12,259 research outputs found
Unimodular measures on the space of all Riemannian manifolds
We study unimodular measures on the space of all pointed
Riemannian -manifolds. Examples can be constructed from finite volume
manifolds, from measured foliations with Riemannian leaves, and from invariant
random subgroups of Lie groups. Unimodularity is preserved under weak* limits,
and under certain geometric constraints (e.g. bounded geometry) unimodular
measures can be used to compactify sets of finite volume manifolds. One can
then understand the geometry of manifolds with large, finite volume by
passing to unimodular limits.
We develop a structure theory for unimodular measures on ,
characterizing them via invariance under a certain geodesic flow, and showing
that they correspond to transverse measures on a foliated `desingularization'
of . We also give a geometric proof of a compactness theorem for
unimodular measures on the space of pointed manifolds with pinched negative
curvature, and characterize unimodular measures supported on hyperbolic
-manifolds with finitely generated fundamental group.Comment: 81 page
Exploring strong-field deviations from general relativity via gravitational waves
Two new observational windows have been opened to strong gravitational
physics: gravitational waves, and very long baseline interferometry. This
suggests observational searches for new phenomena in this regime, and in
particular for those necessary to make black hole evolution consistent with
quantum mechanics. We describe possible features of "compact quantum objects"
that replace classical black holes in a consistent quantum theory, and
approaches to observational tests for these using gravitational waves. This is
an example of a more general problem of finding consistent descriptions of
deviations from general relativity, which can be tested via gravitational wave
detection. Simple models for compact modifications to classical black holes are
described via an effective stress tensor, possibly with an effective equation
of state. A general discussion is given of possible observational signatures,
and of their dependence on properties of the colliding objects. The possibility
that departures from classical behavior are restricted to the near-horizon
regime raises the question of whether these will be obscured in gravitational
wave signals, due to their mutual interaction in a binary coalescence being
deep in the mutual gravitational well. Numerical simulation with such simple
models will be useful to clarify the sensitivity of gravitational wave
observation to such highly compact departures from classical black holes.Comment: 20 pages, 9 figures. v2: references and CERN preprint number adde
Properties of derivations on some convolution algebras
For all the convolution algebras and
, the derivations are of the form for suitable measures , where . We describe the
(weakly) compact as well as the (weakly) Montel derivations on these algebras
in terms of properties of the measure . Moreover, for all these algebras
we show that the extension of to a natural dual space is weak-star
continuous.Comment: 12 page
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