The theory of gravitational radiation in exact
general relativity is based on structures defined at future null
infinity . In particular, associated with every
cross-section of
- which represents a retarded ‘instant
of time’ - there is a well defined notion of mass, introduced by
Bondi [54
], which decreases as
gravitational radiation flows across
. On the other hand,
except for those based on conformal methods, most simulations only
deal with a finite portion of space-time and thus have no direct
access to
. Instead, one usually uses scalars such
as the Weyl tensor component
to define the radiation
waveform. However,
depends on the choice of a null tetrad
as well as coordinates. While a natural tetrad is available for the
perturbation theory on Kerr back ground, this is not true in
general. In this section we will sketch an approach to solve both
these problems using the isolated horizon framework: One can
construct an approximate analog of
for a suitable,
finite portion of space-time, and
introduce a geometrically defined null tetrad and coordinates to
extract gauge invariant, radiative information from
simulations.
One can also define a null tetrad in the
neighborhood in a similar fashion. Let be an arbitrary
complex vector tangent to the good cuts such that
is a null tetrad on
. Using parallel
transport along the null geodesics, we can define a null tetrad in
the neighborhood. This tetrad is unique up to the spin rotations of
:
on the fiducial good cut.
This construction is shown in Figure 8
. The domain of
validity of the coordinate system and tetrads is the space-time
region in which the null geodesics emanating from good cuts do not
cross.
Numerically, it might be possible to adapt
existing event horizon finders to locate the outgoing past null
cone of a cross-section of the horizon. This is because event
horizon trackers also track null surfaces backward in
time [82, 83]; the event horizon is the ingoing past
null surface starting from sufficiently close to future timelike
infinity while here we are interested in the outgoing past directed null surface starting
from an apparent horizon.
In the Schwarzschild solution, it can be shown
analytically that this coordinate system covers an entire
asymptotic region. In the Kerr space-time, the domain of validity
is not explicitly known, but in a numerical implementation, this
procedure does not encounter geodesic crossing in the region of
interest to the simulation [82]. In general space-times, the extraction
of wave forms requires the construction to go through only along
the past light cones of good cuts which lie in the distant future,
whence problems with geodesic-crossing are unlikely to prevent one
from covering a sufficiently large region with these coordinates.
This invariantly defined structure provides a new approach to
extract waveforms. First, the null tetrad presented above can be
used to calculate . There is only a phase factor ambiguity
(which is a function independent of
and
) inherited from the ambiguity in the choice of
on the fiducial good cut. Second, the past null cone
of a good cut at a sufficiently late time can be used as an
approximate null infinity. This should enable one to calculate
dynamical quantities such as the analogs of the Bondi mass and the
rate of energy loss from the black hole, now on the ‘approximate’
. However, a detailed framework to extract the
approximate expressions for fluxes of energy and Bondi mass, with
sufficient control on the errors, is yet to be developed. This is a
very interesting analytical problem since its solution would
provide numerical relativists with an algorithm to extract
waveforms and fluxes of energy in gravitational waves in an
invariant and physically reliable manner. Finally, the invariant
coordinates and tetrads also enable one to compare late time
results of distinct numerical simulations.