60 research outputs found
The arithmetic of genus two curves with (4,4)-split Jacobians
In this paper we study genus 2 curves whose Jacobians admit a polarized
(4,4)-isogeny to a product of elliptic curves. We consider base fields of
characteristic different from 2 and 3, which we do not assume to be
algebraically closed. We obtain a full classification of all principally
polarized abelian surfaces that can arise from gluing two elliptic curves along
their 4-torsion and we derive the relation their absolute invariants satisfy.
As an intermediate step, we give a general description of Richelot isogenies
between Jacobians of genus 2 curves, where previously only Richelot isogenies
with kernels that are pointwise defined over the base field were considered.
Our main tool is a Galois theoretic characterization of genus 2 curves
admitting multiple Richelot isogenies.Comment: 30 page
The Schwarzschild-de Sitter solution in five-dimensional general relativity briefly revisited
We briefly revisit the Schwarzschild-de Sitter solution in the context of
five-dimensional general relativity. We obtain a class of five-dimensional
solutions of Einstein vacuum field equations into which the four-dimensional
Schwarzschild-de Sitter space can be locally and isometrically embedded. We
show that this class of solutions is well-behaved in the limit of lambda
approaching zero. Applying the same procedure to the de Sitter cosmological
model in five dimensions we obtain a class of embedding spaces which are
similarly well-behaved in this limit. These examples demonstrate that the
presence of a non-zero cosmological constant does not in general impose a rigid
relation between the (3+1) and (4+1)-dimensional spacetimes, with degenerate
limiting behaviour.Comment: 7 page
Dynamically generated embeddings of spacetime
We discuss how embeddings in connection with the Campbell-Magaard (CM)
theorem can have a physical interpretation. We show that any embedding whose
local existence is guaranteed by the CM theorem can be viewed as a result of
the dynamical evolution of initial data given in a four-dimensional spacelike
hypersurface. By using the CM theorem, we establish that for any analytic
spacetime, there exist appropriate initial data whose Cauchy development is a
five-dimensional vacuum space into which the spacetime is locally embedded. We
shall see also that the spacetime embedded is Cauchy stable with respect these
the initial data.Comment: (8 pages, 1 figure). A section on Cauchy Stability of the embedding
was added. (To appear in Class. Quant. Grav.
Gauge-Dependent Cosmological "Constant"
When the cosmological constant of spacetime is derived from the 5D
induced-matter theory of gravity, we show that a simple gauge transformation
changes it to a variable measure of the vacuum which is infinite at the big
bang and decays to an astrophysically-acceptable value at late epochs. We
outline implications of this for cosmology and galaxy formation.Comment: 14 pages, no figures, expanded version to be published in Class.
Quantum Gra
On the embedding of branes in five-dimensional spaces
We investigate the embedding of four-dimensional branes in five-dimensional
spaces. We firstly consider the case when the embedding space is a vacuum bulk
whose energy-momentum tensor consists of a Dirac delta function with support in
the brane. We then consider the embedding in the context of
Randall-Sundrum-type models, taking into account symmetry and a
cosmological constant. We employ the Campbell-Magaard theorem to construct the
embeddings and are led to the conclusion that the content of energy-matter of
the brane does not necessarily determine its curvature. Finally, as an
application to illustrate our results, we construct the embedding of Minkowski
spacetime filled with dust.Comment: 12 pages - REVTEX To appear in Classical and Quantum Gravit
Embeddings in Non-Vacuum Spacetimes
A scheme is discussed for embedding n-dimensional, Riemannian manifolds in an
(n+1)-dimensional Einstein space. Criteria for embedding a given manifold in a
spacetime that represents a solution to Einstein's equations sourced by a
massless scalar field are also discussed. The embedding procedures are
illustrated with a number of examples.Comment: 17 pages, Plain Latex. Extended discussion on embeddings with scalar
fields and further examples included. In press, Classical and Quantum Gravit
On Applications of Campbell's Embedding Theorem
A little known theorem due to Campbell is employed to establish the local
embedding of a wide class of 4-dimensional spacetimes in 5-dimensional
Ricci-flat spaces. An embedding for the class of n-dimensional Einstein spaces
is also found. The local nature of Campbell's theorem is highlighted by
studying the embedding of some lower-dimensional spaces.Comment: 17 pages, standard Latex sourc
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