163 research outputs found
Existence of families of spacetimes with a Newtonian limit
J\"urgen Ehlers developed \emph{frame theory} to better understand the
relationship between general relativity and Newtonian gravity. Frame theory
contains a parameter , which can be thought of as , where
is the speed of light. By construction, frame theory is equivalent to general
relativity for , and reduces to Newtonian gravity for .
Moreover, by setting \ep=\sqrt{\lambda}, frame theory provides a framework to
study the Newtonian limit \ep \searrow 0 (i.e. ). A number of
ideas relating to frame theory that were introduced by J\"urgen have
subsequently found important applications to the rigorous study of both the
Newtonian limit and post-Newtonian expansions. In this article, we review frame
theory and discuss, in a non-technical fashion, some of the rigorous results on
the Newtonian limit and post-Newtonian expansions that have followed from
J\"urgen's work
A Metric for Gradient RG Flow of the Worldsheet Sigma Model Beyond First Order
Tseytlin has recently proposed that an action functional exists whose
gradient generates to all orders in perturbation theory the Renormalization
Group (RG) flow of the target space metric in the worldsheet sigma model. The
gradient is defined with respect to a metric on the space of coupling constants
which is explicitly known only to leading order in perturbation theory, but at
that order is positive semi-definite, as follows from Perelman's work on the
Ricci flow. This gives rise to a monotonicity formula for the flow which is
expected to fail only if the beta function perturbation series fails to
converge, which can happen if curvatures or their derivatives grow large. We
test the validity of the monotonicity formula at next-to-leading order in
perturbation theory by explicitly computing the second-order terms in the
metric on the space of coupling constants. At this order, this metric is found
not to be positive semi-definite. In situations where this might spoil
monotonicity, derivatives of curvature become large enough for higher order
perturbative corrections to be significant.Comment: 15 pages; Erroneous sentence in footnote 14 removed; this version
therefore supersedes the published version (our thanks to Dezhong Chen for
the correction
A rigorous formulation of the cosmological Newtonian limit without averaging
We prove the existence of a large class of one-parameter families of
cosmological solutions to the Einstein-Euler equations that have a Newtonian
limit. This class includes solutions that represent a finite, but otherwise
arbitrary, number of compact fluid bodies. These solutions provide exact
cosmological models that admit Newtonian limits but, are not, either implicitly
or explicitly, averaged
A Gradient Flow for Worldsheet Nonlinear Sigma Models
We discuss certain recent mathematical advances, mainly due to Perelman, in
the theory of Ricci flows and their relevance for renormalization group (RG)
flows. We consider nonlinear sigma models with closed target manifolds
supporting a Riemannian metric, dilaton, and 2-form B-field. By generalizing
recent mathematical results to incorporate the B-field and by decoupling the
dilaton, we are able to describe the 1-loop beta-functions of the metric and
B-field as the components of the gradient of a potential functional on the
space of coupling constants. We emphasize a special choice of diffeomorphism
gauge generated by the lowest eigenfunction of a certain Schrodinger operator
whose potential and kinetic terms evolve along the flow. With this choice, the
potential functional is the corresponding lowest eigenvalue, and gives the
order alpha' correction to the Weyl anomaly at fixed points of (g(t),B(t)).
Since the lowest eigenvalue is monotonic along the flow and reproduces the Weyl
anomaly at fixed points, it accords with the c-theorem for flows that remain
always in the first-order regime. We compute the Hessian of the lowest
eigenvalue functional and use it to discuss the linear stability of points
where the 1-loop beta-functions vanish, such as flat tori and K3 manifolds.Comment: Accepted version for publication. Citations added to Friedan and to
Fateev, Onofri, and Zamolodchikov. Introduction modified slightly to discuss
these citations. 25 pages, LaTe
On all possible static spherically symmetric EYM solitons and black holes
We prove local existence and uniqueness of static spherically symmetric
solutions of the Einstein-Yang-Mills equations for any action of the rotation
group (or SU(2)) by automorphisms of a principal bundle over space-time whose
structure group is a compact semisimple Lie group G. These actions are
characterized by a vector in the Cartan subalgebra of g and are called regular
if the vector lies in the interior of a Weyl chamber. In the irregular cases
(the majority for larger gauge groups) the boundary value problem that results
for possible asymptotically flat soliton or black hole solutions is more
complicated than in the previously discussed regular cases. In particular,
there is no longer a gauge choice possible in general so that the Yang-Mills
potential can be given by just real-valued functions. We prove the local
existence of regular solutions near the singularities of the system at the
center, the black hole horizon, and at infinity, establish the parameters that
characterize these local solutions, and discuss the set of possible actions and
the numerical methods necessary to search for global solutions. That some
special global solutions exist is easily derived from the fact that su(2) is a
subalgebra of any compact semisimple Lie algebra. But the set of less trivial
global solutions remains to be explored.Comment: 26 pages, 2 figures, LaTeX, misprints corrected, 1 reference adde
Cosmological post-Newtonian expansions to arbitrary order
We prove the existence of a large class of one parameter families of
solutions to the Einstein-Euler equations that depend on the singular parameter
\ep=v_T/c (0<\ep < \ep_0), where is the speed of light, and is a
typical speed of the gravitating fluid. These solutions are shown to exist on a
common spacetime slab M\cong [0,T)\times \Tbb^3, and converge as \ep
\searrow 0 to a solution of the cosmological Poisson-Euler equations of
Newtonian gravity. Moreover, we establish that these solutions can be expanded
in the parameter \ep to any specified order with expansion coefficients that
satisfy \ep-independent (nonlocal) symmetric hyperbolic equations
Irreversibility of World-sheet Renormalization Group Flow
We demonstrate the irreversibility of a wide class of world-sheet
renormalization group (RG) flows to first order in in string theory.
Our techniques draw on the mathematics of Ricci flows, adapted to
asymptotically flat target manifolds. In the case of somewhere-negative scalar
curvature (of the target space), we give a proof by constructing an entropy
that increases monotonically along the flow, based on Perelman's Ricci flow
entropy. One consequence is the absence of periodic solutions, and we are able
to give a second, direct proof of this. If the scalar curvature is everywhere
positive, we instead construct a regularized volume to provide an entropy for
the flow. Our results are, in a sense, the analogue of Zamolodchikov's
-theorem for world-sheet RG flows on noncompact spacetimes (though our
entropy is not the Zamolodchikov -function).Comment: Minor changes, added one citation, version accepted for publicatio
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