105 research outputs found
Hamiltonian theory for the axial perturbations of a dynamical spherical background
We develop the Hamiltonian theory of axial perturbations around a general
time-dependent spherical background spacetime. Using the fact that the
linearized constraints are gauge generators, we isolate the physical and
unconstrained axial gravitational wave in a Hamiltonian pair of variables.
Then, switching to a more geometrical description of the system, we construct
the only scalar combination of them. We obtain the well-known Gerlach and
Sengupta scalar for axial perturbations, with no known equivalent for polar
perturbations. The strategy suggested and tested here will be applied to the
polar case in a separate article.Comment: 12 pages, accepted by Classical and Quantum Gravit
Manifestly Gauge-Invariant General Relativistic Perturbation Theory: II. FRW Background and First Order
In our companion paper we identified a complete set of manifestly
gauge-invariant observables for general relativity. This was possible by
coupling the system of gravity and matter to pressureless dust which plays the
role of a dynamically coupled observer. The evolution of those observables is
governed by a physical Hamiltonian and we derived the corresponding equations
of motion. Linear perturbation theory of those equations of motion around a
general exact solution in terms of manifestly gauge invariant perturbations was
then developed. In this paper we specialise our previous results to an FRW
background which is also a solution of our modified equations of motion. We
then compare the resulting equations with those derived in standard
cosmological perturbation theory (SCPT). We exhibit the precise relation
between our manifestly gauge-invariant perturbations and the linearly
gauge-invariant variables in SCPT. We find that our equations of motion can be
cast into SCPT form plus corrections. These corrections are the trace that the
dust leaves on the system in terms of a conserved energy momentum current
density. It turns out that these corrections decay, in fact, in the late
universe they are negligible whatever the value of the conserved current. We
conclude that the addition of dust which serves as a test observer medium,
while implying modifications of Einstein's equations without dust, leads to
acceptable agreement with known results, while having the advantage that one
now talks about manifestly gauge-invariant, that is measurable, quantities,
which can be used even in perturbation theory at higher orders.Comment: 51 pages, no figure
On (Cosmological) Singularity Avoidance in Loop Quantum Gravity
Loop Quantum Cosmology (LQC), mainly due to Bojowald, is not the cosmological
sector of Loop Quantum Gravity (LQG). Rather, LQC consists of a truncation of
the phase space of classical General Relativity to spatially homogeneous
situations which is then quantized by the methods of LQG. Thus, LQC is a
quantum mechanical toy model (finite number of degrees of freedom) for LQG(a
genuine QFT with an infinite number of degrees of freedom) which provides
important consistency checks. However, it is a non trivial question whether the
predictions of LQC are robust after switching on the inhomogeneous fluctuations
present in full LQG. Two of the most spectacular findings of LQC are that 1.
the inverse scale factor is bounded from above on zero volume eigenstates which
hints at the avoidance of the local curvature singularity and 2. that the
Quantum Einstein Equations are non -- singular which hints at the avoidance of
the global initial singularity. We display the result of a calculation for LQG
which proves that the (analogon of the) inverse scale factor, while densely
defined, is {\it not} bounded from above on zero volume eigenstates. Thus, in
full LQG, if curvature singularity avoidance is realized, then not in this
simple way. In fact, it turns out that the boundedness of the inverse scale
factor is neither necessary nor sufficient for curvature singularity avoidance
and that non -- singular evolution equations are neither necessary nor
sufficient for initial singularity avoidance because none of these criteria are
formulated in terms of observable quantities.After outlining what would be
required, we present the results of a calculation for LQG which could be a
first indication that our criteria at least for curvature singularity avoidance
are satisfied in LQG.Comment: 34 pages, 16 figure
On the Relation between Operator Constraint --, Master Constraint --, Reduced Phase Space --, and Path Integral Quantisation
Path integral formulations for gauge theories must start from the canonical
formulation in order to obtain the correct measure. A possible avenue to derive
it is to start from the reduced phase space formulation. In this article we
review this rather involved procedure in full generality. Moreover, we
demonstrate that the reduced phase space path integral formulation formally
agrees with the Dirac's operator constraint quantisation and, more
specifically, with the Master constraint quantisation for first class
constraints. For first class constraints with non trivial structure functions
the equivalence can only be established by passing to Abelian(ised) constraints
which is always possible locally in phase space. Generically, the correct
configuration space path integral measure deviates from the exponential of the
Lagrangian action. The corrections are especially severe if the theory suffers
from second class secondary constraints. In a companion paper we compute these
corrections for the Holst and Plebanski formulations of GR on which current
spin foam models are based.Comment: 43 page
Universal Correlations in Pion-less EFT with the Resonating Group Model: Three and Four Nucleons
The Effective Field Theory "without pions" at next-to-leading order is used
to analyze universal bound state and scattering properties of the 3- and
4-nucleon system. Results of a variety of phase shift equivalent nuclear
potentials are presented for bound state properties of 3H and 4He, and for the
singlet S-wave 3He-neutron scattering length a_0(3He-n). The calculations are
performed with the Refined Resonating Group Method and include a full treatment
of the Coulomb interaction and the leading-order 3-nucleon interaction. The
results compare favorably with data and values from AV18(+UIX) model
calculations. A new correlation between a_0(3He-n) and the 3H binding energy is
found. Furthermore, we confirm at next-to-leading order the correlations,
already found at leading-order, between the 3H binding energy and the 3H charge
radius, and the Tjon line. With the 3H binding energy as input, we get
predictions of the Effective Field Theory "without pions" at next-to-leading
order for the root mean square charge radius of 3H of (1.6\pm 0.2) fm, for the
4He binding energy of (28\pm 2.5) MeV, and for Re(a_0(3He-n)) of (7.5\pm
0.6)fm. Including the Coulomb interaction, the splitting in binding energy
between 3H and 3He is found to be (0.66\pm 0.03) MeV. The discrepancy to data
of (0.10\mp 0.03) MeV is model independently attributed to higher order charge
independence breaking interactions. We also demonstrate that different results
for the same observable stem from higher order effects, and carefully assess
that numerical uncertainties are negligible. Our results demonstrate the
convergence and usefulness of the pion-less theory at next-to-leading order in
the 4He channel. We conclude that no 4-nucleon interaction is needed to
renormalize the theory at next-to-leading order in the 4-nucleon sector.Comment: 24 pages revtex4, including 8 figures as .eps files embedded with
includegraphicx, leading-order results added, calculations include the LO
three-nucleon interaction explicitly, comment on Wigner bound added, minor
modification
Manifestly Gauge-Invariant General Relativistic Perturbation Theory: I. Foundations
Linear cosmological perturbation theory is pivotal to a theoretical
understanding of current cosmological experimental data provided e.g. by cosmic
microwave anisotropy probes. A key issue in that theory is to extract the gauge
invariant degrees of freedom which allow unambiguous comparison between theory
and experiment. When one goes beyond first (linear) order, the task of writing
the Einstein equations expanded to n'th order in terms of quantities that are
gauge invariant up to terms of higher orders becomes highly non-trivial and
cumbersome. This fact has prevented progress for instance on the issue of the
stability of linear perturbation theory and is a subject of current debate in
the literature. In this series of papers we circumvent these difficulties by
passing to a manifestly gauge invariant framework. In other words, we only
perturb gauge invariant, i.e. measurable quantities, rather than gauge variant
ones. Thus, gauge invariance is preserved non perturbatively while we construct
the perturbation theory for the equations of motion for the gauge invariant
observables to all orders. In this first paper we develop the general framework
which is based on a seminal paper due to Brown and Kuchar as well as the
realtional formalism due to Rovelli. In the second, companion, paper we apply
our general theory to FRW cosmologies and derive the deviations from the
standard treatment in linear order. As it turns out, these deviations are
negligible in the late universe, thus our theory is in agreement with the
standard treatment. However, the real strength of our formalism is that it
admits a straightforward and unambiguous, gauge invariant generalisation to
higher orders. This will also allow us to settle the stability issue in a
future publication.Comment: 77 pages, no figure
Inflationary scalar spectrum in loop quantum cosmology
In the context of loop quantum cosmology, we consider an inflationary era
driven by a canonical scalar field and occurring in the semiclassical regime,
where spacetime is a continuum but quantum gravitational effects are important.
The spectral amplitude and index of scalar perturbations on an unperturbed de
Sitter background are computed at lowest order in the slow-roll parameters. The
scalar spectrum can be blue-tilted and far from scale invariance, and tuning of
the quantization ambiguities is necessary for agreement with observations. The
results are extended to a generalized quantization scheme including those
proposed in the literature. Quantization of the matter field at sub-horizon
scales can provide a consistency check of such schemes.Comment: 29 pages, 2 figures. v2: typos corrected, discussion improved and
extended, new section added. Conclusions are unchange
Spherically Symmetric Quantum Geometry: Hamiltonian Constraint
Variables adapted to the quantum dynamics of spherically symmetric models are
introduced, which further simplify the spherically symmetric volume operator
and allow an explicit computation of all matrix elements of the Euclidean and
Lorentzian Hamiltonian constraints. The construction fits completely into the
general scheme available in loop quantum gravity for the quantization of the
full theory as well as symmetric models. This then presents a further
consistency check of the whole scheme in inhomogeneous situations, lending
further credence to the physical results obtained so far mainly in homogeneous
models. New applications in particular of the spherically symmetric model in
the context of black hole physics are discussed.Comment: 33 page
Loop Quantum Cosmology: A Status Report
The goal of this article is to provide an overview of the current state of
the art in loop quantum cosmology for three sets of audiences: young
researchers interested in entering this area; the quantum gravity community in
general; and, cosmologists who wish to apply loop quantum cosmology to probe
modifications in the standard paradigm of the early universe. An effort has
been made to streamline the material so that, as described at the end of
section I, each of these communities can read only the sections they are most
interested in, without a loss of continuity.Comment: 138 pages, 15 figures. Invited Topical Review, To appear in Classical
and Quantum Gravity. Typos corrected, clarifications and references adde
- …