  We use $\mathcal N=8$ supergravity as a toy model for understanding the
dynamics of black hole binary systems via the scattering amplitudes approach.
We compute the conservative part of the classical scattering angle of two
extremal (half-BPS) black holes with minimal charge misalignment at $\mathcal
O(G^3)$ using the eikonal approximation and effective field theory, finding
agreement between both methods. We construct the massive loop integrands by
Kaluza-Klein reduction of the known $D$-dimensional massless integrands. To
carry out integration we formulate a novel method for calculating the
post-Minkowskian expansion with exact velocity dependence, by solving velocity
differential equations for the Feynman integrals subject to modified boundary
conditions that isolate conservative contributions from the potential region.
Motivated by a recent result for universality in massless scattering, we
compare the scattering angle to the result found by Bern et. al. in Einstein
gravity and find that they coincide in the high-energy limit, suggesting
graviton dominance at this order.
