12 research outputs found
Nonexistence of spacelike foliations and the dominant energy condition in Lorentzian geometry
We show that many Lorentzian manifolds of dimension >2 do not admit a
spacelike codimension-one foliation, and that almost every manifold of
dimension >2 which admits a Lorentzian metric at all admits one which satisfies
the dominant energy condition and the timelike convergence condition. These two
seemingly unrelated statements have in fact the same origin.
We also discuss the problem of topology change in General Relativity. A
theorem of Tipler says that topology change is impossible via a spacetime
cobordism whose Ricci curvature satisfies the strict lightlike convergence
condition. In his theorem, the boundary of the cobordism is required to be
spacelike. We show that topology change with the strict lightlike convergence
condition and also the dominant energy condition is possible in many cases when
one requires instead only that there exists a timelike vector field which is
transverse to the boundary.Comment: 31 page
A remark on the rigidity case of the positive energy theorem
In their proof of the positive energy theorem, Schoen and Yau showed that
every asymptotically flat spacelike hypersurface M of a Lorentzian manifold
which is flat along M can be isometrically imbedded with its given second
fundamental form into Minkowski spacetime as the graph of a function from R^n
to R; in particular, M is diffeomorphic to R^n. In this short note, we give an
alternative proof of this fact. The argument generalises to the asymptotically
hyperbolic case, works in every dimension n, and does not need a spin
structure.Comment: 7 page
Every conformal class contains a metric of bounded geometry
We show that on every manifold, every conformal class of semi-Riemannian
metrics contains a metric such that each -th-order covariant derivative
of the Riemann tensor of has bounded absolute value . This result is
new also in the Riemannian case, where one can arrange in addition that is
complete with injectivity and convexity radius greater than 1. One can even
make the radii rapidly increasing and the functions rapidly decreasing at
infinity. We prove generalizations to foliated manifolds, where curvature,
second fundamental form and injectivity radius of the leaves can be controlled
similarly. Moreover, we explain a general principle that can be used to obtain
analogous results for Riemannian manifolds equipped with arbitrary other
additional geometric structures instead of foliations.Comment: 22 pages, 1 figure. The journal article differs from this version
only by marginal adaptations required by the publisher's style guidelines,
and by one minor typ
Every conformal class contains a metric of bounded geometry
We show that on every manifold, every conformal class of semi-Riemannian metrics contains a
metric g such that each kth-order covariant derivative of the Riemann tensor of g has bounded absolute
value ak . This result is new also in the Riemannian case, where one can arrange in addition that g
is complete with injectivity and convexity radius ¸ 1. One can even make the radii rapidly increasing
and the functions ak rapidly decreasing at infinity. We prove generalizations to foliated manifolds, where
curvature, second fundamental form and injectivity radius of the leaves can be controlled similarly.
Moreover, we explain a general principle that can be used to obtain analogous results for Riemannian
manifolds equipped with arbitrary other additional geometric structures instead of foliations
Quantum Field Theory and GravityConceptual and Mathematical Advances in the Search for a Unified Framework /
XIV, 382 p.online resource
Conceptual and mathematical advances in the search for a unified framework
One of the most challenging problems of contemporary theoretical physics is the mathematically rigorous construction of a theory which describes gravitation and the other fundamental physical interactions within a common framework. The physical ideas which grew from attempts to develop such a theory require highly advanced mathematical methods and radically new physical concepts. This book presents different approaches to a rigorous unified description of quantum fields and gravity. It contains a carefully selected cross-section of lively discussions which took place in autumn 2010 at the fifth conference "Quantum field theory and gravity - Conceptual and mathematical advances in the search for a unified framework" in Regensburg, Germany. In the tradition of the other proceedings covering this series of conferences, a special feature of this book is the exposition of a wide variety of approaches, with the intention to facilitate a comparison. The book is mainly addressed to mathematicians and physicists who are interested in fundamental questions of mathematical physics. It allows the reader to obtain a broad and up-to-date overview of a fascinating active research area