Bondi-Sachs energy-momentum for the constant mean extrinsic curvature initial value problem

The constraints on the asymptotic behavior of the conformal factor and conformal extrinsic curvature imposed by the initial value equations of general relativity on constant mean extrinsic curvature (CMC) hypersurfaces are analyzed in detail. We derive explicit formulas for the Bondi-Sachs energy and momentum in terms of coefficients of asymptotic expansions on CMC hypersurfaces near future null infinity. Precise numerical results for the Bondi-Sachs energy, momentum, and angular momentum are used to interpret physically Bowen-York initial data on conformally flat CMC hypersurfaces similar to that calculated earlier by Buchman et al. [L. T. Buchman, H. P. Pfeiffer, and J. M. Bardeen, Phys. Rev. D 80, 084024-1 (2009).].

Figure 1

Single spinning unboosted centered black hole: The ratio of the Bondi-Sachs mass ($M_{\mathrm{BS}}$) to the irreducible mass ($M_{\mathrm{irr}}$) is compared to the ratio of the Christodoulou mass ($M_{\mathrm{C}}$) to the irreducible mass, as the spin extremality parameter $\zeta$ is varied from 0 to 0.78.

Figure 2

Single black hole displaced in the $x$-direction with no boost or spin. The velocity of the black hole in the asymptotic Minkowski frame, $|P_{\mathrm{BS}}^x|/E_{\mathrm{BS}}$, is plotted against the relative displacement ${\overline{D}}^x=D^x/R_{+}$. The remaining input parameters ($\overline{C}$, ${\overline{R}}_{\mathrm{ms}}$) have their usual values.

Figure 3

Centered black hole with Bowen-York boosts. Using data from Table  above and Table  of Cook and York [25], we plot (i) the fractional excess of the Bondi-Sachs mass over the irreducible mass (solid black curve), (ii) the corresponding excess ADM mass of Bowen-York initial data on maximal hypersurfaces (dashed red curve), and (iii) the velocity of the black hole $v^z=P_{\mathrm{BS}}^z/E_{\mathrm{BS}}$ (dash-dotted violet curve) in our initial data. All quantities are shown as a function of the ratio of physical momentum, $P_{\mathrm{BS}}$ or $P_{\mathrm{ADM}}$ as appropriate, to the irreducible mass. The curves are interpolations between results for discrete models.

Figure 4

Finding the Bowen-York boost that cancels the Bondi-Sachs momentum associated with the displacement ${\overline{D}}^x=0.1$. The CMC parameters have their usual values, except that ${\overline{R}}_{\mathrm{ms}}$ is varied with the boost to keep ${\overline{M}}_{\mathrm{irr}}$ approximately constant.