45,009 research outputs found
Spinors, Jets, and the Einstein Equations
Many important features of a field theory, {\it e.g.}, conserved currents,
symplectic structures, energy-momentum tensors, {\it etc.}, arise as tensors
locally constructed from the fields and their derivatives. Such tensors are
naturally defined as geometric objects on the jet space of solutions to the
field equations. Modern results from the calculus on jet bundles can be
combined with a powerful spinor parametrization of the jet space of Einstein
metrics to unravel basic features of the Einstein equations. These techniques
have been applied to computation of generalized symmetries and ``characteristic
cohomology'' of the Einstein equations, and lead to results such as a proof of
non-existence of ``local observables'' for vacuum spacetimes and a uniqueness
theorem for the gravitational symplectic structure.Comment: to appear in the proceedings of the Sixth Canadian Conference on
General Relativity and Relativistic Astrophysics, 13 pages, uses AMSTeX and
AMSppt.st
Discrete Symmetry and Stability in Hamiltonian Dynamics
In this tutorial we address the existence and stability of periodic and
quasiperiodic orbits in N degree of freedom Hamiltonian systems and their
connection with discrete symmetries. Of primary importance in our study are the
nonlinear normal modes (NNMs), i.e periodic solutions which represent
continuations of the system's linear normal modes in the nonlinear regime. We
examine the existence of such solutions and discuss different methods for
constructing them and studying their stability under fixed and periodic
boundary conditions. In the periodic case, we employ group theoretical concepts
to identify a special type of NNMs called one-dimensional "bushes". We describe
how to use linear combinations such NNMs to construct s(>1)-dimensional bushes
of quasiperiodic orbits, for a wide variety of Hamiltonian systems and exploit
the symmetries of the linearized equations to simplify the study of their
destabilization. Applying this theory to the Fermi Pasta Ulam (FPU) chain, we
review a number of interesting results, which have appeared in the recent
literature. We then turn to an analytical and numerical construction of
quasiperiodic orbits, which does not depend on the symmetries or boundary
conditions. We demonstrate that the well-known "paradox" of FPU recurrences may
be explained in terms of the exponential localization of the energies Eq of
NNM's excited at the low part of the frequency spectrum, i.e. q=1,2,3,....
Thus, we show that the stability of these low-dimensional manifolds called
q-tori is related to the persistence or FPU recurrences at low energies.
Finally, we discuss a novel approach to the stability of orbits of conservative
systems, the GALIk, k=2,...,2N, by means of which one can determine accurately
and efficiently the destabilization of q-tori, leading to the breakdown of
recurrences and the equipartition of energy, at high values of the total energy
E.Comment: 50 pages, 13 figure
Linearization Instability for Generic Gravity in AdS
In general relativity, perturbation theory about a background solution fails
if the background spacetime has a Killing symmetry and a compact spacelike
Cauchy surface. This failure, dubbed as {\it linearization instability}, shows
itself as non-integrability of the perturbative infinitesimal deformation to a
finite deformation of the background. Namely, the linearized field equations
have spurious solutions which cannot be obtained from the linearization of
exact solutions. In practice, one can show the failure of the linear
perturbation theory by showing that a certain quadratic (integral) constraint
on the linearized solutions is not satisfied. For non-compact Cauchy surfaces,
the situation is different and for example, Minkowski space having a
non-compact Cauchy surface, is linearization stable. Here we study, the
linearization instability in generic metric theories of gravity where
Einstein's theory is modified with additional curvature terms. We show that,
unlike the case of general relativity, for modified theories even in the
non-compact Cauchy surface cases, there are some theories which show
linearization instability about their anti-de Sitter backgrounds. Recent
dimensional critical and three dimensional chiral gravity theories are two such
examples. This observation sheds light on the paradoxical behavior of vanishing
conserved charges (mass, angular momenta) for non-vacuum solutions, such as
black holes, in these theories.Comment: 31 pages, 1 figure, some grammatical typos are correcte
On the uniqueness and global dynamics of AdS spacetimes
We study global aspects of complete, non-singular asymptotically locally AdS
spacetimes solving the vacuum Einstein equations whose conformal infinity is an
arbitrary globally stationary spacetime. It is proved that any such solution
which is asymptotically stationary to the past and future is itself globally
stationary.
This gives certain rigidity or uniqueness results for exact AdS and related
spacetimes.Comment: 18pp, significant revision of v
Implementation of higher-order absorbing boundary conditions for the Einstein equations
We present an implementation of absorbing boundary conditions for the
Einstein equations based on the recent work of Buchman and Sarbach. In this
paper, we assume that spacetime may be linearized about Minkowski space close
to the outer boundary, which is taken to be a coordinate sphere. We reformulate
the boundary conditions as conditions on the gauge-invariant
Regge-Wheeler-Zerilli scalars. Higher-order radial derivatives are eliminated
by rewriting the boundary conditions as a system of ODEs for a set of auxiliary
variables intrinsic to the boundary. From these we construct boundary data for
a set of well-posed constraint-preserving boundary conditions for the Einstein
equations in a first-order generalized harmonic formulation. This construction
has direct applications to outer boundary conditions in simulations of isolated
systems (e.g., binary black holes) as well as to the problem of
Cauchy-perturbative matching. As a test problem for our numerical
implementation, we consider linearized multipolar gravitational waves in TT
gauge, with angular momentum numbers l=2 (Teukolsky waves), 3 and 4. We
demonstrate that the perfectly absorbing boundary condition B_L of order L=l
yields no spurious reflections to linear order in perturbation theory. This is
in contrast to the lower-order absorbing boundary conditions B_L with L<l,
which include the widely used freezing-Psi_0 boundary condition that imposes
the vanishing of the Newman-Penrose scalar Psi_0.Comment: 25 pages, 9 figures. Minor clarifications. Final version to appear in
Class. Quantum Grav
Implementation of higher-order absorbing boundary conditions for the Einstein equations
We present an implementation of absorbing boundary conditions for the
Einstein equations based on the recent work of Buchman and Sarbach. In this
paper, we assume that spacetime may be linearized about Minkowski space close
to the outer boundary, which is taken to be a coordinate sphere. We reformulate
the boundary conditions as conditions on the gauge-invariant
Regge-Wheeler-Zerilli scalars. Higher-order radial derivatives are eliminated
by rewriting the boundary conditions as a system of ODEs for a set of auxiliary
variables intrinsic to the boundary. From these we construct boundary data for
a set of well-posed constraint-preserving boundary conditions for the Einstein
equations in a first-order generalized harmonic formulation. This construction
has direct applications to outer boundary conditions in simulations of isolated
systems (e.g., binary black holes) as well as to the problem of
Cauchy-perturbative matching. As a test problem for our numerical
implementation, we consider linearized multipolar gravitational waves in TT
gauge, with angular momentum numbers l=2 (Teukolsky waves), 3 and 4. We
demonstrate that the perfectly absorbing boundary condition B_L of order L=l
yields no spurious reflections to linear order in perturbation theory. This is
in contrast to the lower-order absorbing boundary conditions B_L with L<l,
which include the widely used freezing-Psi_0 boundary condition that imposes
the vanishing of the Newman-Penrose scalar Psi_0.Comment: 25 pages, 9 figures. Minor clarifications. Final version to appear in
Class. Quantum Grav
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