1 Introduction

The circular geodesic motions of test particles in rotating Kerr black-hole spacetimes have attracted the attention of physicists and mathematicians during the last five decades (see [1,2,3,4,5,6] and references therein). The characteristic radii of these astrophysically important orbits are bounded from below by the radius of the equatorial null circular geodesic which, for a given value of the black-hole angular momentum, is characterized by the smallest possible circumference.

Intriguingly, as discussed by many authors [1,2,3,4,5,6,7,8,9], in the maximally-spinning (extremal) Kerr black-hole spacetime, there exist circular orbits of radius \(r_{\text {circular}}\) which are characterized by the limiting near-horizon behavior \(r_{\text {circular}}/r_{\text {H}}\rightarrow 1^+\) [10, 11].

The main goal of the present compact paper is to provide evidence, which is based on the famous Thorne hoop conjecture [12], for the existence of a non-trivial lower bound \(r_{\text {circular}}>r_{\text {min}}>r_{\text {H}}\) on the radii of circular orbits in the maximally-spinning Kerr black-hole spacetime. In particular, as we shall show below, according to the hoop conjecture the smallest possible circular radius \(r_{\text {min}}=r_{\text {min}}(\mu /M)\) is determined by the dimensionless mass ratio \(\mu /M\) [13] which characterizes the composed extremal-black-hole-orbiting-particle system.

2 Circular orbits in the maximally-spinning Kerr black-hole spacetime

We consider a particle of proper mass \(\mu \) which orbits around a maximally-spinning (extremal) Kerr black-hole of mass M (with \(\mu /M\ll 1\)) and angular momentum \(J=M^2\). The extremal curved black-hole spacetime is characterized by the line element [1],

$$\begin{aligned} ds^2= & {} -(1-2Mr/\Sigma )dt^2-(4M^2r\sin ^2\theta /\Sigma )dt d\phi \nonumber \\&+(\Sigma /\Delta )dr^2+\Sigma d\theta ^2 \nonumber \\&+(r^2+M^2+2M^3r\sin ^2\theta /\Sigma )\sin ^2\theta d\phi ^2, \end{aligned}$$
(1)

where \((t,r,\theta ,\phi )\) are the familiar Boyer–Lindquist spacetime coordinates and the metric functions in (1) are defined by the relations,

$$\begin{aligned} \Delta \equiv (r-M)^2;\quad \Sigma \equiv r^2+M^2\cos ^2\theta . \end{aligned}$$
(2)

The degenerate horizon of the extremal black-hole spacetime is defined by the radial functional relation \(\Delta (r=r_{\text {H}})=0\), which yields the simple relation

$$\begin{aligned} r_{\text {H}}=M. \end{aligned}$$
(3)

The dimensionless energy ratio \(E(r)/\mu \) (energy per unit mass as measured by asymptotic observers) associated with a circular orbit of radius r in the equatorial plane of the extremal (maximally-spinning) Kerr black-hole spacetime is given by the functional expression [1],

$$\begin{aligned} {{E(r)}\over {\mu }}={{{r\pm M^{1/2}r^{1/2}-M}}\over {r^{3/4}(r^{1/2}\pm 2M^{1/2})^{1/2}}}\ . \end{aligned}$$
(4)

The upper/lower signs in the dimensionless energy expression (4) correspond to co-rotating/counter-rotating circular orbits in the black-hole spacetime, respectively. From Eq. (4) one finds the near-horizon co-rotating energy ratio [14],

$$\begin{aligned} {{E(x)}\over {\mu }}={{1}\over {\sqrt{3}}}+{{2}\over {3\sqrt{3}}}\cdot x+O(x^2), \end{aligned}$$
(5)

where the dimensionless parameter x (with the near-horizon property \(x\ll 1\)) is defined by the relation,

$$\begin{aligned} r\equiv M(1+x). \end{aligned}$$
(6)

3 The Thorne hoop conjecture and a lower bound on the radii of circular orbits in the extremal black-hole spacetime

The total energy (as measured by asymptotic observers) of the composed extremal-Kerr-black-hole-orbiting-particle system is given by

$$\begin{aligned} E_{\text {total}}=M+E_{\text {ISCO}}+O(E^2_{\text {ISCO}}/M)\ . \end{aligned}$$
(7)

According to the Thorne hoop conjecture [12], the composed system will form an engulfing horizon if it can be placed inside a ring whose circumference C is equal to (or smaller than) \(4\pi E_{\text {total}}\). That is, Thorne’s famous hoop conjecture asserts that [12]

$$\begin{aligned} C(E_{\text {total}})\le 4\pi E_{\text {total}}\ \ \Longrightarrow \ \ \text {Black-hole horizon exists}. \end{aligned}$$
(8)

As we shall now show, the hoop relation (8) provides compelling evidence for the existence of a lower bound (with \(r_{\text {circular}}>r_{\text {min}}>r_{\text {H}}\)) on the radii of circular orbits in the maximally-spinning (extremal) Kerr black-hole spacetime. In particular, using Eqs. (1) and (2) one obtains the functional expression [15, 16]

$$\begin{aligned} C(r)=2\pi \sqrt{r^2+M^2+2M^3/r} \end{aligned}$$
(9)

for the circumference \(C=C(r)\) of an equatorial circular orbit in the extremal Kerr black-hole spacetime. Substituting (6) into (9), one finds the simple near-horizon relation,

$$\begin{aligned} C(x\ll 1)=4\pi M\cdot \left[ 1+{{3}\over {8}}\cdot x^2+O(x^3)\right] . \end{aligned}$$
(10)

Taking cognizance of Eqs. (5), (7), (8), and (10), one obtains the dimensionless lower bound

$$\begin{aligned} x_{\text {circular}}>x_{\text {min}}=\left( {{8}\over {3\sqrt{3}}}\cdot {{\mu }\over {M}}\right) ^{1/2} \end{aligned}$$
(11)

on the scaled radii of circular orbits in the composed extremal-Kerr-black-hole-orbiting-particle system. In particular, according to the Thorne hoop conjecture [12], composed black-hole-particle configurations whose circular orbits are characterized by the relation \(x_{\text {circular}}\le x_{\text {min}}\) are expected to be engulfed by a larger horizon with \(r_{\text {horizon}}\ge r_{\text {circular}}\) [see Eqs. (8) and (11)].

4 Summary

A remarkable feature of the maximally-spinning (extremal) Kerr black-hole spacetime, which has been discussed in the physics literature by many authors (see [1,2,3,4,5,6,7,8,9] and references therein), is the existence of co-rotating circular orbits which are characterized by the limiting radial behavior \(r_{\text {circular}}/r_{\text {H}}\rightarrow 1^+\).

In the present compact paper we have used the famous Thorne hoop conjecture [12] in order to provide evidence for the possible existence of a larger horizon (with \(r_{\text {horizon}}\ge r_{\text {circular}}\ge r_{\text {H}}\)) that engulfs composed extremal-Kerr-black-hole-orbiting-particle configurations which violate the dimensionless relation (11). In particular, our analysis has revealed the intriguing fact that circular orbits in the extremal Kerr black-hole spacetime are restricted to the radial region [see Eqs. (3), (6), and (11)]

$$\begin{aligned} r_{\text {circular}}>r_{\text {H}}\cdot \left[ 1+\left( {{8}\over {3\sqrt{3}}}\cdot {{\mu }\over {M}}\right) ^{1/2}+O\left( {{\mu }\over {M}}\right) \right] . \end{aligned}$$
(12)