Post-linear Approximation Research Articles

General expansion of time transfer functions in optical spacetime

When dealing with highly accurate modeling of time and frequency transfers into arbitrarily moving dielectrics medium, it may be convenient to work with Gordon's optical spacetime metric rather than the usual physical spacetime metric. Additionally, an accurate modeling of the geodesic evolution of observable quantities (e.g., the range and the Doppler) requires us to know the reception or the emission time transfer functions. In the physical spacetime, these functions can be derived to any post-Minkowskian orders through a recursive procedure. In this work, we show that the time transfer functions can be determined to any order in Gordon's optical spacetime as well. The exact integral forms of the gravitational, the refractive, and the coupling contributions are recursively derived. The expression of the time transfer function is given within the post-linear approximation assuming a stationary optical spacetime covered with geocentric celestial reference system coordinates. The light-dragging effect due to the steady rotation of the neutral atmosphere of the Earth is found to be at the threshold of visibility in many experiments involving accurate modeling of the time and frequency transfers.

Post-linear Schwarzschild solution in harmonic coordinates: Elimination of structure-dependent terms

The paper deals with a special kind of problems that appear in solutions of Einstein's field equations for extended bodies: many structure-dependent terms appear in intermediate calculations that cancel exactly in virtue of the local equations of motion or can be eliminated by appropriate gauge transformations. For a single body at rest these problems are well understood for both the post-Newtonian and the post-Minkowskian cases. However, the situation is still unclear for approximations of higher orders. This paper discusses this problem for a "body" of spherical symmetry to post-linear order. We explicitly demonstrate how the usual Schwarzschild field can be derived directly from the field equations in the post-linear approximation in the harmonic gauge and for an arbitrary spherically symmetric matter distribution. Both external and internal solutions are considered. The case of static incompressible fluid is then compared to the well-known results from the literature. The results of this paper can be applied to generalize the well-known post-Newtonian and post-Minkowskian multipole expansions of the metric in the post-linear approximation.

General post-Minkowskian expansion of time transfer functions

Modeling most of the tests of general relativity requires us to know the function relating light travel time to the coordinate time of reception and to the spatial coordinates of the emitter and the receiver. We call such a function the reception time transfer function. Of course, an emission time transfer function may as well be considered. We present here a recursive procedure enabling us to expand each time transfer function into a perturbative series of ascending powers of the Newtonian gravitational constant G (general post-Minkowskian expansion). Our method is self-sufficient in the sense that neither the integration of null geodesic equations nor the determination of Synge's world function is necessary. To illustrate the method, the time transfer function of a three-parameter family of static, spherically symmetric metrics is derived within the post-linear approximation.

The ADM Hamiltonian at the postlinear approximation

The ADM Hamiltonian for a many-particle system is calculated up to the postlinear approximation, i.e., to the approximation that both the equations of motion for the particles and the equations of motion for the gravitational field in case of no-incoming radiation correctly result up to the postlinear approximation. The relation of this Hamiltonian to the ADM Hamiltonian obtained by a post-Newtonian approximation scheme which was applied up to the first radiation-reaction and radiation levels is discussed. From here the standard formulas for the mechanical angular momentum and energy losses as well as the radiated energy and angular momentum are deduced. Background logarithmic and logarithmic radiative terms are shown to be not present at our approximation if the condition of no-incoming radiation is fulfilled.

Radiation reaction and angular momentum loss in small angle gravitational scattering

Starting from the Poincaré invariant post-linear approximation of general relativity, we present newtonian-like equations of motion for two slowly moving point masses. They include radiation reaction forces and imply a net mechanical angular momentum loss which agrees with the quadrupole formula in the small angle scattering limit.

Poincar�-invariant gravitational field and equations of motion of two pointlike objects: The postlinear approximation of general relativity

Using a fast-motion approximation method we obtain the second-order gravitational field and equations of motion for two pointlike objects in algebraically closed form. A regularization procedure is used which is shown to guarantee the consistency of the approximation scheme. The equations of motion are then transformed within the framework of relativistic predictive mechanics into a system of ordinary differential equations.

Gravitational scattering of two relativistic particles in post-linear approximation

Gravitational scattering of two relativistic particles in post-linear approximation