574 research outputs found
Universal properties of the near-horizon optical geometry
We make use of the fact that the optical geometry near a static
non-degenerate Killing horizon is asymptotically hyperbolic to investigate
universal features of black hole physics. We show how the Gauss-Bonnet theorem
allows certain lensing scenarios to be ruled in or out. We find rates for the
loss of scalar, vector and fermionic `hair' as objects fall quasi- statically
towards the horizon. In the process we find the Lienard-Wiechert potential for
hyperbolic space and calculate the force between electrons mediated by
neutrinos, extending the flat space result of Feinberg and Sucher. We use the
enhanced conformal symmetry of the Schwarzschild and Reissner-Nordstrom
backgrounds to re-derive the electrostatic field due to a point charge in a
simple fashion
Applications of the Gauss-Bonnet theorem to gravitational lensing
In this geometrical approach to gravitational lensing theory, we apply the
Gauss-Bonnet theorem to the optical metric of a lens, modelled as a static,
spherically symmetric, perfect non-relativistic fluid, in the weak deflection
limit. We find that the focusing of the light rays emerges here as a
topological effect, and we introduce a new method to calculate the deflection
angle from the Gaussian curvature of the optical metric. As examples, the
Schwarzschild lens, the Plummer sphere and the singular isothermal sphere are
discussed within this framework.Comment: 10 pages, 1 figure, IoP styl
A Note on the Instability of Lorentzian Taub-NUT-Space
I show that there are no SU(2)-invariant (time-dependent) tensorial
perturbations of Lorentzian Taub-NUT space. It follows that the spacetime is
unstable at the linear level against generic perturbations. I speculate that
this fact is responsible for so far unsuccessful attempts to define a sensible
thermodynamics for NUT-charged spacetimes.Comment: 13 pages, no figure
Semi-classical stability of AdS NUT instantons
The semi-classical stability of several AdS NUT instantons is studied.
Throughout, the notion of stability is that of stability at the one-loop level
of Euclidean Quantum Gravity. Instabilities manifest themselves as negative
eigenmodes of a modified Lichnerowicz Laplacian acting on the transverse
traceless perturbations. An instability is found for one branch of the
AdS-Taub-Bolt family of metrics and it is argued that the other branch is
stable. It is also argued that the AdS-Taub-NUT family of metrics are stable. A
component of the continuous spectrum of the modified Lichnerowicz operator on
all three families of metrics is found.Comment: 18 pages, 3 figures; references adde
Simple generalizations of Anti-de Sitter space-time
We consider new cosmological solutions which generalize the cosmological
patch of the Anti-de Sitter (AdS) space-time, allowing for fluids with
equations of state such that . We use them to derive the associated
full manifolds. We find that these solutions can all be embedded in flat
five-dimensional space-time with signature, revealing deformed
hyperboloids. The topology and causal-structure of these spaces is therefore
unchanged, and closed time-like curves are identified, before a covering space
is considered. However the structure of Killing vector fields is entirely
different and so we may expect a different structure of Killing horizons in
these solutions.Comment: 6 Pages, 5 Figures, Corrections and additions made for publication in
Journal of Classical and Quantum Gravit
Non-existence of Skyrmion-Skyrmion and Skyrmion-anti-Skyrmion static equilibria
We consider classical static Skyrmion-anti-Skyrmion and Skyrmion-Skyrmion
configurations, symmetric with respect to a reflection plane, or symmetric up
to a -parity transformation respectively. We show that the stress tensor
component completely normal to the reflection plane, and hence its integral
over the plane, is negative definite or positive definite respectively.
Classical Skyrmions always repel classical Skyrmions and classical Skyrmions
always attract classical anti-Skyrmions and thus no static equilibrium, whether
stable or unstable, is possible in either case. No other symmetry assumption is
made and so our results also apply to multi-Skyrmion configurations. Our
results are consistent with existing analyses of Skyrmion forces at large
separation, and with numerical results on Skymion-anti-Skyrmion configurations
in the literature which admit a different reflection symmetry. They also hold
for the massive Skyrme model. We also point out that reflection symmetric
self-gravitating Skyrmions or black holes with Skyrmion hair cannot rest in
symmetric equilibrium with self-gravitating anti-Skyrmions.Comment: v2 Typos corrected, refs added. v3 Journal versio
Stationary Metrics and Optical Zermelo-Randers-Finsler Geometry
We consider a triality between the Zermelo navigation problem, the geodesic
flow on a Finslerian geometry of Randers type, and spacetimes in one dimension
higher admitting a timelike conformal Killing vector field. From the latter
viewpoint, the data of the Zermelo problem are encoded in a (conformally)
Painleve-Gullstrand form of the spacetime metric, whereas the data of the
Randers problem are encoded in a stationary generalisation of the usual optical
metric. We discuss how the spacetime viewpoint gives a simple and physical
perspective on various issues, including how Finsler geometries with constant
flag curvature always map to conformally flat spacetimes and that the Finsler
condition maps to either a causality condition or it breaks down at an
ergo-surface in the spacetime picture. The gauge equivalence in this network of
relations is considered as well as the connection to analogue models and the
viewpoint of magnetic flows. We provide a variety of examples.Comment: 37 pages, 6 figure
Light-bending in Schwarzschild-de-Sitter: projective geometry of the optical metric
We interpret the well known fact that the equations for light rays in the
Kottler or Schwarzschild-de Sitter metric are independent of the cosmological
constant in terms of the projective equivalence of the optical metric for any
value of \Lambda. We explain why this does not imply that lensing phenomena are
independent of \Lambda. Motivated by this example, we find a large collection
of one-parameter families of projectively equivalent metrics including both the
Kottler optical geometry and the constant curvature metrics as special cases.
Using standard constructions for geodesically equivalent metrics we find
classical and quantum conserved quantities and relate these to known
quantities.Comment: 8 page
The helical phase of chiral nematic liquid crystals as the Bianchi VII(0) group manifold
We show that the optical structure of the helical phase of a chiral nematic
is naturally associated with the Bianchi VII(0) group manifold, of which we
give a full account. The Joets-Ribotta metric governing propagation of the
extraordinary rays is invariant under the simply transitive action of the
universal cover of the three dimensional Euclidean group of two dimensions.
Thus extraordinary light rays are geodesics of a left-invariant metric on this
Bianchi type VII(0) group. We are able to solve by separation of variables both
the wave equation and the Hamilton-Jacobi equation for this metric. The former
reduces to Mathieu's equation and the later to the quadrantal pendulum
equation. We discuss Maxwell's equations for uniaxial optical materials where
the configuration is invariant under a group action and develop a formalism to
take advantage of these symmetries. The material is not assumed to be impedance
matched, thus going beyond the usual scope of transformation optics. We show
that for a chiral nematic in its helical phase Maxwell's equations reduce to a
generalised Mathieu equation. Our results may also be relevant to helical
phases of some magnetic materials and to light propagation in certain
cosmological models.Comment: 15 pages, 1 figure; Version 2 updated and expanded, to appear in
Phys. Rev.
The Simplicial Ricci Tensor
The Ricci tensor (Ric) is fundamental to Einstein's geometric theory of
gravitation. The 3-dimensional Ric of a spacelike surface vanishes at the
moment of time symmetry for vacuum spacetimes. The 4-dimensional Ric is the
Einstein tensor for such spacetimes. More recently the Ric was used by Hamilton
to define a non-linear, diffusive Ricci flow (RF) that was fundamental to
Perelman's proof of the Poincare conjecture. Analytic applications of RF can be
found in many fields including general relativity and mathematics. Numerically
it has been applied broadly to communication networks, medical physics,
computer design and more. In this paper, we use Regge calculus (RC) to provide
the first geometric discretization of the Ric. This result is fundamental for
higher-dimensional generalizations of discrete RF. We construct this tensor on
both the simplicial lattice and its dual and prove their equivalence. We show
that the Ric is an edge-based weighted average of deficit divided by an
edge-based weighted average of dual area -- an expression similar to the
vertex-based weighted average of the scalar curvature reported recently. We use
this Ric in a third and independent geometric derivation of the RC Einstein
tensor in arbitrary dimension.Comment: 19 pages, 2 figure
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