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Computational Relativity
The largest field of enquiry historically has been the field of exact solutions of Einstein's field equations. The Einstein field equations constitute an
extremely complicated set of non-linear partial differential equations. It came as something of a surprise when Schwarzschild found an exact solution
within a year of the field equations being published. In the ensuing years very few exact solutions were found until new invariant techniques were
introduced in the mid 1960s. This led to an explosion of exact solutions being discovered.
Many of the calculations associated with exact solutions are straightforward but
extremely long and complicated to perform which can easily lead to errors.
Around the mid 1960s the field of Algebraic Computing in General Relativity
came into existence and it soon became possible to undertake calculations with
computers that would take more than a lifetime to complete by hand. An example
(and one which has been used as a standard for comparing algebraic computing
systems) is the calculation of the Ricci tensor for Bondi's radiating metric and
was first undertaken by Ray d’Inverno using his algebraic computing system
LAM (which eventually developed into the system SHEEP). The original hand
calculation was undertaken over a period of some six months and now takes less
than a second on a reasonable spec PC.
The advent of Algebraic Computing led to a new attack on the famous
equivalence problem of General Relativity, namely: given two metrics, does there
exist a local transformation which transforms one into the other? Cartan showed
in some classic work that the problem can be solved but depends on computing
the 10th covariant derivative of the Riemann tensor of each metric. Even with
modern computer algebra systems this is out of the question. The work of
Karlhede significantly improved the situation. Karlhede's approach provides an
invariant classification of a metric. Thus, if two metrics have different
classifications then they are necessarily inequivalent, whereas if they have the
same classification then they are candidates for equivalence. The problem then
reduces to solving four algebraic equations. Karlhede's algorithm reduces the
derivative bounds significantly below the original 10 of Cartan. There is particular
interest in the Southampton group in reducing the bounds which might occur in
various cases, for if they can be reduced sufficiently then it becomes possible to
use algebraic computing systems to classify exact solutions.
Part of the SHEEP output for the famous Bondi radiating metric.
The user asks SHEEP to make and write (wmake) the line element
which is called ds2 and then to make the Ricci tensor and the first
23 terms in the output are displayed above.
The advent of the Karlhede classification algorithm has lead to the setting up of a computer database of exact solutions through collaborative work between
UERJ, Rio de Janeiro, Brazil and Southampton. At present some 200 metrics exist in the computer database of exact solutions. The ultimate hope is that it
will contain all known solutions, fully documented and classified. Then any `newly' discovered solution can be compared with the contents of the database
and, if indeed it is new, then the database can be updated ccordingly. Were this to be fully realised then it would provide a valuable resource for the
international community of relativists
While several hundred exact solutions to the field equations are known to exist, few of these solutions describe physically relevant situations. If we
want to solve the equations in scenarios of astrophysical interest, such as the birth of a neutron star in a supernova or the collision of two black holes
in a binary system, we need to perform large scale numerical simulations. In the last ten years or so, there has been an international effort to develop
Numerical Relativity to a level where simulations can provide reliable information about the most violent events in the Universe. This development
was largely motivated by the construction of new generations of gravitational wave detectors. In addition, it is only recently that computers have
become powerful enough that one can hope to achieve the desired computational precision (in a reasonable computing time). After all, Einstein's
equations are extremely complicated (they involve over 100,000 terms in the general case) and solving them numerically makes enormous demands
on the processing power and memory of a computer. Because of this, the development of Numerical Relativity has proceeded in stages first solving
1-dimensional problems (that is 1 spatial dimension), then 2-dimensional problems and only very recently have fully 3-dimensional problems been
tackled.
One of the major problems in NR is that there is no known local
expression for gravitational waves - they can only be properly
described asymptotically (that is out at infinity). In most previous
work, numerical simulations are carried out on a central finite grid
extending into the vacuum region surrounding the sources present
and ad-hoc conditions are imposed at the edge of the grid to
prevent incoming waves (which would be unphysical).
Unfortunately, these ad-hoc conditions themselves generate
spurious reflected numerical waves. The basic formalism that the
numerical codes employ is the Arnowitt-Deser-Misner 3+1
formalism which decomposes 4-dimensional space-time into a
family of constant time 3-dimensional spatial slices. Although this
A Teukolsky gravitational wave on a finite 3+1 grid instead of dissipating off of the grid
formalism is well adapted to central regions it does not work well
leads to a small spurious reflection (see second diagram). The Southampton CCM code
in asymptotic regions. Instead, the d’Inverno-Stachel-Smallwood
provides a global code in which the wave reaches infinity unaltered and where it can be
2+2 formalism decomposes space-time into two families of 2
unambiguously characterised
dimensional spacelike surfaces. The formalism encompasses 6
different cases, one of which is called the null-timelike case in
which one of the families can be taken to form null 3-surfaces and
the other to form timelike 3-surfaces. Null surfaces are especially
important in General Relativity because they are ruled by null
geodesics and these are the curves along which gravitational
information is propagated. The idea behind the Southampton
CCM (Cauchy-Characteristic Matching) project is to combine a
central 3+1 numerical code with an exterior null-timelike 2+2 code
connected across a timelike interface residing in the vacuum. In
addition, the exterior region is compactified so as to incorporate
null infinity where gravitational radiation can be unambiguously
defined. The Southampton project has to date used CCM to
investigate cylindrically symmetric and axially symmetric systems
Left: results from the US Binary Black Hole Grand Challenge supercomputer simulations
as a prelude to investigating fully 3 dimensional systems. The
of the coalescence of two black holes : the merger of the two horizons lead to the famous
significance of this work is that it leads to wave forms
`pair of pants' picture. Right: NASA supported Netron Star Grand Challenge: the
asymptotically, and it is these exact templates which are needed
simulation of the merger of two Neutron stars emitting gravitational waves.
in the search for gravitational waves.