522 research outputs found
The Positivity of Energy for Asymptotically Anti-de Sitter Spacetimes
We use the formulation of asymptotically anti-de Sitter boundary conditions
given by Ashtekar and Magnon to obtain a coordinate expression for the general
asymptotically AdeS metric in a neighbourhood of infinity. From this, we are
able to compute the time delay of null curves propagating near infinity. If the
gravitational mass is negative, so will be the time delay (relative to null
geodesics at infinity) for certain null geodesics in the spacetime. Following
closely an argument given by Penrose, Sorkin, and Woolgar, who treated the
asymptotically flat case, we are then able to argue that a negative time delay
is inconsistent with non-negative matter-energies in spacetimes having good
causal properties. We thereby obtain a new positive mass theorem for these
spacetimes. The theorem may be applied even when the matter flux near the
boundary-at-infinity falls off so slowly that the mass changes, provided the
theorem is applied in a time-averaged sense. The theorem also applies in
certain spacetimes having local matter-energy that is sometimes negative, as
can be the case in semi-classical gravity.Comment: (Plain TeX - figures not included
A uniqueness theorem for the adS soliton
The stability of physical systems depends on the existence of a state of
least energy. In gravity, this is guaranteed by the positive energy theorem.
For topological reasons this fails for nonsupersymmetric Kaluza-Klein
compactifications, which can decay to arbitrarily negative energy. For related
reasons, this also fails for the AdS soliton, a globally static, asymptotically
toroidal spacetime with negative mass. Nonetheless, arguing from
the AdS/CFT correspondence, Horowitz and Myers (hep-th/9808079) proposed a new
positive energy conjecture, which asserts that the AdS soliton is the unique
state of least energy in its asymptotic class. We give a new structure theorem
for static spacetimes and use it to prove uniqueness of the AdS
soliton. Our results offer significant support for the new positive energy
conjecture and add to the body of rigorous results inspired by the AdS/CFT
correspondence.Comment: Revtex, 4 pages; Matches published version. More detail in Abstract
and one equation corrected. For details of proofs and further results, see
hep-th/020408
Bounds on area and charge for marginally trapped surfaces with cosmological constant
We sharpen the known inequalities and between the area and the electric charge of a stable marginally
outer trapped surface (MOTS) of genus g in the presence of a cosmological
constant . In particular, instead of requiring stability we include
the principal eigenvalue of the stability operator. For we obtain a lower and an upper bound for in terms of as well as the upper bound for the charge, which reduces to in the stable case . For
there remains only a lower bound on . In the spherically symmetric, static,
stable case one of the area inequalities is saturated iff the surface gravity
vanishes. We also discuss implications of our inequalities for "jumps" and
mergers of charged MOTS.Comment: minor corrections to previous version and to published versio
New Five Dimensional Black Holes Classified by Horizon Geometry, and a Bianchi VI Braneworld
We introduce two new families of solutions to the vacuum Einstein equations
with negative cosmological constant in 5 dimensions. These solutions are static
black holes whose horizons are modelled on the 3-geometries nilgeometry and
solvegeometry. Thus the horizons (and the exterior spacetimes) can be foliated
by compact 3-manifolds that are neither spherical, toroidal, hyperbolic, nor
product manifolds, and therefore are of a topological type not previously
encountered in black hole solutions. As an application, we use the
solvegeometry solutions to construct Bianchi VI braneworld cosmologies.Comment: LaTeX, 20 pages, 2 figures Typographical errors corrected, and
references to printed matter added in favour of preprints where possibl
Theorems on gravitational time delay and related issues
Two theorems related to gravitational time delay are proven. Both theorems
apply to spacetimes satisfying the null energy condition and the null generic
condition. The first theorem states that if the spacetime is null geodesically
complete, then given any compact set , there exists another compact set
such that for any , if there exists a ``fastest null
geodesic'', , between and , then cannot enter . As
an application of this theorem, we show that if, in addition, the spacetime is
globally hyperbolic with a compact Cauchy surface, then any observer at
sufficiently late times cannot have a particle horizon. The second theorem
states that if a timelike conformal boundary can be attached to the spacetime
such that the spacetime with boundary satisfies strong causality as well as a
compactness condition, then any ``fastest null geodesic'' connecting two points
on the boundary must lie entirely within the boundary. It follows from this
theorem that generic perturbations of anti-de Sitter spacetime always produce a
time delay relative to anti-de Sitter spacetime itself.Comment: 15 pages, 1 figure. Example of gauge perturbation changed/corrected.
Two footnotes added and one footnote remove
The Cosmic Censor Forbids Naked Topology
For any asymptotically flat spacetime with a suitable causal structure
obeying (a weak form of) Penrose's cosmic censorship conjecture and satisfying
conditions guaranteeing focusing of complete null geodesics, we prove that
active topological censorship holds. We do not assume global hyperbolicity, and
therefore make no use of Cauchy surfaces and their topology. Instead, we
replace this with two underlying assumptions concerning the causal structure:
that no compact set can signal to arbitrarily small neighbourhoods of spatial
infinity (``-avoidance''), and that no future incomplete null geodesic is
visible from future null infinity. We show that these and the focusing
condition together imply that the domain of outer communications is simply
connected. Furthermore, we prove lemmas which have as a consequence that if a
future incomplete null geodesic were visible from infinity, then given our
-avoidance assumption, it would also be visible from points of spacetime
that can communicate with infinity, and so would signify a true naked
singularity.Comment: To appear in CQG, this improved version contains minor revisions to
incorporate referee's suggestions. Two revised references. Plain TeX, 12
page
Compactness of the space of causal curves
We prove that the space of causal curves between compact subsets of a
separable globally hyperbolic poset is itself compact in the Vietoris topology.
Although this result implies the usual result in general relativity, its proof
does not require the use of geometry or differentiable structure.Comment: 15 page
A Causal Order for Spacetimes with Lorentzian Metrics: Proof of Compactness of the Space of Causal Curves
We recast the tools of ``global causal analysis'' in accord with an approach
to the subject animated by two distinctive features: a thoroughgoing reliance
on order-theoretic concepts, and a utilization of the Vietoris topology for the
space of closed subsets of a compact set. We are led to work with a new causal
relation which we call , and in terms of it we formulate extended
definitions of concepts like causal curve and global hyperbolicity. In
particular we prove that, in a spacetime \M which is free of causal cycles,
one may define a causal curve simply as a compact connected subset of \M
which is linearly ordered by . Our definitions all make sense for
arbitrary metrics (and even for certain metrics which fail to be
invertible in places). Using this feature, we prove for a general metric,
the familiar theorem that the space of causal curves between any two compact
subsets of a globally hyperbolic spacetime is compact. We feel that our
approach, in addition to yielding a more general theorem, simplifies and
clarifies the reasoning involved. Our results have application in a recent
positive energy theorem, and may also prove useful in the study of topology
change. We have tried to make our treatment self-contained by including proofs
of all the facts we use which are not widely available in reference works on
topology and differential geometry.Comment: Two small revisions to accomodate errors brought to our attention by
R.S. Garcia. No change to chief results. 33 page
On the Geometry and Mass of Static, Asymptotically AdS Spacetimes, and the Uniqueness of the AdS Soliton
We prove two theorems, announced in hep-th/0108170, for static spacetimes
that solve Einstein's equation with negative cosmological constant. The first
is a general structure theorem for spacetimes obeying a certain convexity
condition near infinity, analogous to the structure theorems of Cheeger and
Gromoll for manifolds of non-negative Ricci curvature. For spacetimes with
Ricci-flat conformal boundary, the convexity condition is associated with
negative mass. The second theorem is a uniqueness theorem for the negative mass
AdS soliton spacetime. This result lends support to the new positive mass
conjecture due to Horowitz and Myers which states that the unique lowest mass
solution which asymptotes to the AdS soliton is the soliton itself. This
conjecture was motivated by a nonsupersymmetric version of the AdS/CFT
correspondence. Our results add to the growing body of rigorous mathematical
results inspired by the AdS/CFT correspondence conjecture. Our techniques
exploit a special geometric feature which the universal cover of the soliton
spacetime shares with familiar ``ground state'' spacetimes such as Minkowski
spacetime, namely, the presence of a null line, or complete achronal null
geodesic, and the totally geodesic null hypersurface that it determines. En
route, we provide an analysis of the boundary data at conformal infinity for
the Lorentzian signature static Einstein equations, in the spirit of the
Fefferman-Graham analysis for the Riemannian signature case. This leads us to
generalize to arbitrary dimension a mass definition for static asymptotically
AdS spacetimes given by Chru\'sciel and Simon. We prove equivalence of this
mass definition with those of Ashtekar-Magnon and Hawking-Horowitz.Comment: Accepted version, Commun Math Phys; Added Remark IV.3 and supporting
material dealing with non-uniqueness arising from choice of special cycle on
the boundary at infinity; 2 new citations added; LaTeX 27 page
A poor man's positive energy theorem: II. Null geodesics
We show that positivity of energy for stationary, or strongly uniformly
Schwarzschildian, asymptotically flat, non-singular domains of outer
communications can be proved using Galloway's null rigidity theorem.Comment: Latex2e, 24 A4 pages, minor change
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