Direction of light propagation to order G^2 in static, spherically symmetric spacetimes: a new derivation

TL;DR

This paper presents a new method to determine the direction of light propagation in static, spherically symmetric spacetimes without integrating null geodesic equations, extending gravitational deflection formulas to second order in G.

Contribution

A novel derivation method for light propagation direction in static spherically symmetric spacetimes avoiding geodesic integration, applicable to second-order gravitational deflection calculations.

Findings

Derived simple expression for the angle between two light rays in these coordinates.

Obtained vector functions for light propagation at emission and reception points.

Extended the gravitational deflection formula used in VLBI astrometry to second order in G.

Abstract

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.