14 research outputs found
Kinematic relative velocity with respect to stationary observers in Schwarzschild spacetime
We study the kinematic relative velocity of general test particles with
respect to stationary observers (using spherical coordinates) in Schwarzschild
spacetime, obtaining that its modulus does not depend on the observer, unlike
Fermi, spectroscopic and astrometric relative velocities. We study some
fundamental particular cases, generalizing some results given in other work
about stationary and radial free-falling test particles. Moreover, we give a
new result about test particles with circular geodesic orbits: the modulus of
their kinematic relative velocity with respect to any stationary observer
depends only on the radius of the circular orbit, and so, it remains constant.Comment: 8 pages, 2 figure
Existence and uniqueness of nontrivial collocation solutions of implicitly linear homogeneous Volterra integral equations
We analyze collocation methods for nonlinear homogeneous Volterra-Hammerstein
integral equations with non-Lipschitz nonlinearity. We present different kinds
of existence and uniqueness of nontrivial collocation solutions and we give
conditions for such existence and uniqueness in some cases. Finally we
illustrate these methods with an example of a collocation problem, and we give
some examples of collocation problems that do not fit in the cases studied
previously.Comment: 18 pages, 4 figure
Lightlike simultaneity, comoving observers and distances in general relativity
We state a condition for an observer to be comoving with another observer in
general relativity, based on the concept of lightlike simultaneity. Taking into
account this condition, we study relative velocities, Doppler effect and light
aberration. We obtain that comoving observers observe the same light ray with
the same frequency and direction, and so gravitational redshift effect is a
particular case of Doppler effect. We also define a distance between an
observer and the events that it observes, that coincides with the known affine
distance. We show that affine distance is a particular case of radar distance
in the Minkowski space-time and generalizes the proper radial distance in the
Schwarzschild space-time. Finally, we show that affine distance gives us a new
concept of distance in Robertson-Walker space-times, according to Hubble law.Comment: 17 pages, 5 figures. Since "lightlike distance" is in fact the known
"affine distance", the notation has been change
Relative velocities for radial motion in expanding Robertson-Walker spacetimes
The expansion of space, and other geometric properties of cosmological
models, can be studied using geometrically defined notions of relative
velocity. In this paper, we consider test particles undergoing radial motion
relative to comoving (geodesic) observers in Robertson-Walker cosmologies,
whose scale factors are increasing functions of cosmological time. Analytical
and numerical comparisons of the Fermi, kinematic, astrometric, and the
spectroscopic relative velocities of test particles are given under general
circumstances. Examples include recessional comoving test particles in the de
Sitter universe, the radiation-dominated universe, and the matter-dominated
universe. Three distinct coordinate charts, each with different notions of
simultaneity, are employed in the calculations. It is shown that the
astrometric relative velocity of a radially receding test particle cannot be
superluminal in any expanding Robertson-Walker spacetime. However, necessary
and sufficient conditions are given for the existence of superluminal Fermi
speeds, and it is shown how the four concepts of relative velocity determine
geometric properties of the spacetime.Comment: 27 pages, 6 figure
A note on the computation of geometrically defined relative velocities
We discuss some aspects about the computation of kinematic, spectroscopic,
Fermi and astrometric relative velocities that are geometrically defined in
general relativity. Mainly, we state that kinematic and spectroscopic relative
velocities only depend on the 4-velocities of the observer and the test
particle, unlike Fermi and astrometric relative velocities, that also depend on
the acceleration of the observer and the corresponding relative position of the
test particle, but only at the event of observation and not around it, as it
would be deduced, in principle, from the definition of these velocities.
Finally, we propose an open problem in general relativity that consists on
finding intrinsic expressions for Fermi and astrometric relative velocities
avoiding terms that involve the evolution of the relative position of the test
particle. For this purpose, the proofs given in this paper can serve as
inspiration.Comment: 8 pages, 2 figure
Intrinsic definitions of "relative velocity" in general relativity
Given two observers, we define the "relative velocity" of one observer with
respect to the other in four different ways. All four definitions are given
intrinsically, i.e. independently of any coordinate system. Two of them are
given in the framework of spacelike simultaneity and, analogously, the other
two are given in the framework of observed (lightlike) simultaneity. Properties
and physical interpretations are discussed. Finally, we study relations between
them in special relativity, and we give some examples in Schwarzschild and
Robertson-Walker spacetimes.Comment: 29 pages, 12 figures. New proofs in special relativity and a new open
problem in general relativity (see Remark 5.2). An Appendix has been added,
studying the relative velocities in Schwarzschild, with new figures. Some
spelling erros fixe
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