3,694 research outputs found
Small object limit of Casimir effect and the sign of the Casimir force
We show a simple way of deriving the Casimir Polder interaction, present some
general arguments on the finiteness and sign of mutual Casimir interactions and
finally we derive a simple expression for Casimir radiation from small
accelerated objects.Comment: 13 pages, late
General relativity and satellite orbits
The general relativistic correction to the position of a satellite is found by retaining Newtonian physics for an observer on the satellite and introducing a potential. The potential is expanded in terms of the Keplerian elements of the orbit and substituted in Lagrange's equations. Integration of the equations shows that a typical earth satellite with small orbital eccentricity is displaced by about 17 cm. from its unperturbed position after a single orbit, while the periodic displacement over the orbit reaches a maximum of about 3 cm. The moon is displaced by about the same amounts. Application of the equations to Mercury gives a total displacement of about 58 km. after one orbit and a maximum periodic displacement of about 12 km
On geodesic envelopes and caustics
We give a global description of envelopes of geodesic tangents of regular
curves in (not necessarily convex) Riemannian surfaces. We prove that such an
envelope is the union of the curve itself, its inflectional geodesics and its
tangential caustics (formed by the conjugate points to those of the initial
curve along the tangent geodesics). Stable singularities of tangential caustics
and geodesic envelopes are discussed. We also prove the (global) stability of
tangential caustics of close curves in convex closed surfaces under small
deformations of the initial curve and of the ambient metric.Comment: 7 pp. 1 figure. 2nd versio
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
Accurate light-time correction due to a gravitating mass
This work arose as an aftermath of Cassini's 2002 experiment \cite{bblipt03},
in which the PPN parameter was measured with an accuracy
and found consistent with the prediction
of general relativity. The Orbit Determination Program (ODP) of
NASA's Jet Propulsion Laboratory, which was used in the data analysis, is based
on an expression for the gravitational delay which differs from the standard
formula; this difference is of second order in powers of -- the sun's
gravitational radius -- but in Cassini's case it was much larger than the
expected order of magnitude , where is the ray's closest approach
distance. Since the ODP does not account for any other second-order terms, it
is necessary, also in view of future more accurate experiments, to
systematically evaluate higher order corrections and to determine which terms
are significant. Light propagation in a static spacetime is equivalent to a
problem in ordinary geometrical optics; Fermat's action functional at its
minimum is just the light-time between the two end points A and B. A new and
powerful formulation is thus obtained. Asymptotic power series are necessary to
provide a safe and automatic way of selecting which terms to keep at each
order. Higher order approximations to the delay and the deflection are
obtained. We also show that in a close superior conjunction, when is much
smaller than the distances of A and B from the Sun, of order , say, the
second-order correction has an \emph{enhanced} part of order , which
corresponds just to the second-order terms introduced in the ODP. Gravitational
deflection of the image of a far away source, observed from a finite distance
from the mass, is obtained to .Comment: 4 figure
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