715 research outputs found
On the differential geometry of curves in Minkowski space
We discuss some aspects of the differential geometry of curves in Minkowski
space. We establish the Serret-Frenet equations in Minkowski space and use them
to give a very simple proof of the fundamental theorem of curves in Minkowski
space. We also state and prove two other theorems which represent Minkowskian
versions of a very known theorem of the differential geometry of curves in
tridimensional Euclidean space. We discuss the general solution for torsionless
paths in Minkowki space. We then apply the four-dimensional Serret-Frenet
equations to describe the motion of a charged test particle in a constant and
uniform electromagnetic field and show how the curvature and the torsions of
the four-dimensional path of the particle contain information on the
electromagnetic field acting on the particle.Comment: 10 pages. Typeset using REVTE
The Generalized Jacobi Equation
The Jacobi equation in pseudo-Riemannian geometry determines the linearized
geodesic flow. The linearization ignores the relative velocity of the
geodesics. The generalized Jacobi equation takes the relative velocity into
account; that is, when the geodesics are neighboring but their relative
velocity is arbitrary the corresponding geodesic deviation equation is the
generalized Jacobi equation. The Hamiltonian structure of this nonlinear
equation is analyzed in this paper. The tidal accelerations for test particles
in the field of a plane gravitational wave and the exterior field of a rotating
mass are investigated. In the latter case, the existence of an attractor of
uniform relative radial motion with speed is pointed
out. The astrophysical implications of this result for the terminal speed of a
relativistic jet is briefly explored.Comment: LaTeX file, 4 PS figures, 28 pages, revised version, accepted for
publication in Classical and Quantum Gravit
Schwarzschild and Synge once again
We complete the historical overview about the geometry of a Schwarzschild
black hole at its horizon by emphasizing the contribution made by J. L. Synge
in 1950 to its clarification.Comment: 2 pages, LaTeX, submitted for publication; 2 references, one Note,
and an Acknowledgement are adde
Resonant Metalenses for Breaking the Diffraction Barrier
We introduce the resonant metalens, a cluster of coupled subwavelength
resonators. Dispersion allows the conversion of subwavelength wavefields into
temporal signatures while the Purcell effect permits an efficient radiation of
this information in the far-field. The study of an array of resonant wires
using microwaves provides a physical understanding of the underlying mechanism.
We experimentally demonstrate imaging and focusing from the far-field with
resolutions far below the diffraction limit. This concept is realizable at any
frequency where subwavelength resonators can be designed.Comment: 4 pages, 3 figure
A Snapshot of J. L. Synge
A brief description is given of the life and influence on relativity theory
of Professor J. L. Synge accompanied by some technical examples to illustrate
his style of work
Mode-sum regularization of the scalar self-force: Formulation in terms of a tetrad decomposition of the singular field
We examine the motion in Schwarzschild spacetime of a point particle endowed
with a scalar charge. The particle produces a retarded scalar field which
interacts with the particle and influences its motion via the action of a
self-force. We exploit the spherical symmetry of the Schwarzschild spacetime
and decompose the scalar field in spherical-harmonic modes. Although each mode
is bounded at the position of the particle, a mode-sum evaluation of the
self-force requires regularization because the sum does not converge: the
retarded field is infinite at the position of the particle. The regularization
procedure involves the computation of regularization parameters, which are
obtained from a mode decomposition of the Detweiler-Whiting singular field;
these are subtracted from the modes of the retarded field, and the result is a
mode-sum that converges to the actual self-force. We present such a computation
in this paper. There are two main aspects of our work that are new. First, we
define the regularization parameters as scalar quantities by referring them to
a tetrad decomposition of the singular field. Second, we calculate four sets of
regularization parameters (denoted schematically by A, B, C, and D) instead of
the usual three (A, B, and C). As proof of principle that our methods are
reliable, we calculate the self-force acting on a scalar charge in circular
motion around a Schwarzschild black hole, and compare our answers with those
recorded in the literature.Comment: 38 pages, 2 figure
The Problem of Inertia in Friedmann Universes
In this paper we study the origin of inertia in a curved spacetime,
particularly the spatially flat, open and closed Friedmann universes. This is
done using Sciama's law of inertial induction, which is based on Mach's
principle, and expresses the analogy between the retarded far fields of
electrodynamics and those of gravitation. After obtaining covariant expressions
for electromagnetic fields due to an accelerating point charge in Friedmann
models, we adopt Sciama's law to obtain the inertial force on an accelerating
mass by integrating over the contributions from all the matter in the
universe. The resulting inertial force has the form , where
depends on the choice of the cosmological parameters such as ,
, and and is also red-shift dependent.Comment: 10 page
Quasi-local contribution to the scalar self-force: Geodesic Motion
We consider a scalar charge travelling in a curved background spacetime. We
calculate the quasi-local contribution to the scalar self-force experienced by
such a particle following a geodesic in a general spacetime. We also show that
if we assume a massless field and a vacuum background spacetime, the expression
for the self-force simplifies significantly. We consider some specific cases
whose gravitational analog are of immediate physical interest for the
calculation of radiation reaction corrected orbits of binary black hole
systems. These systems are expected to be detectable by the LISA space based
gravitational wave observatory. We also investigate how alternate techniques
may be employed in some specific cases and use these as a check on our own
results.Comment: Final Phys. Rev. D version. 24 pages, revtex4. Minor typos correcte
H-theorem for classical matter around a black hole
We propose a classical solution for the kinetic description of matter falling
into a black hole, which permits to evaluate both the kinetic entropy and the
entropy production rate of classical infalling matter at the event horizon. The
formulation is based on a relativistic kinetic description for classical
particles in the presence of an event horizon. An H-theorem is established
which holds for arbitrary models of black holes and is valid also in the
presence of contracting event horizons
Rudiments of Holography
An elementary introduction to Maldacena's AdS/CFT correspondence is given,
with some emphasis in the Fefferman-Graham construction. This is based on
lectures given by one of us (E.A.) at the Universidad Autonoma de Madrid.Comment: 60 pages, additional misprints corrected, references adde
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