6,336 research outputs found
The Geometry of Small Causal Diamonds
The geometry of causal diamonds or Alexandrov open sets whose initial and
final events and respectively have a proper-time separation
small compared with the curvature scale is a universal. The corrections from
flat space are given as a power series in whose coefficients involve the
curvature at the centre of the diamond. We give formulae for the total 4-volume
of the diamond, the area of the intersection the future light cone of
with the past light cone of and the 3-volume of the hyper-surface of
largest 3-volume bounded by this intersection valid to .
The formula for the 4-volume agrees with a previous result of Myrheim.
Remarkably, the iso-perimetric ratio depends only on the energy density at the centre and is bigger
than unity if the energy density is positive. These results are also shown to
hold in all spacetime dimensions. Formulae are also given, valid to next
non-trivial order, for causal domains in two spacetime dimensions. We suggest a
number of applications, for instance, the directional dependence of the volume
allows one to regard the volumes of causal diamonds as an observable providing
a measurement of the Ricci tensor.Comment: 17 pages, no figures; Misprints in eqs.(62), (65), (66) and (81)
corrected; a new note on page 13 (with 2 new equations) adde
Time-Dependent Multi-Centre Solutions from New Metrics with Holonomy Sim(n-2)
The classifications of holonomy groups in Lorentzian and in Euclidean
signature are quite different. A group of interest in Lorentzian signature in n
dimensions is the maximal proper subgroup of the Lorentz group, SIM(n-2).
Ricci-flat metrics with SIM(2) holonomy were constructed by Kerr and Goldberg,
and a single four-dimensional example with a non-zero cosmological constant was
exhibited by Ghanam and Thompson. Here we reduce the problem of finding the
general -dimensional Einstein metric of SIM(n-2) holonomy, with and without
a cosmological constant, to solving a set linear generalised Laplace and
Poisson equations on an (n-2)-dimensional Einstein base manifold. Explicit
examples may be constructed in terms of generalised harmonic functions. A
dimensional reduction of these multi-centre solutions gives new time-dependent
Kaluza-Klein black holes and monopoles, including time-dependent black holes in
a cosmological background whose spatial sections have non-vanishing curvature.Comment: Typos corrected; 29 page
Compactifications of Deformed Conifolds, Branes and the Geometry of Qubits
We present three families of exact, cohomogeneity-one Einstein metrics in
dimensions, which are generalizations of the Stenzel construction of
Ricci-flat metrics to those with a positive cosmological constant. The first
family of solutions are Fubini-Study metrics on the complex projective spaces
, written in a Stenzel form, whose principal orbits are the Stiefel
manifolds divided by . The second family are
also Einstein-K\"ahler metrics, now on the Grassmannian manifolds
, whose principal orbits are the
Stiefel manifolds (with no factoring in this case). The
third family are Einstein metrics on the product manifolds , and are K\"ahler only for . Some of these metrics are believed
to play a role in studies of consistent string theory compactifications and in
the context of the AdS/CFT correspondence. We also elaborate on the geometric
approach to quantum mechanics based on the K\"ahler geometry of Fubini-Study
metrics on , and we apply the formalism to study the quantum
entanglement of qubits.Comment: 31 page
Thermodynamics of Magnetised Kerr-Newman Black Holes
The thermodynamics of a magnetised Kerr-Newman black hole is studied to all
orders in the appended magnetic field . The asymptotic properties of the
metric and other fields are dominated by the magnetic flux that extends to
infinity along the axis, leading to subtleties in the calculation of conserved
quantities such as the angular momentum and the mass. We present a detailed
discussion of the implementation of a Wald-type procedure to calculate the
angular momentum, showing how ambiguities that are absent in the usual
asymptotically-flat case may be resolved by the requirement of gauge
invariance. We also present a formalism from which we are able to obtain an
expression for the mass of the magnetised black holes. The expressions for the
mass and the angular momentum are shown to be compatible with the first law of
thermodynamics and a Smarr type relation. Allowing the appended magnetic field
to vary results in an extra term in the first law of the form
where is interpreted as an induced magnetic moment. Minimising the total
energy with respect to the total charge at fixed values of the angular
momentum and energy of the seed metric allows an investigation of Wald's
process. The Meissner effect is shown to hold for electrically neutral extreme
black holes. We also present a derivation of the angular momentum for black
holes in the four-dimensional STU model, which is supergravity
coupled to three vector multiplets.Comment: 27 page
The Action of Instantons with Nut Charge
We examine the effect of a non-trivial nut charge on the action of
non-compact four-dimensional instantons with a U(1) isometry. If the instanton
action is calculated by dimensionally reducing along the isometry, then the nut
charge is found to make an explicit non-zero contribution. For metrics
satisfying AF, ALF or ALE boundary conditions, the action can be expressed
entirely in terms of quantities (including the nut charge) defined on the fixed
point set of the isometry. A source (or sink) of nut charge also implies the
presence of a Misner string coordinate singularity, which will have an
important effect on the Hamiltonian of the instanton.Comment: 25 page
Bohm and Einstein-Sasaki Metrics, Black Holes and Cosmological Event Horizons
We study physical applications of the Bohm metrics, which are infinite
sequences of inhomogeneous Einstein metrics on spheres and products of spheres
of dimension 5 <= d <= 9. We prove that all the Bohm metrics on S^3 x S^2 and
S^3 x S^3 have negative eigenvalue modes of the Lichnerowicz operator and by
numerical methods we establish that Bohm metrics on S^5 have negative
eigenvalues too. We argue that all the Bohm metrics will have negative modes.
These results imply that higher-dimensional black-hole spacetimes where the
Bohm metric replaces the usual round sphere metric are classically unstable. We
also show that the stability criterion for Freund-Rubin solutions is the same
as for black-hole stability, and hence such solutions using Bohm metrics will
also be unstable. We consider possible endpoints of the instabilities, and show
that all Einstein-Sasaki manifolds give stable solutions. We show how Wick
rotation of Bohm metrics gives spacetimes that provide counterexamples to a
strict form of the Cosmic Baldness conjecture, but they are still consistent
with the intuition behind the cosmic No-Hair conjectures. We show how the
Lorentzian metrics may be created ``from nothing'' in a no-boundary setting. We
argue that Lorentzian Bohm metrics are unstable to decay to de Sitter
spacetime. We also argue that noncompact versions of the Bohm metrics have
infinitely many negative Lichernowicz modes, and we conjecture a general
relation between Lichnerowicz eigenvalues and non-uniqueness of the Dirichlet
problem for Einstein's equations.Comment: 53 pages, 11 figure
Gravitational Instantons, Confocal Quadrics and Separability of the Schr\"odinger and Hamilton-Jacobi equations
A hyperk\"ahler 4-metric with a triholomorphic SU(2) action gives rise to a
family of confocal quadrics in Euclidean 3-space when cast in the canonical
form of a hyperk\"ahler 4-metric metric with a triholomorphic circle action.
Moreover, at least in the case of geodesics orthogonal to the U(1) fibres, both
the covariant Schr\"odinger and the Hamilton-Jacobi equation is separable and
the system integrable.Comment: 10 pages Late
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