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Numerical evolutions of nonrotating spherical
relativistic star in stable equilibrium have been
performed and small radial pulsations excited by the
first-order truncation errors that occur at the center and at
the surface of the star have been detected. These pulsations
modulate the evolution of the central rest-mass density. This
figure hows the evolution of the
central rest-mass density for different grid resolutions of a
polytropic star. The evolutions shown in this figure extend to
7 ms.
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Small-amplitude radial pulsations of an initially static
relativistic star> have also been studied. These
pulsations are in the linear regime and are produced by the
truncation errors of the hydrodynamical schemes while evolving
a nonrotating relativistic star in a fixed spacetime.
The evolution of the central rest-mass density is a
superposition of several normal modes of pulsation. Modes
with higher frequencies are damped faster by the numerical
viscosity of the code so that after a certain time the
evolution continues in the fundamental mode of pulsation. This
evolution is shown here.
The frequencies of these small radial pulsations have been
compared with that computed by linear perturbation theory and
with 2D hydrodynamical simulations obtaining an agreement of
1% for the fundamental normal mode and the next four higher
frequency modes.
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Highly non-linear oscillations about an stable
configuration are produced after the migration of a
nonrotating relativistic unstable star to the stable
branch. Numerical truncation errors can perturb the unstable
equilibrium of a relativistic star which will expand and
evolve to a smaller central rest-mass density until a
equilibrium configuration within the stable branch is
reached.
Though this migration cannot take place in an astrophysical
scenario, it can be used as initial data for large amplitude
simulations of relativistic stars. This figure shows the evolution of the
central rest-mass density of an unstable relativistic star
during the migration to the stable branch. The dotted line
represents the evolution of a star with an adiabatic equation
of state and the solid line represents the evolution of a star
with an ideal fluid equation of state.
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Simulations of collapse to Black Hole of nonrotating
unstable spherical relativistic star have also been performed
with GR Hydro Code by introducing a small radial perturbation
in the central rest-mass density of an unstable
configuration.
This figure shows the profiles
along the x-axis of the lapse function, the gxx metric
component and the normalized rest-mass density, where
different times of the evolution are represented by different
lines. The solid line indicates the initial profile and the
thick dashed line the final timeslice at t=0.29ms.
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Long-term evolutions of rapidly rotating relativistic
star in stable equilibrium have also been performed. In
the case of rapidly rotating stars, the truncation errors
produce quasi-radial oscillations causing a shift of the
central rest-mass density towards higher values during the
evolution. We show here the profiles
of the gxx metric component along the x-axis and z-axis at two
coordinate times. The solid line refers to t=0 and the dashed
line to t=3.78 ms, corresponding to 3 rotational periods,
while might appear a short time scale, they represent a
considerable achievement in numerical relativity.
A long-standing problem in relativistic astrophysics, the
mode-frequencies of rotating relativistic stars in full
general relativity and rapid rotation have been calculated
during the long term evolution of the rapidly rotating neutron
stars. This figure shows the
quasi-radial pulsation frequencies for a sequence of
relativistic stars and different rotation rates; the
frequencies of the fundamental mode are represented by squares
and the first higher frequency mode by circles.
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Gravitational waves emitted by a nonrotating
relativistic star pulsating mainly in the fundamental
quadrupolar mode of oscillation have been extracted by means a
perturbative technique.
This figure shows the waveforms as
extracted at 17.7 km and at 23.6 km (top figure), and the
figure below shows the amplitude of the l=2, m=0 component of
the Zerilli function extracted at 23.6 km for two different
resolutions.
