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

Abstract

A field-theory-based framework for discussing and interpreting experimental tests of relativistic gravity, notably at the second post-Newtonian (2PN) level, is introduced. Contrary to previous frameworks which attempted at parametrizing any conceivable phenomenological deviation from general relativity, we focus on the most general class of gravity models of the type suggested by unified theories: namely, models in which gravity is mediated by a tensor field together with one or several scalar fields. The 2PN approximation of these "tensor-multiscalar" 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, within this tensor-multiscalar framework, the 2PN deviations from general relativity can be fully described by introducing only two new 2PN parameters $\ensuremath{\varepsilon}$ and $\ensuremath{\zeta}$ beyond the usual (Eddington) 1PN parameters $\overline{\ensuremath{\beta}}\ensuremath{\equiv}\ensuremath{\beta}\ensuremath{-}1$ and $\overline{\ensuremath{\gamma}}\ensuremath{\equiv}\ensuremath{\gamma}\ensuremath{-}1$. It follows from the basic tenets of field theory, notably the absence of negative-energy excitations, that $\overline{\ensuremath{\beta}}$, $\ensuremath{\varepsilon}$, and $\ensuremath{\zeta}$ (as well as all the further parameters entering higher post-Newtonian orders) must tend to zero with $\overline{\ensuremath{\gamma}}$. It is also found that $\ensuremath{\varepsilon}$ and $\ensuremath{\zeta}$ do not enter the 2PN equations of motion of light. Therefore, within our field-theory framework, second-order light-deflection or time-delay experiments cannot probe any 2PN deviation from general relativity. On the other hand, these experiments can give a clean access to $\overline{\ensuremath{\gamma}}$, 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 relativistic gravity. A simplified analysis of current data on four binary-pulsar systems already leads to significant constraints on the two 2PN parameters $|\ensuremath{\varepsilon}|<7\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}2}$, $|\ensuremath{\zeta}|<6\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$.