Testing gravity to second post-Newtonian order: a field-theory approach

TL;DR

This paper introduces a theory-driven framework for interpreting gravity tests at the second post-Newtonian level within tensor–multi-scalar theories. By a diagrammatic elimination of field degrees of freedom, it derives a structured 2PN N-body Lagrangian and the corresponding metric, showing that only two new 2PN parameters, ε and ζ, can arise beyond the 1PN Eddington parameters. It finds that light-deflection experiments probe only γ̄, while ε and ζ enter predominantly in strong-field, multi-body interactions, making binary pulsars the most sensitive tests of 2PN deviations; current pulsar data already constrain |ε| < 7×10^-2 and |ζ| < 6×10^-3. The work highlights that solar-system tests are unlikely to reveal 2PN deviations but remain crucial for measuring the fundamental scalar coupling γ̄, with pulsars providing complementary strong-field constraints.

Abstract

A new, field-theory-based framework for discussing and interpreting tests of gravity, notably at the second post-Newtonian (2PN) level, is introduced. Contrary to previous frameworks which attempted at parametrizing any conceivable deviation from general relativity, we focus on the best motivated class of models, in which gravity is mediated by a tensor field together with one or several scalar fields. The 2PN approximation of these "tensor-multi-scalar" theories is obtained thanks to a diagrammatic expansion which allows us to compute the Lagrangian describing the motion of N bodies. In contrast with previous studies which had to introduce many phenomenological parameters, we find that the 2PN deviations from general relativity can be fully described by only two new 2PN parameters, epsilon and zeta, beyond the usual (Eddington) 1PN parameters beta and gamma. It follows from the basic tenets of field theory, notably the absence of negative-energy excitations, that (beta-1), epsilon and zeta (as well as any new parameter entering higher post-Newtonian orders) must tend to zero with (gamma-1). It is also found that epsilon and zeta do not enter the 2PN equations of motion of light. Therefore, light-deflection or time-delay experiments cannot probe any theoretically motivated 2PN deviation from general relativity, but they can give a clean access to (gamma-1), which is of greatest significance as it measures the basic coupling strength of matter to the scalar fields. Because of the importance of self-gravity effects in neutron stars, binary-pulsar experiments are found to constitute a unique testing ground for the 2PN structure of gravity. A simplified analysis of four binary pulsars already leads to significant constraints: |epsilon| < 7x10^-2, |zeta| < 6x10^-3.

Figures (11)

• Figure 1: Diagrammatic interpretation of the effective gravitational constant $\widetilde{G} = G_* A_0^2 (1+\alpha_0^2)$.
• Figure 2: Diagrammatic interpretation of the contraction $(\alpha\beta\alpha)_0$ involved in the Eddington parameter $\overline\beta = {1\over 2} (\alpha\beta\alpha)_0/(1+\alpha_0^2)^2$.
• Figure 3: Diagrammatic notations for the material sources, the fields, and their propagator.
• Figure 4: Diagrammatic expression of the $\Phi^i$-linear terms of the total action (\ref{['eq3.6']}), for $i=1,2,3,4$.
• Figure 5: Equation (\ref{['eq3.2a']}) satisfied by the field $\overline\Phi[\sigma]$.