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Numerical relativity

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Numerical relativity is a subfield of computational physics that aims to establish numerical solutions to Einstein's field equations in general relativity. Numerical relativists use computers to study black holes, gravitational waves, and other phenomena predicted by Einstein's Theory of General Relativity. Though the main thrust of current research in numerical relativity is the simulation of binary black holes, other branches are still quite active.

Overview

The goal of numerical relativity is to study spacetimes that cannot be studied by analytic means. The focus is therefore primarily on dynamical systems. Numerical relativity has been applied in many areas: cosmological models, critical phenomena, perturbed black holes and neutron stars, and the coalescence of black holes 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 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.

Despite promising results, accurate and validated algorithms for Einstein's equations remain elusive. The size and complexity of the equations along with persisting inquiries in fundamental issues of relativity theory are attributed the cause of thus far unsuccessful attempts at resolution. Nonetheless, the field has prodigiously expanded in recent years as engaging research continues.

Numerical relativity attempts to acquire a comprehensive understanding of the complex nature of strong dynamical gravitational fields. Another topic under investigation in numerical relativity is the initial value problem in vacuum relativity. This involves partial differential equations, discretization techniques for these equations, treatment of black hole spacetimes, and the imposition of boundary conditions.

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.

The "3+1" Approach to Numerical Relativity

Most active research in NR has involved a 3+1, or "space × {\displaystyle \times } time" decomposition of Einstein's equations. Here the spacetime of interest is sliced up into a stack (or "foliation") of spacelike three-dimensional hypersurfaces. That is, cross-sections are taken of the spacetime, so that each point in the spacetime lies on exactly one slice. Successive slices then represent--loosely speaking--different instants in time. More specifically, the hypersurfaces are parametrized by a coordinate usually denoted by t, which can be interpreted as a universal coordinate time. (Note that this time does not necessarily coincide with the proper time of any observer.)

If we now restrict attention to a single hypersurface, we see that it can be treated just like any other geometric space. For example, the space has some Riemannian metric. This three-dimensional metric is usually written as γ i j {\displaystyle \gamma _{ij}} , so that the line element is given by
d l 2 = γ i j d x i d x j   . {\displaystyle dl^{2}=\gamma _{ij}dx^{i}dx^{j}\ .}
Here, the coordinates x i {\displaystyle x^{i}} are just some coordinates placed on the hypersurface, in some smooth way, and the indices i and j run over the three dimensions of the hypersurface.

Next, we need some way to relate the successive hypersurfaces. We imagine a set of observers moving through time along the direction normal to the hypersurface. If we choose one such observer, and denote the proper time by τ {\displaystyle \tau } , we can relate it to the time coordinate t by way of the "lapse function" α {\displaystyle \alpha } :
d τ = α ( t , x k ) d t   . {\displaystyle d\tau =\alpha (t,x^{k})dt\ .}
Similarly, we need to relate the position of the observer on one slice to its position on the next slice, using coordinates. If the change in time coordinate is dt, and the change in spatial coordinates is d x i {\displaystyle dx^{i}} , we can relate the two as:
d x i = − β i ( t , x k ) d t   . {\displaystyle dx^{i}=-\beta ^{i}(t,x^{k})\,dt\ .}
The functions β i {\displaystyle \beta ^{i}} are known as the "shift vector".

The way in which we foliate a spacetime is certainly not unique, as there is clearly no need (other than smoothness) for the spatial coordinates on neighboring slices to be related to each other uniquely. This means that the functions α ( t , x k ) {\displaystyle \alpha (t,x^{k})} and β i ( t , x k ) {\displaystyle \beta ^{i}(t,x^{k})} can be freely chosen. These two functions determine our choice of coordinate system, and are known as "gauge functions" for this reason.

The metric of the full spacetime can now be written in the form
d s 2 = ( − α 2 + β i β j γ i j ) d t 2 + 2 β i γ i j d t d x j + γ i j d x i d x j   . {\displaystyle ds^{2}=\left(-\alpha ^{2}+\beta ^{i}\beta ^{j}\gamma _{ij}\right)dt^{2}+2\beta ^{i}\gamma _{ij}dtdx^{j}+\gamma _{ij}dx^{i}dx^{j}\ .}
In computations for General Relativity, the desired "solution" is just the metric, given as a function of the coordinates. From the above equation, then, we see that objective is to find γ i j ( t , x k ) {\displaystyle \gamma _{ij}(t,x^{k})} .

The mathematics of the 3+1 split of Einstein's equations was laid out in a series of papers by Arnowitt, Deser and Misner (ADM) in the early 1960s; the form actually used for most simulation efforts until the mid-1990s was described by York (1979). ADM showed that data could be specified on an initial hypersurface, and Einstein's equations then show how that initial data evolves onto later hypersurfaces. More specifically, ADM showed that Einstein's equations separate into four "constraint" equations, and a set of "evolution" equations.

The four constraint equations are equations which only refer to quantities defined on each individual hypersurface, and must be satisified on each hypersurface--including the initial slice. The evolution equations then describe how those quantities change in time. (In technical terms, ADM showed that Einstein's equations form a constrained hyperbolic system.)

Computational Difficulties

It can be shown that if the constraint equations are satisfied initially, and the variables are evolved exactly according to the evolution equations, then the constraints will remain satisfied. In theory, this means that all we need to do is specify initial data which satisfy the constraints, and evolve according to the evolution equations, and we will have a correct solution to Einstein's equations. Unfortunately, putting the problem on computers brings up serious complications.

First, since a computer only has finite precision, we cannot give it initial data which exactly satisfy the constraint equations in general. This means that the equations which we will be solving will not actually be Einstein's equations. If the constraints are small enough, the hope is that the equations we happen to be solving will be very nearly Einstein's equations, but there is no practical guarantee of this.

Second, the evolution equations are very non-linear partial differential equations. This means that solving them is very difficult. For any interesting spacetime, the dynamics and non-linearity will cause difficulties for any numerical algorithm

Because a computer only has finite memory, we also have to restrict attention to some finite-sized spacetime. Usually, this means that we have to introduce some boundaries into our computation, outside of which we do not know the solution, or simply ignore it. This also requires the introduction of boundary conditions. These boundary conditions affect what happens inside of our simulation domain, and should be dictated by physics. Unfortunately, it is not known how to impose physical boundary conditions. This means that the solution obtained will not be entirely physically accurate.

These complications are just a few of the problems encountered in numerical relativity.

Numerical Methods

In general, an evolution equation expresses the time derivative of a variable in terms of spatial derivatives of that variable and other variables. Implementing these equations on a computer, then, requires methods to take spatial derivatives of the functions (spatial differentiation), and use the resulting time derivative to evolve the variables (time stepping).

Spatial Differentiation

For D spatial dimensions in a simulation where physical phenomena are reduced to a set of difference equations (restricting attention to finite-difference methods) with a corresponding D-dimensional array of N "mesh points", the CPU time is of order O ( N D + 1 ) {\displaystyle {\mathcal {O}}(N^{D+1})} and the memory required is of order O ( N D ) {\displaystyle {\mathcal {O}}(N^{D})} . Typically, the accuracy of finite-difference methods improves with the number of points as N 2 {\displaystyle N^{2}} (second-order), N 4 {\displaystyle N^{4}} (fourth-order), or some times N 6 {\displaystyle N^{6}} (sixth-order).

An alternative technique is to approximate each variable in a spectral series. Examples of spectral series include Fourier series, expansion in terms of Chebyshev polynomials, and spherical harmonics. These approximations can then be differentiated exactly. The advantage of this technique is the speed with which its accuracy improves (its convergence rate); the accuracy of spectral methods typically improves as e N {\displaystyle e^{N}} . Compared to N 2 {\displaystyle N^{2}} or N 4 {\displaystyle N^{4}} for finite differencing, this is a vast improvement.

The disadvantage of spectral methods is their sensitivity to smoothness of the functions they are used with. Especially in situations involving matter, shocks and discontinuities can arise easily. In those cases, spectral methods are not worth the trouble, and finite differencing methods are preferred. Nonetheless, many simulations of solutions of the vacuum Einstein equations can be significantly improved by using spectral methods. The great majority of Numerical Relativity research groups currently use finite differencing methods, for reasons of simplicity.

Time Stepping

The most common techniques for time stepping are Iterated Crank-Nicolson and Runge-Kutta. Iterated Crank-Nicolson is a second-order convergent method, which means that its errors decrease with time step Δ t {\displaystyle \Delta t} as 1 / ( Δ t ) 2 {\displaystyle 1/(\Delta t)^{2}} . Runge-Kutta methods are somewhat more general. The "classic" Runge-Kutta algorithm is fourth-order convergent, but related algorithms can be made arbitrarily convergent.

There is a penalty, however, for moving to higher-order methods. These methods require more computations, and therefore run more slowly. The method used in a given simulation depends on balancing the accuracy with the time required. Many simulations which use second-order finite differencing, for example, use Iterated Crank-Nicolson time stepping because higher accuracy is not always needed.

Elliptic Equations

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Many equations arising from Numerical Relativity are not hyperbolic, but elliptic equations. For example, the constraint equations for initial data take an elliptic form. (Constraint projection methods and certain gauge conditions also require the solution of elliptic equations.)

See also

Notes

  1. York, J W "Kinematics and Dynamics of General Relativity" from "Sources of Gravitational Radiation: Proceedings of the Battelle Seattle Workshop" ed. Larry Smarr, pp. 83-126 (Cambridge University Press 1979)

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