the amount of radiation produced. We end with a discussion of the results and their implication for the choice of frame of reference one uses in perturbative evolutions of black hole collisions with linear momentum.

Conformal boosted black hole as a perturbation of Schwarzschild
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Initial data in the conformal approach
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The families of initial data that we will consider in this paper are obtained via the “conformal approach” to the initial value problem in general relativity. In it, one assumes the metric to be conformally flat $g_{ab} =\phi^4 \delta_{ab}$ and defines the conformal extrinsic curvature $\widehat{K}_{ab} = \phi^{2} K_{ab}$. In terms of these variables the initial value constraint equations (assuming maximal slicing ${\rm Tr} K =0$) read, $$\begin{aligned}
\nabla_a \widehat{K}^{ab} &=& 0\\
\nabla^2 \phi &=& -{1\over 8} {\widehat{K}^{ab} \widehat{K}_{ab} 
\over \phi^{7}}
\label{Hamil}\end{aligned}$$ where all the derivatives are with respect to flat space.

One can construct [@BoYo] solutions to the first set of equations (momentum constraint) for a single black hole centered at $R=0$, with linear momentum $P_a$, $$\label{onehole}
\widehat{K}_{ab} = {3 \over  2 R^2} \left[ 2 P_{(a} n_{b)} -(\delta_{ab}
-n_a n_b)P^c n_c\right]\ .  \label{boyok}$$ where $R$ is a spherical radial coordinate and $n_a$ a radial unit vector, and 