13,147 research outputs found
Rolling Stiefel manifolds equipped with -metrics
We discuss the rolling, without slip and without twist, of Stiefel manifolds
equipped with -metrics, from an intrinsic and an extrinsic point of
view. We, however, start with a more general perspective, namely by
investigating intrinsic rolling of normal naturally reductive homogeneous
spaces. This gives evidence why a seemingly straightforward generalization of
intrinsic rolling of symmetric spaces to normal naturally reductive homogeneous
spaces is not possible, in general. For a given control curve, we derive a
system of explicit time-variant ODEs whose solutions describe the desired
rolling. These findings are applied to obtain the intrinsic rolling of Stiefel
manifolds, which is then extended to an extrinsic one. Moreover, explicit
solutions of the kinematic equations are obtained provided that the development
curve is the projection of a not necessarily horizontal one-parameter subgroup.
In addition, our results are put into perspective with examples of rolling
Stiefel manifolds known from the literature.Comment: 48 page
Fuzzy Euclidean wormholes in anti-de Sitter space
This paper is devoted to an investigation of Euclidean wormholes made by
fuzzy instantons. We investigate the Euclidean path integral in anti-de Sitter
space. In Einstein gravity, we introduce a scalar field with a potential.
Because of the analyticity, there is a contribution of complex-valued
instantons, so-called fuzzy instantons. If we have a massless scalar field,
then we obtain Euclidean wormholes, where the probabilities become smaller and
smaller as the size of the throat becomes larger and larger. If we introduce a
non-trivial potential, then in order to obtain a non-zero tunneling rate, we
need to tune the shape of the potential. With the symmetry, after the
analytic continuation to the Lorentzian time, the wormhole throat should expand
to infinity. However, by adding mass, one may obtain an instant wormhole that
should eventually collapse to the event horizon. The existence of Euclidean
wormholes is related to the stability or unitarity issues of anti-de Sitter
space. We are not conclusive yet, but we carefully comment on these physical
problems.Comment: 20 pages, 9 figure
MacDowell-Mansouri gravity and Cartan geometry
The geometric content of the MacDowell-Mansouri formulation of general
relativity is best understood in terms of Cartan geometry. In particular,
Cartan geometry gives clear geometric meaning to the MacDowell-Mansouri trick
of combining the Levi-Civita connection and coframe field, or soldering form,
into a single physical field. The Cartan perspective allows us to view physical
spacetime as tangentially approximated by an arbitrary homogeneous "model
spacetime", including not only the flat Minkowski model, as is implicitly used
in standard general relativity, but also de Sitter, anti de Sitter, or other
models. A "Cartan connection" gives a prescription for parallel transport from
one "tangent model spacetime" to another, along any path, giving a natural
interpretation of the MacDowell-Mansouri connection as "rolling" the model
spacetime along physical spacetime. I explain Cartan geometry, and "Cartan
gauge theory", in which the gauge field is replaced by a Cartan connection. In
particular, I discuss MacDowell-Mansouri gravity, as well as its more recent
reformulation in terms of BF theory, in the context of Cartan geometry.Comment: 34 pages, 5 figures. v2: many clarifications, typos correcte
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