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

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

From geodesics of the multipole solutions to the perturbed Kepler problem

A static and axisymmetric solution of the Einstein vacuum equations with a finite number of Relativistic Multipole Moments (RMM) is written in MSA coordinates up to certain order of approximation, and the structure of its metric components is explicitly shown. From the equation of equatorial geodesics we obtain the Binet equation for the orbits and it allows us to determine the gravitational potential that leads to the equivalent classical orbital equations of the perturbed Kepler problem. The relativistic corrections to Keplerian motion are provided by the different contributions of the RMM of the source starting from the Monopole (Schwarzschild correction). In particular, the perihelion precession of the orbit is calculated in terms of the quadrupole and 2$^4$-pole moments. Since the MSA coordinates generalize the Schwarzschild coordinates, the result obtained allows measurement of the relevance of the quadrupole moment in the first order correction to the perihelion frequency-shift

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

Accelerating relativistic reference frames in Minkowski space-time

We study accelerating relativistic reference frames in Minkowski space-time under the harmonic gauge. It is well-known that the harmonic gauge imposes constraints on the components of the metric tensor and also on the functional form of admissible coordinate transformations. These two sets of constraints are equivalent and represent the dual nature of the harmonic gauge. We explore this duality and show that the harmonic gauge allows presenting an accelerated metric in an elegant form that depends only on two harmonic potentials. It also allows reconstruction of the spatial structure of the post-Galilean coordinate transformation functions relating inertial and accelerating frames. The remaining temporal dependence of these functions together with corresponding equations of motion are determined from dynamical conditions, obtained by constructing the relativistic proper reference frame of an accelerated test particle. In this frame, the effect of external forces acting on the observer is balanced by the fictitious frame-reaction force that is needed to keep the test particle at rest with respect to the frame, conserving its relativistic linear momentum. We find that this approach is sufficient to determine all the terms of the coordinate transformation. The same method is then used to develop the inverse transformations. The resulting post-Galilean coordinate transformations extend the Poincar\'e group on the case of accelerating observers. We present and discuss the resulting coordinate transformations, relativistic equations of motion, and the structure of the metric tensors corresponding to the relativistic reference frames involved.Comment: revtex4, 21 page

INPOP08, a 4-D planetary ephemeris: From asteroid and time-scale computations to ESA Mars Express and Venus Express contributions

The latest version of the planetary ephemerides developed at the Paris Observatory and at the Besancon Observatory is presented here. INPOP08 is a 4-dimension ephemeris since it provides to users positions and velocities of planets and the relation between TT and TDB. Investigations leading to improve the modeling of asteroids are described as well as the new sets of observations used for the fit of INPOP08. New observations provided by the European Space Agency (ESA) deduced from the tracking of the Mars Express (MEX) and Venus Express (VEX) missions are presented as well as the normal point deduced from the Cassini mission. We show the huge impact brought by these observations in the fit of INPOP08, especially in terms of Venus, Saturn and Earth-Moon barycenter orbits.Comment: 14 pages. submitted to A&A. accepted in A&

On the Lense-Thirring test with the Mars Global Surveyor in the gravitational field of Mars

I discuss some aspects of the recent test of frame-dragging performed by me by exploiting the Root-Mean-Square (RMS) orbit overlap differences of the out-of-plane component N of the orbit of the Mars Global Surveyor (MGS) spacecraft in the gravitational field of Mars. A linear fit of the full time series of the entire MGS data (4 February 1999-14 January 2005) yields a normalized slope 1.03 +/- 0.41 (with 95% confidence bounds). Other linear fits to different data sets confirm the agreement with general relativity. The huge systematic effects induced by the mismodeling in the martian gravitational field claimed by some authors are absent in the MGS out-of-plane record. The non-gravitational forces affect at the same level of the gravitomagnetic one the in-plane orbital components of MGS, not the out-of-plane one. Moreover, they experience high-frequency variations which does not matter in the present case in which secular effects are relevant.Comment: LaTex2e, 8 pages, no figures, no tables, 17 references. It refers to K. Krogh, Class. Quantum Grav., 24, 5709-5715, 2007 based on astro-ph/0701653. Final version to appear in CEJP (Central European Journal of Physics