28 research outputs found
On the applicability of constrained symplectic integrators in general relativity
The purpose of this note is to point out that a naive application of
symplectic integration schemes for Hamiltonian systems with constraints such as
SHAKE or RATTLE which preserve holonomic constraints encounters difficulties
when applied to the numerical treatment of the equations of general relativity.Comment: 13 pages, change the title to be more descriptive, typos corrected,
added referenc
Hamiltonian structure and quantization of 2+1 dimensional gravity coupled to particles
It is shown that the reduced particle dynamics of 2+1 dimensional gravity in
the maximally slicing gauge has hamiltonian form. This is proved directly for
the two body problem and for the three body problem by using the Garnier
equations for isomonodromic transformations. For a number of particles greater
than three the existence of the hamiltonian is shown to be a consequence of a
conjecture by Polyakov which connects the auxiliary parameters of the fuchsian
differential equation which solves the SU(1,1) Riemann-Hilbert problem, to the
Liouville action of the conformal factor which describes the space-metric. We
give the exact diffeomorphism which transforms the expression of the spinning
cone geometry in the Deser, Jackiw, 't Hooft gauge to the maximally slicing
gauge. It is explicitly shown that the boundary term in the action, written in
hamiltonian form gives the hamiltonian for the reduced particle dynamics. The
quantum mechanical translation of the two particle hamiltonian gives rise to
the logarithm of the Laplace-Beltrami operator on a cone whose angular deficit
is given by the total energy of the system irrespective of the masses of the
particles thus proving at the quantum level a conjecture by 't Hooft on the two
particle dynamics. The quantum mechanical Green's function for the two body
problem is given.Comment: 34 pages LaTe
The Hamiltonian formulation of General Relativity: myths and reality
A conventional wisdom often perpetuated in the literature states that: (i) a
3+1 decomposition of space-time into space and time is synonymous with the
canonical treatment and this decomposition is essential for any Hamiltonian
formulation of General Relativity (GR); (ii) the canonical treatment
unavoidably breaks the symmetry between space and time in GR and the resulting
algebra of constraints is not the algebra of four-dimensional diffeomorphism;
(iii) according to some authors this algebra allows one to derive only spatial
diffeomorphism or, according to others, a specific field-dependent and
non-covariant four-dimensional diffeomorphism; (iv) the analyses of Dirac
[Proc. Roy. Soc. A 246 (1958) 333] and of ADM [Arnowitt, Deser and Misner, in
"Gravitation: An Introduction to Current Research" (1962) 227] of the canonical
structure of GR are equivalent. We provide some general reasons why these
statements should be questioned. Points (i-iii) have been shown to be incorrect
in [Kiriushcheva et al., Phys. Lett. A 372 (2008) 5101] and now we thoroughly
re-examine all steps of the Dirac Hamiltonian formulation of GR. We show that
points (i-iii) above cannot be attributed to the Dirac Hamiltonian formulation
of GR. We also demonstrate that ADM and Dirac formulations are related by a
transformation of phase-space variables from the metric to lapse
and shift functions and the three-metric , which is not canonical. This
proves that point (iv) is incorrect. Points (i-iii) are mere consequences of
using a non-canonical change of variables and are not an intrinsic property of
either the Hamilton-Dirac approach to constrained systems or Einstein's theory
itself.Comment: References are added and updated, Introduction is extended,
Subsection 3.5 is added, 83 pages; corresponds to the published versio
Creating Statistically Anisotropic and Inhomogeneous Perturbations
In almost all structure formation models, primordial perturbations are
created within a homogeneous and isotropic universe, like the one we observe.
Because their ensemble averages inherit the symmetries of the spacetime in
which they are seeded, cosmological perturbations then happen to be
statistically isotropic and homogeneous. Certain anomalies in the cosmic
microwave background on the other hand suggest that perturbations do not
satisfy these statistical properties, thereby challenging perhaps our
understanding of structure formation. In this article we relax this tension. We
show that if the universe contains an appropriate triad of scalar fields with
spatially constant but non-zero gradients, it is possible to generate
statistically anisotropic and inhomogeneous primordial perturbations, even
though the energy momentum tensor of the triad itself is invariant under
translations and rotations.Comment: 20 pages, 1 figure. Uses RevTeX
The Cotton tensor in Riemannian spacetimes
Recently, the study of three-dimensional spaces is becoming of great
interest. In these dimensions the Cotton tensor is prominent as the substitute
for the Weyl tensor. It is conformally invariant and its vanishing is
equivalent to conformal flatness. However, the Cotton tensor arises in the
context of the Bianchi identities and is present in any dimension. We present a
systematic derivation of the Cotton tensor. We perform its irreducible
decomposition and determine its number of independent components for the first
time. Subsequently, we exhibit its characteristic properties and perform a
classification of the Cotton tensor in three dimensions. We investigate some
solutions of Einstein's field equations in three dimensions and of the
topologically massive gravity model of Deser, Jackiw, and Templeton. For each
class examples are given. Finally we investigate the relation between the
Cotton tensor and the energy-momentum in Einstein's theory and derive a
conformally flat perfect fluid solution of Einstein's field equations in three
dimensions.Comment: 27 pages, revtex
Gravity in the 3+1-Split Formalism I: Holography as an Initial Value Problem
We present a detailed analysis of the 3+1-split formalism of gravity in the
presence of a cosmological constant. The formalism helps revealing the intimate
connection between holography and the initial value formulation of gravity. We
show that the various methods of holographic subtraction of divergences
correspond just to different transformations of the canonical variables, such
that the initial value problem is properly set up at the boundary. The
renormalized boundary energy momentum tensor is a component of the Weyl tensor.Comment: 28 pages; v2: minor improvements, references adde
Perturbations of Self-Accelerated Universe
We discuss small perturbations on the self-accelerated solution of the DGP
model, and argue that claims of instability of the solution that are based on
linearized calculations are unwarranted because of the following: (1) Small
perturbations of an empty self-accelerated background can be quantized
consistently without yielding ghosts. (2) Conformal sources, such as radiation,
do not give rise to instabilities either. (3) A typical non-conformal source
could introduce ghosts in the linearized approximation and become unstable,
however, it also invalidates the approximation itself. Such a source creates a
halo of variable curvature that locally dominates over the self-accelerated
background and extends over a domain in which the linearization breaks down.
Perturbations that are valid outside the halo may not continue inside, as it is
suggested by some non-perturbative solutions. (4) In the Euclidean continuation
of the theory, with arbitrary sources, we derive certain constraints imposed by
the second order equations on first order perturbations, thus restricting the
linearized solutions that could be continued into the full nonlinear theory.
Naive linearized solutions fail to satisfy the above constraints. (5) Finally,
we clarify in detail subtleties associated with the boundary conditions and
analytic properties of the Green's functions.Comment: 39 LaTex page
Three Dimensional Canonical Quantum Gravity
General aspects of vielbein representation, ADM formulation and canonical
quantization of gravity are reviewed using pure gravity in three dimensions as
a toy model. The classical part focusses on the role of observers in general
relativity, which will later be identified with quantum observers. A precise
definition of gauge symmetries and a classification of inequivalent solutions
of Einstein's equations in dreibein formalism is given as well. In the quantum
part the construction of the physical Hilbert space is carried out explicitly
for a torus and cylinder type space manifold, which has not been done so far.
Some conceptual problems of quantum gravity are discussed from the point of
view of an observer sitting inside the universe.Comment: 82 pages, uses LaTeX2e and PiCTe
Modified F(R) Horava-Lifshitz gravity: a way to accelerating FRW cosmology
We propose a general approach for the construction of modified gravity which
is invariant under foliation-preserving diffeomorphisms. Special attention is
paid to the formulation of modified Ho\v{r}ava-Lifshitz gravity (FRHL),
whose Hamiltonian structure is studied. It is demonstrated that the
spatially-flat FRW equations of FRHL are consistent with the constraint
equations. The analysis of de Sitter solutions for several versions of FRHL
indicates that the unification of the early-time inflation with the late-time
acceleration is possible. It is shown that a special choice of parameters for
FRHL leads to the same spatially-flat FRW equations as in the case of
traditional -gravity. Finally, an essentially most general modified
Ho\v{r}ava-Lifshitz gravity is proposed, motivated by its fully
diffeomorphism-invariant counterpart, with the restriction that the action does
not contain derivatives higher than the second order with respect to the time
coordinate.Comment: LaTeX 11 pages. v4: Some errors have been correcte