105 research outputs found
The Calabi complex and Killing sheaf cohomology
It has recently been noticed that the degeneracies of the Poisson bracket of
linearized gravity on constant curvature Lorentzian manifold can be described
in terms of the cohomologies of a certain complex of differential operators.
This complex was first introduced by Calabi and its cohomology is known to be
isomorphic to that of the (locally constant) sheaf of Killing vectors. We
review the structure of the Calabi complex in a novel way, with explicit
calculations based on representation theory of GL(n), and also some tools for
studying its cohomology in terms of of locally constant sheaves. We also
conjecture how these tools would adapt to linearized gravity on other
backgrounds and to other gauge theories. The presentation includes explicit
formulas for the differential operators in the Calabi complex, arguments for
its local exactness, discussion of generalized Poincar\'e duality, methods of
computing the cohomology of locally constant sheaves, and example calculations
of Killing sheaf cohomologies of some black hole and cosmological Lorentzian
manifolds.Comment: tikz-cd diagrams, 69 page
Topology, rigid cosymmetries and linearization instability in higher gauge theories
We consider a class of non-linear PDE systems, whose equations possess
Noether identities (the equations are redundant), including non-variational
systems (not coming from Lagrangian field theories), where Noether identities
and infinitesimal gauge transformations need not be in bijection. We also
include theories with higher stage Noether identities, known as higher gauge
theories (if they are variational). Some of these systems are known to exhibit
linearization instabilities: there exist exact background solutions about which
a linearized solution is extendable to a family of exact solutions only if some
non-linear obstruction functionals vanish. We give a general, geometric
classification of a class of these linearization obstructions, which includes
as special cases all known ones for relativistic field theories (vacuum
Einstein, Yang-Mills, classical N=1 supergravity, etc.). Our classification
shows that obstructions arise due to the simultaneous presence of rigid
cosymmetries (generalized Killing condition) and non-trivial de Rham cohomology
classes (spacetime topology). The classification relies on a careful analysis
of the cohomologies of the on-shell Noether complex (consistent deformations),
adjoint Noether complex (rigid cosymmetries) and variational bicomplex
(conserved currents). An intermediate result also gives a criterion for
identifying non-linearities that do not lead to linearization instabilities.Comment: v2: 33 pages, added an important reference to earlier work of
Arms-Anderson, close to published versio
Quantum astrometric observables I: time delay in classical and quantum gravity
A class of diffeomorphism invariant, physical observables, so-called
astrometric observables, is introduced. A particularly simple example, the time
delay, which expresses the difference between two initially synchronized proper
time clocks in relative inertial motion, is analyzed in detail. It is found to
satisfy some interesting inequalities related to the causal structure of
classical Lorentzian spacetimes. Thus it can serve as a probe of causal
structure and in particular of violations of causality. A quantum model of this
observable as well as the calculation of its variance due to vacuum
fluctuations in quantum linearized gravity are sketched. The question of
whether the causal inequalities are still satisfied by quantized gravity, which
is pertinent to the nature of causality in quantum gravity, is raised, but it
is shown that perturbative calculations cannot provide a definite answer. Some
potential applications of astrometric observables in quantum gravity are
discussed.Comment: revtex4-1, 21 pages, 7 figures (published version); added journal re
Quantum astrometric observables II: time delay in linearized quantum gravity
A clock synchronization thought experiment is modeled by a diffeomorphism
invariant "time delay" observable. In a sense, this observable probes the
causal structure of the ambient Lorentzian spacetime. Thus, upon quantization,
it is sensitive to the long expected smearing of the light cone by vacuum
fluctuations in quantum gravity. After perturbative linearization, its mean and
variance are computed in the Minkowski Fock vacuum of linearized gravity. The
na\"ive divergence of the variance is meaningfully regularized by a length
scale , the physical detector resolution. This is the first time vacuum
fluctuations have been fully taken into account in a similar calculation.
Despite some drawbacks this calculation provides a useful template for the
study of a large class of similar observables in quantum gravity. Due to their
large volume, intermediate calculations were performed using computer algebra
software. The resulting variance scales like , where
is the Planck length and is the distance scale separating the ("lab" and
"probe") clocks. Additionally, the variance depends on the relative velocity of
the lab and the probe, diverging for low velocities. This puzzling behavior may
be due to an oversimplified detector resolution model or a neglected second
order term in the time delay.Comment: 30 pages, 8 figures, revtex4-1; v3: minor updates and corrections,
close to published versio
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