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In the work The relativistic problem of several bodies, Am. J. Math. 59(1), 9 (1937), Levi-Civita addresses the problem of the motion of $n$ bodies in general relativity.

He solves simultaneously Einstein's equations for the gravitational field in the presence of $n$ bodies, and the geodesic equations for the $n$ bodies in the presence of the gravitational field above. This is done in perturbation theory, by means of an expansion in the reciprocal of the speed of light.

The result, in Levi-Civita's notation, is that the Lagrangian for body $h$ ($h=1,....,n$) contains the following terms:

  • the zeroth-order, Newtonian Lagrangian $N_h = \frac{1}{2}{{\bf \dot x}}^2 + \gamma_h$, where $\bf x$ is the position of the center of mass of body $h$, and $\gamma_h = G\sum_{\nu \neq h} \frac{m_\nu}{r_{h\nu}}$ is (minus) the Newtonian potential at body $h$, $m_\nu$ is the mass of body $\nu$, and $r_{\nu h}$ is the distance between the centers of $\nu$ and $h$.
  • a relativistic correction $S_h = - \frac{3}{2} \tilde{\omega}_h {{\bf \dot x}}^2 - G\sum_{\nu \neq h} \frac{m_\nu}{r_{h\nu}}(\eta_\nu + 2 \tilde{\omega}_h)$, where $\eta$ and $\tilde \omega$ are constants. This correction is named $S_h$ because, roughly speaking, it represents the self-contribution to the gravitational potential between a body and itself.

The author states that `[...] by comparing $S_h$ and $N_h$, we see that $S_h$ is built by exactly the same inertial and gravitational terms as $N_h$, each affected by a small coefficient $\tilde \omega$ or $\eta$. The presence of $S_h$ [...] has thus barely the effect of altering the intertial and gravitational masses of the bodies'.

And, most importantly, `The equality for each body of these two masses is inherent to the Newtonian term $N_h$, but does no more necessarily hold for the self-term $S_h$'.

The latter statement seems to imply that inertial and gravitational masses are no longer equal when ones takes account of the relativistic corrections.

Where do we stand today on this?

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