366 research outputs found
Light Propagation in the Gravitational Field of Moving Bodies by means of Lorentz Transformation. I. Mass monopoles moving with constant velocities
We show how to derive the equations of light propagation in the gravitational
field of uniformly moving mass monopoles without formulating and integrating
the differential equations of light propagation in that field. The well-known
equations of light propagation in the gravitational field of a motionless mass
monopole are combined with a suitable Lorentz transformation. The possibility
to generalize this technique for the more complicated case of uniformly moving
body of arbitrary multipole structure is discussed.Comment: 10 page
Numerical versus analytical accuracy of the formulas for light propagation
Numerical integration of the differential equations of light propagation in
the Schwarzschild metric shows that in some situations relevant for practical
observations the well-known post-Newtonian solution for light propagation has
an error up to 16 microarcsecond. The aim of this work is to demonstrate this
fact, identify the reason for this error and to derive an analytical formula
accurate at the level of 1 microarcsecond as needed for high-accuracy
astrometric projects (e.g., Gaia).
An analytical post-post-Newtonian solution for the light propagation for both
Cauchy and boundary problems is given for the Schwarzschild metric augmented by
the PPN and post-linear parameters , and . Using
analytical upper estimates of each term we investigate which
post-post-Newtonian terms may play a role for an observer in the solar system
at the level of 1 microarcsecond and conclude that only one post-post-Newtonian
term remains important for this numerical accuracy. In this way, an analytical
solution for the boundary problem for light propagation is derived. That
solution contains terms of both post-Newtonian and post-post-Newtonian order,
but is valid for the given numerical level of 1 microarcsecond. The derived
analytical solution has been verified using the results of a high-accuracy
numerical integration of differential equations of light propagation and found
to be correct at the level well below 1 microarcsecond for arbitrary observer
situated within the solar system. Furthermore, the origin of the
post-post-Newtonian terms relevant for the microarcsecond accuracy is
elucidated. We demonstrate that these terms result from an inadequate choice of
the impact parameter in the standard post-Newtonian formulas
A generalized lens equation for light deflection in weak gravitational fields
A generalized lens equation for weak gravitational fields in Schwarzschild
metric and valid for finite distances of source and observer from the light
deflecting body is suggested. The magnitude of neglected terms in the
generalized lens equation is estimated to be smaller than or equal to 15 Pi/4
(m/d')^2, where m is the Schwarzschild radius of massive body and d' is
Chandrasekhar's impact parameter. The main applications of this generalized
lens equation are extreme astrometrical configurations, where 'Standard
post-Newtonian approach' as well as 'Classical lens equation' cannot be
applied. It is shown that in the appropriate limits the proposed lens equation
yields the known post-Newtonian terms, 'enhanced' post-post-Newtonian terms and
the Classical lens equation, thus provides a link between these both essential
approaches for determining the light deflection.Comment: 11 pages, 3 figure
Consistent modeling of the geodetic precession in Earth rotation
A highly precise model for the motion of a rigid Earth is indispensable to
reveal the effects of non-rigidity in the rotation of the Earth from
observations. To meet the accuracy goal of modern theories of Earth rotation of
1 microarcsecond (muas) it is clear, that for such a model also relativistic
effects have to be taken into account. The largest of these effects is the so
called geodetic precession.
In this paper we will describe this effect and the standard procedure to deal
with it in modeling Earth rotation up to now. With our relativistic model of
Earth rotation Klioner et al. (2001) we are able to give a consistent
post-Newtonian treatment of the rotational motion of a rigid Earth in the
framework of General Relativity. Using this model we show that the currently
applied standard treatment of geodetic precession is not correct. The
inconsistency of the standard treatment leads to errors in all modern theories
of Earth rotation with a magnitude of up to 200 muas for a time span of one
century.Comment: 6 pages, 4 figures, 1 table, published in the Proceedings of the VII
Hotine-Marussi Symposium, Chapter 4
Direction of light propagation to order G^2 in static, spherically symmetric spacetimes: a new derivation
A procedure avoiding any integration of the null geodesic equations is used
to derive the direction of light propagation in a three-parameter family of
static, spherically symmetric spacetimes within the post-post-Minkowskian
approximation. Quasi-Cartesian isotropic coordinates adapted to the symmetries
of spacetime are systematically used. It is found that the expression of the
angle formed by two light rays as measured by a static observer staying at a
given point is remarkably simple in these coordinates. The attention is mainly
focused on the null geodesic paths that we call the "quasi-Minkowskian light
rays". The vector-like functions characterizing the direction of propagation of
such light rays at their points of emission and reception are firstly obtained
in the generic case where these points are both located at a finite distance
from the centre of symmetry. The direction of propagation of the
quasi-Minkowskian light rays emitted at infinity is then straightforwardly
deduced. An intrinsic definition of the gravitational deflection angle relative
to a static observer located at a finite distance is proposed for these rays.
The expression inferred from this definition extends the formula currently used
in VLBI astrometry up to the second order in the gravitational constant G.Comment: 19 pages; revised introduction; added references for introduction;
corrected typos; published in Class. Quantum Gra
Relativistic Celestial Mechanics with PPN Parameters
Starting from the global parametrized post-Newtonian (PPN) reference system
with two PPN parameters and we consider a space-bounded
subsystem of matter and construct a local reference system for that subsystem
in which the influence of external masses reduces to tidal effects. Both the
metric tensor of the local PPN reference system in the first post-Newtonian
approximation as well as the coordinate transformations between the global PPN
reference system and the local one are constructed in explicit form. The terms
proportional to reflecting a violation of the
equivalence principle are discussed in detail. We suggest an empirical
definition of multipole moments which are intended to play the same role in PPN
celestial mechanics as the Blanchet-Damour moments in General Relativity.
Starting with the metric tensor in the local PPN reference system we derive
translational equations of motion of a test particle in that system. The
translational and rotational equations of motion for center of mass and spin of
each of extended massive bodies possessing arbitrary multipole structure
are derived. As an application of the general equations of motion a
monopole-spin dipole model is considered and the known PPN equations of motion
of mass monopoles with spins are rederived.Comment: 71 page
Units of relativistic time scales and associated quantities
This note suggests nomenclature for dealing with the units of various
astronomical quantities that are used with the relativistic time scales TT,
TDB, TCB and TCG. It is suggested to avoid wordings like "TDB units" and "TT
units" and avoid contrasting them to "SI units". The quantities intended for
use with TCG, TCB, TT or TDB should be called "TCG-compatible",
"TCB-compatible", "TT-compatible" or "TDB-compatible", respectively. The names
of the units second and meter for numerical values of all these quantities
should be used with out any adjectives. This suggestion comes from a special
discussion forum created within IAU Commission 52 "Relativity in Fundamental
Astronomy"
Post-Newtonian limitations on measurement of the PPN parameters caused by motion of gravitating bodies
We derive explicit Lorentz-invariant solution of the Einstein and null
geodesic equations for data processing of the time delay and ranging
experiments in gravitational field of moving gravitating bodies of the solar
system - the Sun and major planets. We discuss general-relativistic
interpretation of these experiments and the limitations imposed by motion of
the massive bodies on measurement of the parameters gamma_{PPN}, beta_{PPN} and
delta_{PPN} of the parameterized post-Newtonian formalism.Comment: 17 pages, 1 figure; accepted for publication to MNRA
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