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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 supercomputers to study black holes, gravitational waves, and other phenomena predicted by Einstein's Theory of General Relativity. 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 x time" decomposition of Einstein's equations. Here the spacetime of interest is sliced up into a stack (or "foliation") of spacelike hypersurfaces. Einstein's equations are then used to evolve initial data (fields on the "earliest" hypersurface) forward in time. Typically, the initial slice chosen represents a system of gravitating bodies before they enter a highly dynamical interaction.
The mathematics of the 3+1 split of Einstein's equations was laid out in a series of papers by Arnowitt, Deser and Misner in the early 1960s; the form actually used for most simulation efforts until the mid-1990s was described by York (1979). In the "standard ADM" decomposition, the space-time four-metric gab(the central geometrical field of interest) is written in the form
ds = gab dx dx = (- α + γijββ)dt + 2βi dt dx + γijdx dx
Here the rank-two space-like tensor γij -- called the three-metric -- describes the geometry on a "slice" t = constant of the full space-time. The lapse function α determines the infinitesimal separation between slice "t" and slice "t + Δ t", while the shift vector β describes the change in spatial coordinates x going from "t" to "t + Δ t".
With the four-metric written in this way, Einstein's equations of general relativity also decompose into two types: (a) constraint equations, which must be satisfied on each hypersurface by the three-metric and its first time derivative, and (b) evolution equations, which determine how the three-metric itself develops from slice to slice.
Computational aspects
For D spatial dimensions in a simulation of, for example, a black hole spacetime where physical phenomena are reduced to a set of difference equations (restricted to finite difference methods) with a corresponding two dimensional NxN array of mesh points and a mesh size of h, the CPU time has order O(N) and the memory required has order O(N). Coupled, non-linear, elliptic systems have O(N) computational work, where N is the number of discrete unknowns.
See also
External links
- Initial Data for Numerical Relativity — A review article which includes a nice (technical) discussion of numerical relativity.
- Rotating Stars in Relativity — A (technical) review article about rotating stars, with a section on numerical relativity applications.
- A Relativity Tutorial at Caltech — A basic introduction to concepts of Numerical Relativity.
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