Spinning black holes as cosmic string factories

In this work we focus on the case where the invariant length of a string loop is much greater than the black-hole gravitational radius, $L\gg R=GM/c^{2}$, but at the same time its mass is much smaller than that of the black hole, $G\mu L/c^{4}\ll R$; here $\mu$ is the string tension force, $L=mc^{2}/\mu$ is the invariant length of the string, and $m$ is the mass of the string. From here on we will be using geometric units with ${\bf G=c=1}$; in these units $\mu$ is dimensionless.

The best observational bound on the string tension $\mu$ is based on the lack of detection of gravitational waves from freely oscillating string loops by Pulsar Timing Arrays. A somewhat model-dependent constraint $\mu<1.5\times 10^{-11}$ has been obtained in Blanco-Pillado et al. (2018). This small value of the dimensionless tension allows a large range of possible string length that satisfies our constraints, $1\ll L/R\ll(\mu)^{-1}$.

In this section we focus on the kinematics of string motion and we will neglect its influence on the black hole position. In this approximation of an infinitely light string, we model the influence of the black hole by rigidly fixing a point on the loop in space. The rest of the loop is assumed to move in Minkowski space; we neglect the general-relativistic character of the string motion through the curved spacetime at distances $\sim R$ from the black hole, since $R\ll L$.

The motion of the free part of the string (i.e., all of it except for the pinned point) is given by

$${\bf r}(\sigma,t)={1\over 2}\left[{\bf a}(\sigma-t)+{\bf b}(\sigma+t)\right].$$

(1)

Here $\sigma$ is the invariant length coordinate marking points along the string, ${\bf r}=(x,y,z)$ is the position of a string point $\sigma$ at time $t$, and ${\bf a}$ and ${\bf b}$ are vector functions of a single variable, such that $\left|{\bf a}^{\prime}\right|=\left|{\bf b}^{\prime}\right|=1$; see Vilenkin and Shellard (2000) for derivation. We shall take $\sigma=0$ and $\sigma=L$ at the black hole location ${\bf r}=0$. The boundary conditions ${\bf r}(0,t)={\bf r}(L,t)=0$ lead to the following constraints:

	$\displaystyle{\bf a}(\eta)=-{\bf b}(-\eta),$
	$\displaystyle{\bf a}(\eta)={\bf a}(\eta+2L).$		(2)

Here $\eta$ is a point on a real axis, $-\infty<\eta<\infty$. The general solution for the pinned loop is therefore given by

$${\bf r}(\sigma,t)={1\over 2}[{\bf a}(\sigma-t)-{\bf a}(-\sigma-t)],$$

(3)

with the requirement that ${\bf a}$ is periodic with the period of $2L$. Formally ${\bf r}(\sigma,t)$ is defined for all real values of $\sigma$ and $t$, but the physical loop corresponds to the values of $0\leq\sigma\leq L$. Extending $\sigma$ to the interval between $L$ and $2L$ would produce a “ghost” loop obtained from the original loop by reflection with respect to the origin, since the solution observes the symmetry

$${\bf r}(\sigma,t)=-{\bf r}(2L-\sigma,t).$$

(4)

In other words, a loop pinned at a point can be considered as half of a free loop that self-intersects at the origin and has a reflection symmetry with respect to the origin.

Self-intersections are important because they can lead to reconnections and ejections of daughter loops, which deplete the loop bound to the black hole. The reconnection probability $p$ for a cosmic string that is a solution of a classical gauge field theory is close to unity Vilenkin and Shellard (2000), unless the segments collide at ultra-relativistic velocities Achúcarro and de Putter (2006). On the other hand, the reconnection probability for cosmic super-strings may be smaller than unity by orders of magnitude, plausibly as small as $10^{-3}$ in some superstring models Jackson et al. (2005). In this work we keep an open mind about the value of $p$, but it is clear that reconnections are very important for the loop evolution.

The fact that a general quadrilateral auxiliary curve ${\bf a}(\sigma)$ corresponds to a non-self-intersecting loop makes it plausible that such loops are pretty common, and that a general-shape loop will settle into a non-self-intersecting configuration after several reconnections. We did not manage to find a rigorous mathematical proof to this statement. Instead we carried out numerical experiments with assumed reconnection probability $p=1$, that showed that this is indeed what happens to loops with initially arbitrary shapes.

The reader uninterested in details is urged to accept this statement on faith and skip the rest of this subsection. Methodologically, we are interested in how the number of reconnections and the length of the final loop depends on the complexity of the initial loop. Our task is then to first, introduce some way of initializing loops of variable complexity, and second, to develop a reliable and efficient algorithm that searches for self-intersections of a moving loop.

We construct the initial loop as follows. We discretize the ${\bf a}$ vector function into $2N$ connected line segments. Each segment is defined by its length $L_{i}$ and direction unit vectors ${\bf a}^{\prime}_{i}$ for $i=1,\dots,2N$. For a point on ${\bf a}$ that falls into the $j$th segment, we have:

$${\bf a}(\eta)={\bf a}(0)+\sum_{i<j}L_{i}{\bf a}^{\prime}_{i}+\left(\eta-\sum_{i<j}L_{i}\right){\bf a}^{\prime}_{i}.$$

(6)

Since ${\bf a}$ should be periodic in $2L$, we require that $\sum_{i=1}^{2N}L_{i}=2L$ and $\sum_{i=1}^{2N}L_{i}{\bf a}^{\prime}_{i}={\bf 0}$. We also note that ${\bf r}$ does not change if we add a constant vector to ${\bf a}$. Therefore, we assign ${\bf a}(0)={\bf 0}$ in the simulation.

To make the curve random, the direction vectors ${\bf a}^{\prime}_{i}$ perform a random walk on a unit sphere; the larger is the step of the random walk, the greater is the complexity of the loop. For the self-intersection search, it is convenient for all segment lengths to be equal $L_{i}=L/N$. Our algorithm generates the set of random ${\bf a}^{\prime}_{i}$ which satisfies:

1. 1.

   $|{\bf a}^{\prime}_{i}|=1$ (on unit sphere)

2. 2.

   $\sum{\bf a}^{\prime}_{i}=0$ (periodicity)

3. 3.

   $|{\bf a}^{\prime}_{i+1}-{\bf a}^{\prime}_{i}|\leq B^{\prime}=2\sin(\theta_{m}/2)$ (smoothness)

Here $\theta_{m}$ is the maximum angle between the neighbouring segments; it is assumed that $1/N\ll\theta_{m}<1$. Our iterative procedure is described in Appendix A.

We also design an algorithm for detecting self-intersections and for implementing reconnections that follow. An intersection occurs when, for some $0\leq\sigma_{i}\neq\sigma_{j}<L$, ${\bf r}(\sigma_{i},t)={\bf r}(\sigma_{j},t)$. We can use Eq. (3) and Eq. (6) to formulate a linear equation and solve for $\sigma_{i}$, $\sigma_{j}$ and $t$. After intersection, the relevant portion of ${\bf a}$ will break off from the original function. In practice, since ${\bf a}$ is periodic in $2L$, we search for self-intersection in the time range $0\leq t<2L$. The simulation method is described in Appendix B. The algorithm was tested (a) by running it on loops specified by ${\bf a}(\sigma)$ that were general quadrilaterals, to check that it does not find spurious intersections, and (b) by performing resolution tests in segment lengths and timesteps.

We simulate strings with $L=1$, $N=500$ and a range of different smoothness constraints. Fifty random initializations are used for each $B^{\prime}$. We find that there exists loops that are stable (i.e., non-self-intersecting). A randomly initialized loop will always, after a series of self-intersection and reconnection, reduce to a stable configuration.

Figure 3(a) shows a histogram of the number of reconnections a string undergoes before it stabilizes. A larger $B^{\prime}$ causes more complexity in the initial loop and leads to more self-intersections. As illustrated in Figure 3(c), the average number of intersections scales approximately linearly with $B^{\prime}$. Figure 3(b) shows the distribution of $L_{f}$, the final invariant length of the string after all self-intersections. The result approximately follows an exponential distribution

$$P(L_{f})\sim{\kappa\over L}e^{-\kappa L_{f}/L}$$

(7)

where $\kappa$ is the decay constant. A similar statistical result for free cosmic strings has been shown (York, 1989). We further demonstrate in Figure 3(d) that there is an approximate linear relationship $\kappa\sim 0.01B^{\prime}$; note that this relation is only valid for $N=500$ and $\kappa>1$.

The angular momentum of the black hole ${\bf J}$ evolves due to the torque applied by the string. For the case with a single string entering the horizon, we can use Eq. (31) to write down the evolution equation in vector form

$${d{\bf J}\over dt}=-{\mu\over R}\left[{\bf J}-\left({\bf n}\cdot{\bf J}\right){\bf n}\right],$$

(32)

where ${\bf n}$ is the unit vector along the string at $R\ll r\ll L$. This implies

	$\displaystyle{\bf J}_{\parallel}$	$\displaystyle=$	$\displaystyle{{\bf J}_{\parallel}}_{0},$		(33)
	$\displaystyle{\bf J}_{\perp}$	$\displaystyle=$	$\displaystyle{{\bf J}_{\perp}}_{0}\exp\left[-{\mu\over R}t\right],$		(34)

where ${\bf J}_{\parallel}$ and ${\bf J}_{\perp}$ are components of the angular momentum vector parallel and perpendicular to the string, respectively. Therefore the angular momentum vector of the black hole aligns with the string on a timescale $t_{\mu}=R/\mu$, while keeping ${\bf J}_{\parallel}$ fixed. If the black hole is threaded through by a straight string, the alignment would occur on a timescale $t_{\mu}/2$. We note that while the spin-down of a black hole from a stationary string was noted in, e.g., Boos and Frolov (2018), the alignment of the spin direction with the string is pointed out here for the first time.

As the string is stationary, no energy is extracted from the black hole by the string Hawking and Hartle (1972). The black hole converts its rotational energy into its irreducible mass, by increasing its entropy through dissipation at the horizon. The full mass of the black hole stays fixed.

In the previous subsection we assumed that the tangent unit vector representing the long string at a black hole ${\bf n}$ was stationary. However, since we are exploring a moving string loop, we also consider the case where ${\bf n}$ is rotating slowly about the black hole, with the angular velocity ${\bf o}={\bf n}\times\dot{\bf n}$. If the black hole is not spinning, this creates torques acting on the black hole, which in the limit $\left|{\bf o}\right|\ll 1/R$ must scale linearly as

$${\bf Q}=\beta{\bf o}.$$

(35)

In order to find the coefficient $\beta$, let’s perform a mental experiment in which we add a small spin to the black hole around an axis that is perpendicular to ${\bf n}$. Using Eq. (32), we can express the total torque acting on the black hole to linear order in ${\bf o}$ and ${\bf J}$:

$${\bf Q}=\beta{\bf o}-{\mu\over R}{\bf J}.$$

(36)

For $\left|{\bf J}\right|\ll R^{2}$, the angular velocity of the black hole is given by

$$\mathbf{\Omega}_{\rm BH}={{\bf J}\over 4R^{3}}.$$

(37)

Therefore,

$${\bf Q}=\beta{\bf o}-4\mu R^{2}\mathbf{\Omega_{\rm BH}}.$$

(38)

The torque acting on ${\bf n}$ has to be zero when it is co-rotating with the black hole. Therefore,

$$\beta=4\mu R^{2}.$$

(39)

The expression for the torque ${\bf Q}$ allows us to estimate the timescale on which a loop bound to the non-spinning black hole will dissipate and be swallowed by the black hole. The torques $-{\bf Q}_{1,2}$ applied to the string at its ends $1$ and $2$, combined with the motion of those ends, will result in a loss of energy and length by the string:

$${dE\over dt}=-\beta({\bf o}_{1}^{2}+{\bf o}_{1}^{2})=-\beta\left[\left({d{\bf n}_{1}\over dt}\right)^{2}+\left({d{\bf n}_{2}\over dt}\right)^{2}\right].$$

(40)

To order-of-magnitude,

	$\displaystyle{dE\over dt}$	$\displaystyle\sim$	$\displaystyle-\beta L^{-2}\sim-\mu(R/L)^{2},$
	$\displaystyle{dL\over dt}$	$\displaystyle\sim$	$\displaystyle-(R/L)^{2},$		(41)
	$\displaystyle t_{\rm fr}$	$\displaystyle=$	$\displaystyle{L\over|dL/dt|}\sim L^{3}/R^{2},$

where $t_{\rm fr}$ is the timescale for the loop to be absorbed by the non-spinning black hole through horizon friction.

In this section we show that a circularly polarized tension wave is amplified upon reflection from a spinning black hole, provided its sense of rotation is the same as that of the black hole, and its angular frequency $\omega<\Omega_{\rm BH}\cos\theta$, where $\theta$ is the angle between the string and the spin axis of the black hole. This leads to a very fast growth of short wavelength perturbations on a bound string loop. Our treatment gives an exact answer for $\Omega_{\rm BH},\omega\ll 1/R$, but order-of magnitude extrapolation to greater angular frequencies is warranted. A more general treatment is possible but is beyond the scope of our paper.

Consider a small-amplitude incoming elliptically-polarized wave coming towards the black hole along a string that is straight for $\sigma\gg R$. Mathematically it is given by

$${\bf\delta r}_{\rm in}=A_{1}{\bf e}_{1}\cos[\omega(\sigma+t)]+A_{2}{\bf e}_{2}\sin[\omega(\sigma+t)],$$

(42)

where ${\bf\delta r}$ is the string displacement from the resting position and ${\bf e}_{1,2}$ are two unit vectors chosen so that (${\bf e}_{1},{\bf e}_{2},{\bf n_{0}}$) form a right-handed orthonormal basis, and that the polarization tensor is diagonalized. Here ${\bf n}_{0}$ stands for the asymptotic unit vector of the unperturbed string. To zero’th order, the reflected outgoing wave is given by

$${\bf\delta r}_{\rm out}=-A_{1}{\bf e}_{1}\cos[\omega(\sigma-t)]+A_{2}{\bf e}_{2}\sin[\omega(\sigma-t)].$$

(43)

However, as we presently see, the black hole exchanges energy and angular momentum with the wave, and therefore the amplitudes of the outgoing wave are expected to be slightly different from those of the ingoing wave.

The incoming fluxes of energy and of the ${\bf n_{0}}$-component of angular momentum are given by

	$\displaystyle{dE_{\rm in}\over dt}$	$\displaystyle=$	$\displaystyle{1\over 2}\mu\omega^{2}\left(A_{1}^{2}+A_{2}^{2}\right)$		(44)
	$\displaystyle{\bf n}_{0}\cdot{d{\bf J}_{\rm in}\over dt}$	$\displaystyle=$	$\displaystyle\mu\omega A_{1}A_{2},$		(45)

and similarly for the outgoing wave. We can figure out the incremental changes in the reflected amplitudes by computing the time-averaged power and torque applied by the black hole to the wavy string.

The vector ${\bf n}(t)$ is oscillating periodically, as follows:

	$\displaystyle{\bf n}$	$\displaystyle=$	$\displaystyle{\bf n}_{0}-2\omega\left[A_{1}{\bf e}_{1}\sin(\omega t)-A_{2}{\bf e_{2}}\cos(\omega t)\right]$		(46)
			$\displaystyle-2\omega^{2}\left[A_{1}^{2}\sin^{2}(\omega t)+A_{2}^{2}\cos^{2}(\omega t)\right]{\bf n}_{0}.$

The last term ensures that $|{\bf n}|=1$ up to the second order in $A_{1},A_{2}$. The torque applied to the string by the black hole is given by

$${\bf Q_{s}}=4\mu R^{2}\left[\mathbf{\Omega}_{\rm BH}-\left({\bf n}\cdot\mathbf{\Omega}_{\rm BH}\right){\bf n}-{\bf n}\times{d{\bf n}\over dt}\right].$$

(47)

The last term on the right-hand side represents horizon friction. When integrating it over the wave period, we get twice the directed area of the contour drawn by the ${\bf n}$-vector on the unit sphere:

$$\int_{0}^{2\pi/\omega}{\bf n}\times{d{\bf n}\over dt}dt=8\pi\omega^{2}A_{1}A_{2}{\bf n}_{0}.$$

(48)

Integrating over one cycle, we get

	$\displaystyle\int_{0}^{2\pi/\omega}{\bf n}_{0}\cdot{\bf Q}_{s}dt$	$\displaystyle=$	$\displaystyle 16\pi\mu R^{2}\omega$
			$\displaystyle\times\left[\Omega_{\rm BH}\cos\theta\mathcal{A}_{+}-\omega\mathcal{A}_{\times}\right],$

where

	$\displaystyle\mathcal{A}_{+}$	$\displaystyle=$	$\displaystyle A_{1}^{2}+A_{2}^{2},$		(50)
	$\displaystyle\mathcal{A}_{\times}$	$\displaystyle=$	$\displaystyle 2A_{1}A_{2}.$

The time-averaged torque about the string line is given by

	$\displaystyle{\bf n}_{0}\cdot\langle{\bf Q}_{s}\rangle$	$\displaystyle=$	$\displaystyle 8\mu R^{2}\omega^{2}$
			$\displaystyle\times\left[\Omega_{\rm BH}\cos\theta\mathcal{A}_{+}-\omega\mathcal{A}_{\times}\right].$

Comparing Eqs. (IV.3) and (45), we see that upon reflection

$$\Delta\mathcal{A}_{\times}=16R^{2}\omega\left[\Omega_{\rm BH}\cos\theta\mathcal{A}_{+}-\omega\mathcal{A}_{\times}\right].$$

(52)

The second relation is obtained from the conservation of energy. The work done by the black hole on the string over one cycle equals

$$\Delta E_{s}=\int_{0}^{2\pi/\omega}{\bf Q}_{s}\cdot\left({\bf n}\times{d{\bf n}\over dt}\right)dt.$$

(53)

Evaluating this using Eq. (47), we obtain

$$\Delta E_{s}=16\pi\mu R^{2}\omega^{2}\left[\Omega_{\rm BH}\cos\theta\mathcal{A}_{\times}-\omega\mathcal{A}_{+}\right].$$

(54)

Comparing this with Eq. (44), we get

$$\Delta\mathcal{A}_{+}=16R^{2}\omega\left[\Omega_{\rm BH}\cos\theta\mathcal{A}_{\times}-\omega\mathcal{A}_{+}\right].$$

(55)

It is of interest to consider the eigenmodes of the system, i.e. the waves that do not change their polarization state upon reflection from the black hole. This implies $\Delta\mathcal{A}_{+}/\mathcal{A}_{+}=\Delta\mathcal{A}_{\times}/\mathcal{A}_{\times}$. From Eqs. (52) and (55) we see that this is equivalent to

$$\mathcal{A}_{+}=\pm\mathcal{A}_{\times},$$

(56)

and

$$A_{1}=\pm A_{2},$$

(57)

which corresponds to circularly polarized waves. For the string in the northern hemisphere, with $\cos\theta>0$, the $A_{1}=-A_{2}$ wave is partially absorbed by the black hole, with

$${\Delta\mathcal{A}_{+}\over\mathcal{A}_{+}}=-16R^{2}\omega\left[\Omega_{\rm BH}\cos\theta+\omega\right].$$

(58)

On the other hand, the wave with $A_{1}=A_{2}$ is amplified if $\omega<\Omega_{\rm BH}\cos\theta$, and is partially absorbed if $\omega>\Omega_{\rm BH}\cos\theta$:

$${\Delta\mathcal{A}_{+}\over\mathcal{A}_{+}}=16R^{2}\omega\left[\Omega_{\rm BH}\cos\theta-\omega\right].$$

(59)

Intuitively this makes sense. If the wave is rotating in the direction opposite that of the BH, horizon friction dominates. The greater the black hole spin, the stronger is the absorption. If, on the other hand, the wave is rotating in the same direction as the black hole, it is amplified if the black hole spins faster than the wave (measured after projection of $\mathbf{\Omega}_{\rm BH}$ onto the string) and is partially absorbed if the wave spins faster than the black hole. Qualitatively similar results can be derived for an ordinary string viscously interacting with a rotating sphere, in the spirit of the original rotational superradiance proposal by Zeldovich (Zel’Dovich (1971)). The maximum growth rate is achieved for

$$\omega={1\over 2}\Omega_{\rm BH}\cos\theta,$$

(60)

with the relative amplitude increase of

	$\displaystyle\left[{\Delta A_{1,2}/A_{1,2}}\right]_{\rm max}$	$\displaystyle\simeq$	$\displaystyle{1\over 2}\left({\Delta\mathcal{A}_{+}/\mathcal{A}_{+}}\right)_{\rm max}$
		$\displaystyle=$	$\displaystyle 2R^{2}\Omega_{\rm BH}^{2}\cos^{2}\theta.$

We emphasize again that this result is exact in the limit $\Omega_{\rm BH}R\ll 1$, but we expect that both the criterion for superradiance and the expression for the maximum growth rate are correct to order of magnitude for all values of $\Omega_{\rm BH}$.

A wave reflected from the black hole along a straight string will never come into contact with the black hole again. This is not so if the string is a bound loop with both ends attached to the black hole. Consider a pulse of circularly polarized waves with angular frequency $\omega<\Omega_{\rm BH}\cos\theta$ that is reflected from the black hole when the corresponding string end is making a polar angle $\theta$ with the black hole spin axis. If the waves in the pulse are rotating in the same sense as the black hole, then the pulse is amplified upon reflection from the black hole. As the pulse travels to the other end of the string, it approaches the black hole along the same direction along which it was traveling upon the first reflection. This is because ${\bf n}_{2}(t+L)=-{\bf n}_{1}(t)$. Since the helicity of the wave is conserved as it travels along the loop, the pulse is rotating in the same sense as the black hole upon the second approach, and therefore it is amplified upon the second reflection also. The amplification repeats with each bounce, leading to an exponential growth of the pulse with the timescale

$$t_{\rm sr}=\left[8R^{2}\omega(\Omega_{\rm BH}\cos\theta-\omega)\right]^{-1}L.$$

(62)

The fastest growth will take place for the wave’s angular frequency

$$\omega_{0}={1\over 2}\Omega_{\rm BH}\cos\theta_{\rm min}$$

(63)

where $\theta_{\rm min}$ is the smallest angle between the spin axis and the string end. The pulse of wavelength $\lambda_{0}=2\pi/\omega_{0}=4\pi/(\Omega_{\rm BH}\cos\theta_{\rm min})$ and timed to reflect from the black hole when $\theta=\theta_{\rm min}$, will grow on a timescale

$$t_{\rm bomb}={L\over 2(R\Omega_{\rm BH}\cos\theta_{\rm min})^{2}}.$$

(64)

The subscript in the equation above refers to the concept of a black-hole bomb, originally conceived by Press & Teukolsky (Press and Teukolsky (1972)). They pointed out that if a spinning black hole was placed in a cavity with reflecting walls, then there would be exponentially amplified superradiant modes. In our case, the string loop itself plays a role of the cavity, causing repeated superradiant interaction between the black hole and pulses of tension waves.

The growth timescale in Eq. (62) is likely to be shorter than all other evolutionary timescale for the loop, and therefore one can expect the short-wavelength waves to quickly reach a non-linear amplitude on the string. What happens afterwards is not clear from the arguments given above, and a new approach is needed. In the next section we make a step in this direction, by exploring the evolution of the auxiliary curve. We finish this section by a discussion of angular momentum exchange between the black hole and the string loop.

Since the loop has two ends attached to the black hole, the angular momentum equation (32) is modified to

$${d{\bf J}\over dt}=-{\mu\over R}\mathcal{N}{\bf J},$$

(65)

where $\mathcal{N}$ is a linear evolution operator defined by

$$\mathcal{N}{\bf J}=2{\bf J}-\left(\bf{n}_{1}\cdot{\bf J}\right){\bf n}_{1}-\left(\bf{n}_{2}\cdot{\bf J}\right){\bf n}_{2}.$$

(66)

Recall that ${\bf n}_{1}(t)$ and ${\bf n}_{2}(t)$ are the unit vectors at the 2 ends of the string that are pointing away from the black hole, which are given by

	$\displaystyle{\bf n}_{1}(t)$	$\displaystyle=$	$\displaystyle{\bf a}^{\prime}(-t)$
	$\displaystyle{\bf n}_{2}(t)$	$\displaystyle=$	$\displaystyle-{\bf a}^{\prime}(L-t).$		(67)

As was already noted above, ${\bf n}_{1}$ and $-{\bf n}_{2}$ trace out the same trajectory during the loop’s oscillation period, but with a half-period delay. Thus the oscillation-averaged angular momentum evolution operator is given by

$$\langle\mathcal{N}\rangle{\bf J}=2\left\{{\bf J}-{1\over 2L}\int\limits_{0}^{2L}\left[{\bf a}^{\prime}(t)\cdot{\bf J}\right]{\bf a}^{\prime}(t)dt\right\}.$$

(68)

If vectors ${\bf n}_{1}$ and ${\bf n}_{2}$ are not pointing along the same line, the operator $\mathcal{N}$ is positive-definite with no eigenvalues that are equal $0$. Therefore all three components of angular momentum ${\bf J}$ will reduce exponentially, on a timescale

$$t_{\rm spindown}\sim R/\mu.$$

(69)

Note that this timescale is similar to that of $t_{\rm shrink}$ from Eq. (28), which is the timescale for loop depletion due to the black hole’s finite mass.

The lost black hole angular momentum will be acquired by the string loop; its angular momentum ${\mathbf{\Lambda}}$ will evolve according to the equation

$${d{{\mathbf{\Lambda}}}\over dt}={\mu\over R}\langle\mathcal{N}\rangle{\bf J},$$

(70)

where we averaged over the loop’s oscillation period. Let $z$-axis point in the direction of the black hole spin. It is clear that

$${d\Lambda_{z}\over dt}\sim{\mu R\alpha}>0,$$

(71)

where $\alpha=a/R$ is the dimensionless spin of the black hole. Since $\Lambda\sim\mu L^{2}$, the loop’s angular momentum will change on a timescale

$$t_{\rm am}\sim{L^{2}\over\alpha R}.$$

(72)

Over time $t>t_{\rm am}$, the component $\Lambda_{z}$ becomes positive.

We note that the computation above did not consider the angular momentum drained from the black hole by a superradiant wave. It is easy to check, however, that

$${(dJ/dt)_{\rm wave}\over(dJ/dt)_{\rm loop}}\sim(A\omega)^{2},$$

(73)

and thus the two contributions become comparable only when $A\omega\sim 1$, i.e. in a strongly non-linear regime. Similarly, the change of rotational energy of the black hole due to a superradiant wave is related to the total loss of rotational energy by

$${(dE_{\rm rot}/dt)_{\rm wave}\over(dE_{\rm rot}/dt)_{\rm total}}\sim(A\omega)^{2}$$

(74)

for $A\omega\lesssim 1$. Therefore, the fraction of the black hole’s rotational energy that gets converted into string is determined by the nonlinear saturation of the string superradiance. The rest of the rotational energy is converted into the irreducible mass of the black hole¹¹1It is easy to see that the most efficient, albeit the slowest spin energy conversion into string length takes place if the string nearly co-rotates with the black hole. In that case the black hole is spun down nearly adiabatically, with vanishing increase in its area/entropy..

To make further progress, we need to consider the non-linear evolution of the string loop due to its interaction with a spinning black hole.

Equation (77) describes the non-linear evolution of the auxiliary curve, with the non-linearity implicit due to $\sigma$ being understood as the length coordinate. It represents a one-dimensional geometric flow, and as such is of considerable interest to mathematicians. In fact, when $\Omega_{\rm BH}=0$, the equation of motion

$${\bf v}\propto{\bf a}^{\prime\prime}$$

(79)

describes a famous and extensively studied curve-shortening flow (see, e.g., Gage (1984),Gage and Hamilton (1986),Grayson (1989)). For practical purposes, the main result of this exploration is something of which one can immediately convince oneself through qualitative arguments and simple numerical experiments (we have done both): asymptotically, the curve becomes a shrinking planar circle and disappears in a singularity after a finite time. A circular auxiliary curve of length $2L$ corresponds to a degenerate physical loop that extends radially from the black hole to radius $L/\pi$ and then traces the same radial line back to the black hole. This double-line is rotating around the black hole with the angular velocity $\pi/L$; the tip of the double-line is moving with the speed of light and thus the double-line is a prolific emitter of gravitational waves. The length of the line is shrinking due to horizon friction and the line disappears after the time

$$t_{\rm short}\simeq{L^{3}\over 24\pi^{2}R^{2}},$$

(80)

cf. Eq. (41). In the process the double-line makes

$$N_{\rm short}\simeq{1\over 2}\left({L\over 4\pi R}\right)^{2}$$

(81)

turns around the black hole. The approximate equality in the equation above is due to the fact that our curve-shortening formalism is only valid for $L\gg R$.

Consider now a non-self-intersecting auxiliary curve in the plane perpendicular to $\mathbf{\Omega}_{\rm BH}$, with the circulation in the same sense as rotation of the black hole. Clearly the effect of the first term on the right in Eq. (77) is to expand the curve outwards and make it increasingly more round. If only the first term is included in the evolution equation, some of the initial configurations develop kinks and singularities, however the inclusion of the second term smooths them out.

If one can neglect the second term (a good approximation when ${\Omega}_{\rm BH}\gg 1/L$), then one can prove the following. Suppose the auxiliary curve is convex and everywhere differentiable. Choose the point of origin inside the curve and find maximum and minimum radii $a_{\rm max}$ and $a_{\rm min}$. As the curve lengthens, the diffarence $a_{\rm max}-a_{\rm min}$ remains constant. Therefore the ellipticity

$$\epsilon\equiv{a_{\rm max}-a_{\rm min}\over a_{\rm max}+a_{\rm min}}\sim 1/L,$$

(82)

and therefore the loop becomes increasingly circular. The latter expands according to

$$L=\sqrt{L_{0}^{2}+16\pi R^{2}\Omega_{\rm BH}t},$$

(83)

where $L_{0}$ is the initial invariant length of the loop with circular auxiliary curve.

A circular auxiliary curve corresponds to a double-line which in this case is rotating around the black hole in its equatorial plane and is growing in length, while reducing its angular velocity of rotation. If the double-line is rotating at the same angular velocity as the black hole (and thus has $L=\pi/\Omega_{\rm BH}$), there is no energy and angular momentum exchange with the black hole and $L$ does not change. This equilibrium, however, is unstable: double lines longer/shorter than $\pi/\Omega_{\rm BH}$ will lengthen/shorten in their extent.

It is straightforward to compute the growth rate for superradiant modes by considering a helical short-wavelength perturbation on a part of the auxiliary curve that can be considered locally straight. For simplicity, let’s assume that the straight part is along the spin axis, and choose $z$-axis to be also aligned with the spin. The helical wave can be written as

	$\displaystyle{\bf a}_{x}(\sigma)$	$\displaystyle=$	$\displaystyle A\cos\left({\omega\over\sqrt{1+A^{2}\omega^{2}}}\sigma\right),$
	$\displaystyle{\bf a}_{y}(\sigma)$	$\displaystyle=$	$\displaystyle A\sin\left({\omega\over\sqrt{1+A^{2}\omega^{2}}}\sigma\right),$		(84)
	$\displaystyle{\bf a}_{z}(\sigma)$	$\displaystyle=$	$\displaystyle{1\over\sqrt{1+A^{2}\omega^{2}}}\sigma,$

where it is understood that the above expression is valid only in some small part of the auxiliary loop (please note that ${|\bf a}^{\prime}|=1$). The amplitude of the helical perturbation evolves according to the following equation:

$${dA\over dt}={8R^{2}\omega A\over L}\left[{\Omega_{\rm BH}\over\sqrt{1+A^{2}\omega^{2}}}-{\omega\over 1+A^{2}\omega^{2}}\right].$$

(85)

In the linear case $A\omega\ll 1$, the amplitude grows exponentially with the timescale

$$t_{\rm sr}=\left[8R^{2}\omega(\Omega_{\rm BH}-\omega)\right]^{-1}L,$$

(86)

which is identical to Eq. (62) for the case $\theta=0$. In the non-linear case $A\omega\gg 1$, the auxiliary curve is tightly winding up the $z$-axis with its tangent nearly horizontal. For $\Omega_{\rm BH}\gg 1/A$, the amplitude grows at a constant rate so long as the overall-length of the curve has not changed much,

$${dA\over dt}={8R^{2}\Omega_{\rm BH}\over L},$$

(87)

with remarkable independence from $\omega$.