational pull becomes so great as to make escape impossible. The most common case...

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
Causality
Causality is the relationship between an event and a second event , where the second event is understood as a consequence of the first....

and properties of the event horizon (i.e. nothing physical inside the black hole can influence any of the physics outside the horizon). 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.g., using harmonic coordinates coordinate conditions). 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. The first stable, long-term evolution of the orbit and merger of two black holes using this technique was published in 2005.

#### Punctures

In the puncture method the solution is factored into an analytical part, which contains the singularity of the black hole, and a numerically constructed part, which is then singularity free. This is a generalization of the Brill-Lindquist prescription for initial data of black holes at rest and can be generalized to the Bowen-York prescription for spinning and moving black hole initial data. Until 2005, 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 lead to numerical instabilities at some stage of the simulation.

In 2005 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.
By choosing appropriate coordinate conditions and making crude analytic assumption about the fields near the singularity (since no physical effects can propagate out of the black hole, the crudeness of the approximations does not matter), numerical solutions could be obtained to the problem of two black holes orbiting each other, as well as accurate computation of gravitational radiation
Gravitational wave
In physics, gravitational waves are theoretical ripples in the curvature of spacetime which propagates as a wave, traveling outward from the source. Predicted to exist by Albert Einstein in 1916 on the basis of his theory of general relativity, gravitational waves theoretically transport energy as...

(ripples in spacetime) emitted by them.

#### The Lazarus Project

The Lazarus project (1998–2005) 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. All previous attempts to numerically integrate in supercomputers the Hilbert-Einstein equations describing the gravitational field around binary black holes led to software failure before a single orbit was completed.

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, the linear momentum radiated by unequal mass holes, and the final mass and spin of the remnant black hole. The method also computed detailed gravitational waves emitted by the merger process and predicted that the collision of black holes is the most energetic single event in the Universe, releasing more energy in a fraction of a second in the form of gravitational radiation than an entire galaxy in its lifetime.

In numerical analysis, adaptive mesh refinement is a method of adaptive meshing. Central to any Eulerian method is the manner in which it discretizes the continuous domain of interest into a grid of many individual elements...

(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 1980s through the work of Choptuik in his studies of critical collapse
Critical phenomena
In physics, critical phenomena is the collective name associated with thephysics of critical points. Most of them stem from the divergence of thecorrelation length, but also the dynamics slows down...

of scalar fields. The original work was in one dimension, but it was subsequently extended to two dimensions. In two dimensions, AMR has also been applied to the study of inhomogeneous cosmologies
Cosmology
Cosmology is the discipline that deals with the nature of the Universe as a whole. Cosmologists seek to understand the origin, evolution, structure, and ultimate fate of the Universe at large, as well as the natural laws that keep it in order...

, and to the study of Schwarzschild black holes
Schwarzschild metric
In Einstein's theory of general relativity, the Schwarzschild solution describes the gravitational field outside a spherical, uncharged, non-rotating mass such as a star, planet, or black hole. It is also a good approximation to the gravitational field of a slowly rotating body like the Earth or...

. The techniques has now become a standard tool in numerical relativity and has been used to study the merger of black holes and other compact objects in addition to the propagation of gravitational radiation
Gravitational wave
In physics, gravitational waves are theoretical ripples in the curvature of spacetime which propagates as a wave, traveling outward from the source. Predicted to exist by Albert Einstein in 1916 on the basis of his theory of general relativity, gravitational waves theoretically transport energy as...

generated by such astronomical events.

## Recent developments

In the past few years, hundreds of research papers have been published leading to a wide spectrum of mathematical relativity, gravitational wave, and astrophysical results for the orbiting black hole problem. This technique extended to astrophysical binary systems involving neutron stars and black holes, and multiple black holes. One of the most surprising predictions is that the merger of two black holes can give the remnant hole a kick of up to 4000 km/s that can allow them to escape from any known galaxy
. The simulations also predict an enormous release of gravitational energy in this merger process, amounting to up to 8% of its total rest mass.

• Mathematics of general relativity
Mathematics of general relativity
The mathematics of general relativity refers to various mathematical structures and techniques that are used in studying and formulating Albert Einstein's theory of general relativity. The main tools used in this geometrical theory of gravitation are tensor fields defined on a Lorentzian manifold...

• Post-Newtonian expansion
Post-Newtonian expansion
Post-Newtonian expansions in general relativity are used for finding an approximate solution of the Einstein equations for the metric tensor. The post-Newtonian approximations are expansions in a small parameter, which is the ratio of the velocity of matter, forming the gravitational field, to the...

• Spin-flip
Spin-flip
A black hole spin-flip occurs when the spin axis of a rotating black hole undergoes a sudden change in orientation due to absorption of a second black hole....

• Cactus Framework
Cactus Framework
Cactus is an open-source, problem-solving environment designed for scientists and engineers. Its modular structure easily enables parallel computation across different architectures and collaborative code development between different groups...