Specifying Angular Momentum and Center of Mass for Vacuum Initial Data Sets

Abstract

Abstract. We show that it is possible to perturb arbitrary vacuum asymptotically flat spacetimes to new ones having exactly the same energy and linear momentum, but with center of mass and angular momentum equal to any preassigned values measured with respect to a fixed affine frame at infinity. This is in contrast to the axisymmetric situation where a bound on the angular momentum by the mass has been shown to hold for black hole solutions. Our construction involves changing the solution at the linear level in a shell near infinity, and perturbing to impose the vacuum constraint equations. The procedure involves the perturbation correction of an approximate solution which is given explicitly.

The positive mass theorem for multiple rotating charged black holes

“…For these inequalities, the axisymmetric condition is necessary as it is related to conservation of angular momentum, without which the motivating heuristic arguments would no longer apply. In fact, counterexamples exist [16] without the axisymmetric hypothesis. In this setting, and with the addition of supplementary hypotheses to be discussed below, the massangular momentum inequality was established for a single black hole by Dain in [13], and was later extended and improved upon by Schoen and Zhou [22].…”

Geometric inequalities for black holes

“…Note that the global inequality (30) only holds in axial symmetry. This is clear from the physical point of view (see the discussion in [34]) and in [54] highly non-trivial counter examples have been constructed. Finally, concerning the second problem there have been recently some results in [35].…”

Long time existence from interior gluing

“…It further follows from the calculations in [20] and the estimates above that for R ≥ 4k the angular momentum integrals at R ≥ 2k of (g, K) approach the vector J as k tends to infinity. One can then determine the constants c ij algebraically through J, which leads to c ij = O(k 2−2αc ) .…”