[Paper Review] Asymptotic expansions for relativistic celestial mechanics

Mayeul Arminjon|ArXiv.org|Mar 16, 2000

Geophysics and Gravity Measurements2 references3 citations

TL;DR

This paper develops an asymptotic expansion framework for relativistic celestial mechanics using a scalar theory of gravitation, introducing a one-parameter family of similar systems scaled by gravitational field strength. By rescaling units to make the small parameter ε = 1/c² explicit, it derives consistent post-Newtonian and post-Minkowskian approximations, showing that the standard post-Newtonian approximation is incompatible with conventional asymptotic expansion methods due to its selective expansion of only the gravitational field, not matter fields.

ABSTRACT

The method of asymptotic expansions is used to build an approximation scheme relevant to celestial mechanics in relativistic theories of gravitation. A scalar theory is considered, both as a simple example and for its own sake. This theory is summarized, then the relevant boundary problem is seen to be the full initial-value problem. It is shown that, with any given system of gravitating bodies, one may associate a one-parameter family of similar systems, the parameter measuring the gravitational field-strength. After a specific change of units, the derivation of asymptotic expansions becomes straightforward. Two hypotheses could be made as to which time variable has to be used in the expansion. The first one leads to an "asymptotic" post-Newtonian approximation (PNA) with instantaneous propagation, differing from the standard PNA in that, in the asymptotic PNA, all fields are expanded. The second hypothese could lead to an "asymptotic" post-Minkowskian approximation (PMA) allowing to describe propagation effects, but it is not compatible with the Newtonian limit. It is shown that the standard PNA is not compatible with the application of the usual method of asymptotic expansions as envisaged here.

Motivation & Objective

• To develop a rigorous asymptotic expansion framework applicable to relativistic celestial mechanics in scalar theories of gravitation.
• To address the inconsistency of the standard post-Newtonian approximation (PNA) with the formal method of asymptotic expansions, which requires expansion of all fields.
• To construct a one-parameter family of gravitating systems scaled by gravitational field strength, enabling systematic approximation.
• To compare the standard PNA with an 'asymptotic' PNA that expands both gravitational and matter fields, showing the latter is more mathematically consistent.
• To explore the compatibility of asymptotic expansions with the Newtonian limit and propagation effects in relativistic gravity.

Proposed method

• Introduces a one-parameter family of gravitating systems (Sε) where ε measures gravitational field strength, derived from a given physical system with compact support.
• Applies a specific change of units depending on ε to simplify the asymptotic expansion process, making it straightforward and elementary.
• Expands all fields—gravitational and matter—simultaneously in powers of ε = 1/c², ensuring consistency with standard asymptotic expansion theory.
• Derives the initial-value problem as the relevant boundary problem, and analyzes how boundary conditions transform under the expansion.
• Considers two time-variable hypotheses: one leading to an 'asymptotic' PNA with instantaneous propagation, and another potentially yielding a post-Minkowskian scheme with wave propagation.
• Uses the scalar theory as a model to clarify conceptual issues in relativistic celestial mechanics, avoiding the complexities of general relativity.

Experimental results

Research questions

• RQ1 Can a consistent asymptotic expansion framework be constructed for relativistic celestial mechanics that expands both gravitational and matter fields?
• RQ2 Why is the standard post-Newtonian approximation incompatible with the formal method of asymptotic expansions as conventionally applied?
• RQ3 How does the introduction of a one-parameter family of similar systems (Sε) relate to the initial-value problem in relativistic gravity?
• RQ4 What are the implications of expanding matter fields in the asymptotic scheme, particularly for the Newtonian limit and mass definitions?
• RQ5 Can a post-Minkowskian approximation be consistently derived from the asymptotic expansion method, and does it preserve propagation effects?

Key findings

• The standard post-Newtonian approximation (PNA) is incompatible with the usual method of asymptotic expansions because it expands only the gravitational field while leaving matter fields unexpanded.
• An 'asymptotic' PNA is constructed by expanding both gravitational and matter fields, which is mathematically consistent and leads to a scheme with instantaneous propagation.
• An alternative hypothesis could lead to an 'asymptotic' post-Minkowskian approximation (PMA) that includes wave propagation, but it fails to recover the Newtonian limit.
• The scalar theory used in the paper predicts no non-Newtonian effects below order 1/c², and expansions in even powers of 1/c are both consistent and natural in the chosen units.
• The Newtonian mass of a celestial body is not uniquely defined in the asymptotic scheme; instead, it is represented by multiple coefficients (e.g., M₀ₐ and M₁ₐ), which can be expressed in terms of initial data at T=0.
• The additional mass parameters (e.g., M₁ₐ) are theory-dependent and not identical to Newtonian masses, implying that Newtonian masses are only approximately valid in relativistic mechanics, with corrections of the same order as second-order terms.