858 research outputs found
The Torricelli-Fermat Point Generalised
The Torricelli-Fermat point (TF-point) of a triangle is that point which minimises the sum of its distances from the vertices. I generalise this definition, replacing the triangle by a set of M+1 points in E^N. Using the theory of convex functions, I show that the TF-point is unique and find explicit conditions to to determine whether it coincides with any of the given points. If it does not, it may be found by solving a set of ordinary differential equations
Relativistic analysis of the LISA long range optical links
The joint ESA/NASA LISA mission consists in three spacecraft on heliocentric
orbits, flying in a triangular formation of 5 Mkm each side, linked by infrared
optical beams. The aim of the mission is to detect gravitational waves in a low
frequency band. For properly processing the science data, the propagation
delays between spacecraft must be accurately known. We thus analyse the
propagation of light between spacecraft in order to systematically derive the
relativistic effects due to the static curvature of the Schwarzschild spacetime
in which the spacecraft are orbiting with time-varying light-distances. In
particular, our analysis allows to evaluate rigorously the Sagnac effect, and
the gravitational (Einstein) redshift.Comment: 6 figures; accepted for publication in PR
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
Relativistic Equilibrium Distribution by Relative Entropy Maximization
The equilibrium state of a relativistic gas has been calculated based on the
maximum entropy principle. Though the relativistic equilibrium state was long
believed to be the Juttner distribution, a number of papers have been published
in recent years proposing alternative equilibrium states. However, some of
these papers do not pay enough attention to the covariance of distribution
functions, resulting confusion in equilibrium states. Starting from a fully
covariant expression to avoid this confusion, it has been shown in the present
paper that the Juttner distribution is the maximum entropy state if we assume
the Lorentz symmetry.Comment: Six pages, no figure
Multipole structure of current vectors in curved spacetime
A method is presented which allows the exact construction of conserved (i.e.
divergence-free) current vectors from appropriate sets of multipole moments.
Physically, such objects may be taken to represent the flux of particles or
electric charge inside some classical extended body. Several applications are
discussed. In particular, it is shown how to easily write down the class of all
smooth and spatially-bounded currents with a given total charge. This
implicitly provides restrictions on the moments arising from the smoothness of
physically reasonable vector fields. We also show that requiring all of the
moments to be constant in an appropriate sense is often impossible; likely
limiting the applicability of the Ehlers-Rudolph-Dixon notion of quasirigid
motion. A simple condition is also derived that allows currents to exist in two
different spacetimes with identical sets of multipole moments (in a natural
sense).Comment: 13 pages, minor changes, accepted to J. Math. Phy
Quantum phase shift and neutrino oscillations in a stationary, weak gravitational field
A new method based on Synge's world function is developed for determining
within the WKB approximation the gravitationally induced quantum phase shift of
a particle propagating in a stationary spacetime. This method avoids any
calculation of geodesics. A detailed treatment is given for relativistic
particles within the weak field, linear approximation of any metric theory. The
method is applied to the calculation of the oscillation terms governing the
interference of neutrinos considered as a superposition of two eigenstates
having different masses. It is shown that the neutrino oscillations are not
sensitive to the gravitomagnetic components of the metric as long as the spin
contributions can be ignored. Explicit calculations are performed when the
source of the field is a spherical, homogeneous body. A comparison is made with
previous results obtained in Schwarzschild spacetime.Comment: 14 pages, no figure. Enlarged version; added references. In the
Schwarzschild case, our results on the non-radial propagation are compared
with the previous work
Geometric transport along circular orbits in stationary axisymmetric spacetimes
Parallel transport along circular orbits in orthogonally transitive
stationary axisymmetric spacetimes is described explicitly relative to Lie
transport in terms of the electric and magnetic parts of the induced
connection. The influence of both the gravitoelectromagnetic fields associated
with the zero angular momentum observers and of the Frenet-Serret parameters of
these orbits as a function of their angular velocity is seen on the behavior of
parallel transport through its representation as a parameter-dependent Lorentz
transformation between these two inner-product preserving transports which is
generated by the induced connection. This extends the analysis of parallel
transport in the equatorial plane of the Kerr spacetime to the entire spacetime
outside the black hole horizon, and helps give an intuitive picture of how
competing "central attraction forces" and centripetal accelerations contribute
with gravitomagnetic effects to explain the behavior of the 4-acceleration of
circular orbits in that spacetime.Comment: 33 pages ijmpd latex article with 24 eps figure
Energy Contents of Gravitational Waves in Teleparallel Gravity
The conserved quantities, that are, gravitational energy-momentum and its
relevant quantities are investigated for cylindrical and spherical
gravitational waves in the framework of teleparallel equivalent of General
Relativity using the Hamiltonian approach. For both cylindrical and spherical
gravitational waves, we obtain definite energy and constant momentum. The
constant momentum shows consistency with the results available in General
Relativity and teleparallel gravity. The angular momentum for cylindrical and
spherical gravitational waves also turn out to be constant. Further, we
evaluate their gravitational energy-momentum fluxes and gravitational pressure.Comment: 14 pages, accepted for publication in Mod. Phys. Lett.
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