Second post-Newtonian approximation of Einstein-aether theory

Abstract

In this paper, second post-Newtonian approximation of Einstein-aether theory is obtained by Chandrasekhar's approach. Five parametrized post-Newtonian parameters in first post-Newtonian approximation are presented after a time transformation and they are identical with previous works, in which $\ensuremath{\gamma}=1$, $\ensuremath{\beta}=1$, and two preferred-frame parameters remain. Meanwhile, in second post-Newtonian approximation, a parameter, which represents third order nonlinearity for gravity, is zero\char22{}the same as in general relativity. For an application for future deep space laser ranging missions, we reduce the metric coefficients for light propagation in a case of $N$ point masses as a simplified model of the Solar System. The resulting light deflection angle in second post-Newtonian approximation poses another constraint on the Einstein-aether theory.