9,788 research outputs found
Periodicity of free subgroup numbers modulo prime powers
We completely characterise when the sequence of free subgroup numbers of a
finitely generated virtually free group is ultimately periodic modulo a given
prime power.Comment: AmS-LaTeX; 18 page
General relativistic hydrodynamics in curvilinear coordinates
In this paper we report on what we believe is the first successful
implementation of relativistic hydrodynamics, coupled to dynamical spacetimes,
in spherical polar coordinates without symmetry assumptions. We employ a
high-resolution shock-capturing scheme, which requires that the equations be
cast in flux-conservative form. One example of such a form is the :Valencia"
formulation, which has been adopted in numerous applications, in particular in
Cartesian coordinates. Here we generalize this formulation to allow for a
reference-metric approach, which provides a natural framework for calculations
in curvilinear coordinates. In spherical polar coordinates, for example, it
allows for an analytical treatment of the singular r and sin(\theta) terms that
appear in the equations. We experiment with different versions of our
generalized Valencia formulation in numerical implementations of relativistic
hydrodynamics for both fixed and dynamical spacetimes. We consider a number of
different tests -- non-rotating and rotating relativistic stars, as well as
gravitational collapse to a black hole -- to demonstrate that our formulation
provides a promising approach to performing fully relativistic astrophysics
simulations in spherical polar coordinates.Comment: 14 pages, 8 figures, version to be published in PR
Numerical Relativity in Spherical Polar Coordinates: Off-center Simulations
We have recently presented a new approach for numerical relativity
simulations in spherical polar coordinates, both for vacuum and for
relativistic hydrodynamics. Our approach is based on a reference-metric
formulation of the BSSN equations, a factoring of all tensor components, as
well as a partially implicit Runge-Kutta method, and does not rely on a
regularization of the equations, nor does it make any assumptions about the
symmetry across the origin. In order to demonstrate this feature we present
here several off-centered simulations, including simulations of single black
holes and neutron stars whose center is placed away from the origin of the
coordinate system, as well as the asymmetric head-on collision of two black
holes. We also revisit our implementation of relativistic hydrodynamics and
demonstrate that a reference-metric formulation of hydrodynamics together with
a factoring of all tensor components avoids problems related to the coordinate
singularities at the origin and on the axes. As a particularly demanding test
we present results for a shock wave propagating through the origin of the
spherical polar coordinate system.Comment: 13 pages, 11 figures; matches version published in PR
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