192 research outputs found
General Relativity and Gravitation: A Centennial Perspective
To commemorate the 100th anniversary of general relativity, the International
Society on General Relativity and Gravitation (ISGRG) commissioned a Centennial
Volume, edited by the authors of this article. We jointly wrote introductions
to the four Parts of the Volume which are collected here. Our goal is to
provide a bird's eye view of the advances that have been made especially during
the last 35 years, i.e., since the publication of volumes commemorating
Einstein's 100th birthday. The article also serves as a brief preview of the 12
invited chapters that contain in-depth reviews of these advances. The volume
will be published by Cambridge University Press and released in June 2015 at a
Centennial conference sponsored by ISGRG and the Topical Group of Gravitation
of the American Physical Society.Comment: 37 page
Near-Constant Mean Curvature Solutions of the Einstein Constraint Equations with Non-Negative Yamabe Metrics
We show that sets of conformal data on closed manifolds with the metric in
the positive or zero Yamabe class, and with the gradient of the mean curvature
function sufficiently small, are mapped to solutions of the Einstein constraint
equations. This result extends previous work which required the conformal
metric to be in the negative Yamabe class, and required the mean curvature
function to be nonzero.Comment: 15 page
Waveless Approximation Theories of Gravity
The analysis of a general multibody physical system governed by Einstein's
equations in quite difficult, even if numerical methods (on a computer) are
used. Some of the difficulties -- many coupled degrees of freedom, dynamic
instability -- are associated with the presence of gravitational waves. We have
developed a number of ``waveless approximation theories'' (WAT) which repress
the gravitational radiation and thereby simplify the analysis. The matter,
according to these theories, evolves dynamically. The gravitational field,
however, is determined at each time step by a set of elliptic equations with
matter sources. There is reason to believe that for many physical systems, the
WAT-generated system evolution is a very accurate approximation to that
generated by the full Einstein theory
Asymptotically Hyperbolic Non Constant Mean Curvature Solutions of the Einstein Constraint Equations
We describe how the iterative technique used by Isenberg and Moncrief to
verify the existence of large sets of non constant mean curvature solutions of
the Einstein constraints on closed manifolds can be adapted to verify the
existence of large sets of asymptotically hyperbolic non constant mean
curvature solutions of the Einstein constraints.Comment: 19 pages, TeX, no figure
The constraint equations for the Einstein-scalar field system on compact manifolds
We study the constraint equations for the Einstein-scalar field system on
compact manifolds. Using the conformal method we reformulate these equations as
a determined system of nonlinear partial differential equations. By introducing
a new conformal invariant, which is sensitive to the presence of the initial
data for the scalar field, we are able to divide the set of free conformal data
into subclasses depending on the possible signs for the coefficients of terms
in the resulting Einstein-scalar field Lichnerowicz equation. For many of these
subclasses we determine whether or not a solution exists. In contrast to other
well studied field theories, there are certain cases, depending on the mean
curvature and the potential of the scalar field, for which we are unable to
resolve the question of existence of a solution. We consider this system in
such generality so as to include the vacuum constraint equations with an
arbitrary cosmological constant, the Yamabe equation and even (all cases of)
the prescribed scalar curvature problem as special cases.Comment: Minor changes, final version. To appear: Classical and Quantum
Gravit
Cosmological spacetimes not covered by a constant mean curvature slicing
We show that there exist maximal globally hyperbolic solutions of the
Einstein-dust equations which admit a constant mean curvature Cauchy surface,
but are not covered by a constant mean curvature foliation.Comment: 11 page
Mixmaster Behavior in Inhomogeneous Cosmological Spacetimes
Numerical investigation of a class of inhomogeneous cosmological spacetimes
shows evidence that at a generic point in space the evolution toward the
initial singularity is asymptotically that of a spatially homogeneous spacetime
with Mixmaster behavior. This supports a long-standing conjecture due to
Belinskii et al. on the nature of the generic singularity in Einstein's
equations.Comment: 4 pages plus 4 figures. A sentence has been deleted. Accepted for
publication in PR
On the area of the symmetry orbits in symmetric spacetimes
We obtain a global existence result for the Einstein equations. We show that
in the maximal Cauchy development of vacuum symmetric initial data with
nonvanishing twist constant, except for the special case of flat Kasner initial
data, the area of the group orbits takes on all positive values. This
result shows that the areal time coordinate which covers these spacetimes
runs from zero to infinity, with the singularity occurring at R=0.Comment: The appendix which appears in version 1 has a technical problem (the
inequality appearing as the first stage of (52) is not necessarily true), and
since the appendix is unnecessary for the proof of our results, we leave it
out. version 2 -- clarifications added, version 3 -- reference correcte
Another positivity proof and gravitational energy localizations
Two locally positive expressions for the gravitational Hamiltonian, one using
4-spinors the other special orthonormal frames, are reviewed. A new quadratic
3-spinor-curvature identity is used to obtain another positive expression for
the Hamiltonian and thereby a localization of gravitational energy and positive
energy proof. These new results provide a link between the other two methods.
Localization and prospects for quasi-localization are discussed.Comment: 14 pages REVTe
Conformal ``thin sandwich'' data for the initial-value problem of general relativity
The initial-value problem is posed by giving a conformal three-metric on each
of two nearby spacelike hypersurfaces, their proper-time separation up to a
multiplier to be determined, and the mean (extrinsic) curvature of one slice.
The resulting equations have the {\it same} elliptic form as does the
one-hypersurface formulation. The metrical roots of this form are revealed by a
conformal ``thin sandwich'' viewpoint coupled with the transformation
properties of the lapse function.Comment: 7 pages, RevTe
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