**D. Gravitational Radiation**

Like GRBs, typical sources of gravitational radiation involve the formation of compact objects. Hence it is reasonable to expect that gravitational waves will accompany GRBs. This association is indirect: the gravitational waves are not directly related to the GRB. Additinionally, GRBs have their own, albeit weak, gravitational radiation pulse which arises during the acceleration of the jets to relativistic velocities. Unfortunately this signal is weak and moreover it is perpendicular to the GRB signal.

To estimate the rates of observed gravitational radiation events
associated with GRB we use the rate of long GRBs. The nearest
(long) GRB detected within a year would be at 1 Gpc. As GRBs are
beamed the nearest (long) event would be at would be much nearer,
at 135
_{0.1}^{2}
Mpc. However, this burst would be directed
away from us. Still a GRB that is beamed away from us is expected
to produce an "orphan" afterglow.

The rate of short bursts is less certain. Schmidt
[381]
estimates that the rate of short GRBs is smaller by a factor of
two than the rate of long ones. In this case the distances
mentioned above should be revised up by a factor of 1.25.
However, if the rate of short GRBs is larger by a factor 10 than
the rate of long ones then the corresponding distances should be
revised downwards by a factor of 10^{-1/3} This would put one
event per year at ~ 80
_{0.1}^{2}
Mpc, but once again this burst won't be pointing towards us. The nearest
event with a burst in our direction would be at ~ 450 Mpc.

**1. Gravitational Radiation from Neutron Star
Mergers**

Binary neutron star mergers are the "canonical" sources of gravitational radiation emission. LIGO and VIRGO both aim in detecting these sources. Specifically the goal of these detectors is to detect the characteristic "chirping" signals arising from the in-spiraling phase of these events. The possibility of detection of such signals has been extensively discussed (see e.g. [63]). Such events could be detected up to a distance of several tens of Mpc with LIGO I and up to ~ 100 Mpc with LIGO II.

Comparing with GRB rates we find that if, as some expect, neutron star mergers are associated with short GRBs and if the rate of short GRBs is indeed large, then we have one event per year within the sensitivity of LIGO II and marginally detectable by LIGO I. However, this burst will be pointing away from us.

The detection of the chirping merger signal is based on fitting the gravitational radiation signal to pre-calculated templets. Kochaneck and Piran [196] suggested that the detection of a merger gravitational radiation signal would require a lower S/N ratio if this signal coincides with a GRB. This would increase somewhat the effective sensitivity of LIGO and VIRGO to such events. Finn et al. [97] suggest using the association of GRBs with sources of gravitational waves in a statistical manner and propose to search for enhanced gravitational radiation activity towards the direction of a GRB during the short periods when GRBs are detected. Given the huge distances of observed GRBs it is not clear if any of these techniques will be useful.

**2. Gravitational Radiation from Collapsars**

The Collapsar model
[242,
287,
443]
is based
on the collapse of the core of a massive star to a black hole
surrounded by a thick massive accretion disk. As far as
gravitational radiation is concerned this system is very similar
to a regular supernova. Rotating gravitational collapse has been
analyzed by Stark and Piran
[397].
They find that the
gravitational radiation emission emitted in a rotating collapse
to a black hole is dominated by the black hole's lowest normal
modes, with a typical frequency of 20*c*^{3} /
*GM*. The total energy emitted is:

(119) |

where *a* is the dimensionaless specific angular momentum and
_{max} is a
maximal efficiency which is of the order
a few × 10^{-4}. The expected amplitude of the
gravitational radiation signal, *h*, would be of the order of
^{1/2}
*GM* / *c*^{2}*d* where *d* is the distance
to the source. Even LIGO II won't be sensitive enough to detect such a
signal from a distance of 1Gpc or even from 100 Mpc.

**3. Gravitational Radiation from Supranova**

According to the Supranova model a GRB arises after a neurton star collapse to a black hole. This collapse takes PLACE several weeks or months after the Supernova that formed the neutron star (see IXE). The expected gravitational waves signal from a Supranova [426] includes two components. First the signal from the initial supernova is similar to the gravtitational waves from the collapsar model. However, here the first collapse (the Supernova) takes place several weeks or months before the GRB. Thus, there won't be any correlation between the gravitational waves emitted by the first collapse and the GRB. A second component may arise from the second collapse from the supramassive neutron star to a black hole. This signal should coincide with the GRB.

**4. Gravitational Radiation from the GRB**

The most efficient generation of gravitational radiation could
take place here during the acceleration phase, in which the mass
is accelerated to a Lorentz factor
. To estimate this
emission I follow Weinberg
[437]
analysis of gravitational
radiation emitted from a relativistic collision between two
particles. Consider the following simple toy model: two particles
at rest with a mass *M* that are accelerated instantly at *t*
= 0 to a Lorentz factor
and energy
*E*. Conservation of energy
requires that some (actually most) of the rest mass is converted
to kinetic energy during the acceleration and the rest mass of
the accelerated particle is
*m* = *E* /
= *M* / . The
energy emitted per unit frequency per unit solid angle in the
direction at an angle
relative to
is:

(120) |

The result is independent of the frequency, implying that the
integral over all frequency will diverge. This nonphysical
divergence arises from the nonphysical assumption that the
acceleration is instantaneous. In reality this acceleration takes
place over a time
*t*, which is of
order 0.01 sec. This would produce a cutoff
_{max} ~
2 /
*t* above
which Eq. 120 is not valid. The angular distribution
found in Eq. 120 is disappointing. The EM emission from
the ultrarelativistic source is beamed forwards into a small angle
1/, enhancing the
emission in the forwards direction by a large factor
(^{2}). The
gravitational radiation from this
relativistic ejecta is spread rather uniformly in almost all
4 steradians. Instead of
beaming there is "anti-beaming"
with no radiation at all emitted within the forward angle
1/ along the
direction of the relativistic motion.

Integration of the energy flux over different directions yields:

(121) |

As expected the total energy emitted is proportional to
*m*^{2}
^{2}.
Further integration over frequencies up to the cutoff
2 /
*t* yields:

(122) |

In reality the situation is much more complicated than the one
presented here. First, the angular width of the emitted blobs is
larger than 1 / . The
superposition of emission from
different directions washes out the no emission effect in the
forward direction. Additionally according to the internal shocks
model the acceleration of different blobs go on independently.
Emission from different blobs should be combined to get the actual
emission. Both effects *reduce* the effective emission of
gravitational radiation and makes the above estimate an upper
limit to the actual emission.

The gravitational signal is spread in all directions (apart from
a narrow beam along the direction of the relativistic motion of
the GRB). It ranges in frequency from 0 to *f*_{max}
100 Hz. The amplitude
of the gravitational radiation signal at the maximal frequency,
*f*_{max}
100 Hz, would be:
*h* (*GM*
^{2} /
*c*^{2} *d* ). For typical values of *E* = *M*
= 10^{51}
ergs, *t* = 0.01
sec and a distance of 500 Mpc,
*h* .5
^{.} 10^{-25}, it is far below the sensitivity
of planned gravitational radiation detectors. Even if we a burst
is ten times nearer this "direct" gravitational radiation signal
would still be undetectable.

Some specific models for GRBs' inner engine predict additional
amount of energy. For example
vanPutten
[417],
vanPutten and Levinson
[418]
suggest a model of a
black hole - accretion torus in which a large fraction of the
emitted energy of the black hole - accretion torus system escapes
as gravitational radiation. The radiation arises due to
instabilities within the torus that break down the axial
symmetry. They estimate that as much as 10^{53} ergs would be
emitted as gravitational radiation which will have a
characteristic signature corresponding to the normal mode of the
black hole - accretion torus system with typical frequencies
around few hundred Hz, conveniently within the frequency range of
LIGO/VIRGO. If correct than GRBs are the most powerful
burst-sources of gravitational waves in the Universe
[417].