Dynamics of the gravitational two-body problem at fourth post-Newtonian order and at quadratic order in the Newton constant

We derive the conservative part of the Lagrangian and the energy of a gravitationally bound two-body system at fourth post-Newtonian order, up to terms quadratic in the Newton constant. We also show that such terms are compatible with Lorentz invariance and we write an ansatz for the center-of-mass position. The remaining terms carrying higher powers of the Newton constant are currently under investigation.

Figure 1

The three diagrams contributing at order $G_N$. The $\varphi$, $A$, and $\sigma$ propagators are represented respectively by blue dashed, red dotted, and green solid lines. Propagators and vertices must be considered at appropriate $v$ order to pick the 4PN contribution of these graphs.

Figure 2

The diagrams contributing at order $G_N^2$ arranged in four lines. Diagrams in the $n$th line enter the dynamics at $n\mathrm{PN}$ order. Propagators and vertices must be considered at appropriate $v$ order to pick the 4PN contribution of these diagrams.