Backreaction of frame dragging

The backreaction on black holes due to dragging heavy, rather than test, objects is discussed. As a case study, a five-dimensional regular black Saturn system where the central black hole has vanishing intrinsic angular momentum, $J^{\mathrm{BH}}=0$, is considered. It is shown that there is a correlation between the sign of two response functions. One is interpreted as a moment of inertia of the black ring in the black Saturn system. The other measures the variation of the black ring horizon angular velocity with the central black hole mass, for fixed ring mass and angular momentum. The two different phases defined by these response functions collapse, for small central black hole mass, to the thin and fat ring phases. In the fat phase, the zero area limit of the black Saturn ring has reduced spin $j^2>1$, which is related to the behavior of the ring angular velocity. Using the “gravitomagnetic clock effect,” for which a universality property is exhibited, it is shown that frame dragging measured by an asymptotic observer decreases, in both phases, when the central black hole mass increases, for fixed ring mass and angular momentum. A close parallelism between the results for the fat phase and those obtained recently for the double Kerr solution is drawn, considering also a regular black Saturn system with $J^{\mathrm{BH}}\ne 0$.

Figure 1

Horizon angular velocity as function of the angular momentum, for fixed mass $M=1$, in the double Kerr system. The dashed line corresponds to the infinite distance limit (i.e. the usual Kerr solution). The black (grey) lines correspond to the corotating (counter-rotating) double Kerr system for different values of the physical distance $d$. Note that in the corotating (counter-rotating) case, $\Omega ={\Omega }_1={\Omega }_2$ ($\Omega ={\Omega }_1=-{\Omega }_2$).

Figure 2

Illustration of a fat black ring (left) to which a small central black hole without intrinsic angular momentum is added (right), for fixed black ring mass and angular momentum. Effect (i) is that ${\Omega }^{\mathrm{BH}}\ne 0$; effect (ii) is that ${\Omega }_{\mathrm{without}}^{\mathrm{BR}}>{\Omega }_{\mathrm{with}}^{\mathrm{BR}}$.

Figure 3

Significance of the parameters ${\kappa }_i$ in terms of the rod structure (including rod directions) of the black Saturn solution. Apparently the solution is determined by five parameters: the three finite rod sizes and the two timelike rod directions. However, the regularity constraint and the $J^{\mathrm{BH}}=0$ condition reduce the number of free parameters to three.

Figure 4

Sign of the function $f({\kappa }_1,{\kappa }_2)$ (18). The function is negative in the dark grey region (“fat phase”) and positive in the light grey region (“thin phase”). The black Saturn solution does not exist outside the colored region. The lines correspond to trajectories of constant ring mass and angular momentum, i.e. only the central black hole mass varies. They have been labeled by the value of the reduced spin $j^2\in [27/32,+\infty [$. These trajectories have the two endpoints on ${\kappa }_1=1$ for $j^2\le 1$, as it should be from the well-known phase diagram of black rings (see e.g. 9). As $j^2\to 27/32$ the trajectories collapse to the point $({\kappa }_1,{\kappa }_2)=(1,8/9)$.

Figure 5

Dimensionless angular velocity of the black ring horizon ${\omega }^{\mathrm{BR}}$, in terms of the mass fraction $m$, along trajectories of constant black ring mass and angular momentum. The behavior in the fat (thin) phase corresponds to the dark (light) grey curves. In the fat (thin) phase, the angular velocity of the ring decreases (increases) as the mass of the black hole increases. Observe that rings in the fat phase may have $j^2>1$, but their angular velocity is always smaller than that of an isolated fat black ring, ${\omega }^{\mathrm{BR}}=1$.

Figure 6

Sign of the function $\tilde{f}({\kappa }_1,{\kappa }_2)$ (27). The function is negative in the dark grey region (“thin phase”) and positive in the light grey region (“fat phase”). The straight lines correspond to trajectories of constant ring mass and black hole mass, i.e. only the ring angular momentum varies. They have been labeled by the value of the mass fraction $m=M^{\mathrm{BH}}/M^{\mathrm{BR}}$.

Figure 7

Dimensionless angular velocity of the black ring horizon ${\omega }^{\mathrm{BR}}$, in terms of the reduced spin $j$, along trajectories of constant black hole and black ring mass. The behavior in the fat (thin) phase corresponds to the light (dark) grey curves. This figure is the analogous to Fig. 1, presented in the introduction for the double Kerr system. Observe that, in the fat case, the slope of the curve decreases as $m$ increases, manifesting the increase of the moment of inertia defined in (3).

Figure 8

Dimensionless angular velocity of the black ring and black hole horizons, in the fat, ${\omega }_f^{\mathrm{BR}}$ and ${\omega }_f^{\mathrm{BH}}$, and thin, ${\omega }_t^{\mathrm{BR}}$ and ${\omega }_t^{\mathrm{BH}}$, phases, in terms of the reduced angular momentum of the black hole. The angular momentum of the black hole is turned on with either the same sign (corotating case, $j^{\mathrm{BH}}>0$) or opposite sign (counter-rotating case, $j^{\mathrm{BH}}<0$) as the black ring angular momentum. The dashed lines correspond to the reduced angular velocity of an isolated fat (top line) and thin (bottom line) black ring with $j^2\simeq 1$.