Momentum flow in black-hole binaries. I. Post-Newtonian analysis of the inspiral and spin-induced bobbing

A brief overview is presented of a new Caltech/Cornell research program that is exploring the nonlinear dynamics of curved spacetime in binary black-hole collisions and mergers, and of an initial project in this program aimed at elucidating the flow of linear momentum in binary black holes (BBHs). The “gauge-dependence” (arbitrariness) in the localization of linear momentum in BBHs is discussed, along with the hope that the qualitative behavior of linear momentum will be gauge-independent. Harmonic coordinates are suggested as a possibly preferred foundation for fixing the gauge associated with linear momentum. For a BBH or other compact binary, the Landau-Lifshitz formalism is used to define the momenta of the binary’s individual bodies in terms of integrals over the bodies’ surfaces or interiors, and define the momentum of the gravitational field (spacetime curvature) outside the bodies as a volume integral over the field’s momentum density. These definitions will be used in subsequent papers that explore the internal nonlinear dynamics of BBHs via numerical relativity. This formalism is then used, in the 1.5 post-Newtonian approximation, to explore momentum flow between a binary’s bodies and its gravitational field during the binary’s orbital inspiral. Special attention is paid to momentum flow and conservation associated with synchronous spin-induced bobbing of the black holes, in the so-called “extreme-kick configuration” (where two identical black holes have their spins lying in their orbital plane and antialigned).

Figure 1

Extreme-kick configuration for a black-hole binary: Identical holes, $A$ and $B$, with masses $m=M/2$ move in a circular orbit with their spin angular momenta ${\mathit{S}}_A$ and ${\mathit{S}}_B$ antialigned and lying in the orbital plane.

Figure 2

Bobbing and kick of binary black holes in the extreme-kick configuration of Fig. 1, as simulated by Campanelli, Lousto, Zlochower, and Merritt [6]. Plotted vertically (as a function of time horizontally) is the identical height $z$ of the two black holes, and then transitioning through merger (presumably at $t/M\sim 170$), the height of the merged hole, above the initial orbital plane. This height versus time is shown for six different initial configurations, each leading to a different orbital phase at merger. In all six configurations, the initial holes’ spins are half the maximum allowed, $a/m=0.5$. The height $z$ and time $t$ are those of the “punctures” that represent the holes’ centers in the CLZM computations, as defined in their computational coordinate system, which becomes Lorentz at large radii. These $z$ and $t$ are measured in units of the system’s total (Arnowitt-Deser-Misner) mass $M\simeq 2m$.

Figure 3

Pretorius’s physical explanation for the holes’ bobbing in the extreme-kick configuration.

Figure 4

The four pictures show the $z$ component of field-momentum density $\delta {\tau }^{0z}$ in the orbital plane at four different times, a quarter orbit apart. Red (light gray) represents positive momentum density (coming out of the paper), and blue (dark gray), negative (going into the paper). Only the piece of momentum density $\delta {\tau }^{0z}$ that flows during bobbing [Eq. (2.10)] is depicted. The yellow (near white) arrows are the black holes’ vectorial spins; the large, black arrowed circle shows the orbital path of the two holes. In the top-left picture, one sees the density of momentum when the black holes are at the top of their bob (maximum $z$) and momentarily stationary [Eqs. (2.2, 2.3)]. The gravitational-field momentum is zero, but the momentum density itself shows rich structure. A quarter orbit later, in the top right, the holes are moving downward (into the paper) at top speed. The momentum between the black holes (blue [dark gray] region) flows into the paper with them, while surrounding momentum (red [light gray] region) flows out of the paper ($+z$ direction). A half orbit after the first picture, in the lower left, the holes are momentarily at rest at the bottom of their bob (minimum $z$), the net field momentum is zero, and the momentum distribution is opposite that in the first picture (as one would expect during sinusoidal bobbing). Similarly, three quarters of the way through the orbit, in the lower right, the holes have reached their maximum upward speed, and the momentum distribution is identical to the second figure, but with the opposite sign.

Figure 5

The regions of space around and inside a compact binary system.