[Paper Review] Angular distances in metric theories

P. Teyssandier, C. Le Poncin-Lafitte|ArXiv.org|Nov 14, 2006

Mathematics and Applications5 references3 citations

TL;DR

This paper presents a rigorous, covariant framework for calculating the angular distance between two point-like light sources as measured by an arbitrarily moving observer in any metric theory of gravity. It unifies relativistic aberration and gravitational light deflection, deriving explicit expressions up to post-Newtonian order $1/c^3$ and post-post-Minkowskian order, enabling high-precision astrometry for future space missions like LATOR and ASTROD.

ABSTRACT

The general expression of the angular distance between two point sources as measured by an arbitrary observer is given. The modelling presented here is rigorous, covariant and valid in any space-time. The sources of light may be located at a finite distance from the observer. The aberration and the gravitational deflection of light are treated in a unified way. Assuming the gravitational field to be weak, an explicit expansion of the angular separation within the post-post-Minkowskian approximation is carried out. The angular separation within the post-Newtonian approximation truncated at the order $1/c^3$ is straightforwardly derived.

Motivation & Objective

• To develop a general, covariant formalism for computing angular distances between light sources as measured by a moving observer in arbitrary space-times.
• To unify the treatment of relativistic aberration and gravitational light deflection without separating them.
• To derive explicit perturbative expansions of angular separation in weak-field, slow-motion regimes relevant to upcoming astrometric missions.
• To provide a foundation for testing post-linear gravity theories using high-precision angular measurements.

Proposed method

• Derives a general expression for angular separation using a coordinate-free, four-vector formalism based on orthogonal projections onto the observer's local three-space.
• Introduces the reception time transfer function as the key unknown, enabling full calculation of angular distance when known.
• Performs a post-post-Minkowskian expansion of the angular separation formula up to second order in $G$, valid for weak fields.
• Specializes to the post-Newtonian approximation truncated at $1/c^3$, yielding explicit expressions for relativistic corrections.
• Uses the parametrized post-Newtonian (PPN) metric with parameters $\beta$ and $\gamma$ to generalize results for alternative metric theories.
• Derives the angular separation formula in terms of observer velocity $\bm{\beta}$, light-ray unit vectors $\bm{N}, \bm{N}'$, and gravitational deflection functions $\bm{\lambda}, \bm{\lambda}'$.

Experimental results

Research questions

• RQ1 How can the angular distance between two distant light sources be rigorously computed in a generally covariant way for an arbitrarily moving observer in a metric theory of gravity?
• RQ2 What is the unified effect of relativistic aberration and gravitational light deflection on observed angular separations in weak gravitational fields?
• RQ3 How do the post-Newtonian corrections at order $1/c^3$ modify the observed angular distance between two sources?
• RQ4 What is the explicit form of the angular separation in the post-post-Minkowskian approximation up to second order in $G$?
• RQ5 How do mass-multipole moments of a planetary body like Jupiter affect the light deflection and thus the observed angular distance?

Key findings

• The angular separation is derived in a fully covariant, coordinate-free manner, avoiding tetrad formalism and unifying aberration and gravitational deflection.
• The fundamental formula depends only on the reception time transfer function, reducing the astrometric problem to determining a single unknown function.
• In the post-post-Minkowskian approximation, the angular separation is expanded up to second order in $G$, with explicit terms involving $w/c^2$ and velocity-dependent corrections.
• Within the post-Newtonian approximation truncated at $1/c^3$, the angular separation receives corrections proportional to $w/c^2$, $\bm{\beta} \cdot (\bm{\lambda} + \bm{\lambda}')$, and $({\bm{\lambda}}' - {\bm{\lambda}}) \cdot \bm{s}$, with $\gamma$-dependence in the deflection.
• The derived formula (95) explicitly quantifies the influence of observer motion and gravitational potential on the observed angular distance, with $\phi_u^{(PN)}$ depending on $\bm{N} \cdot \bm{N}'$, $\bm{\beta}$, and $\bm{\lambda}$.
• The framework enables precise modeling of light deflection by planetary multipole moments, with applications to Jupiter's gravitational field in future astrometric tests.