Lower bound on the radii of circular orbits in the extremal Kerr black-hole spacetime

It is often stated in the physics literature that maximally-spinning Kerr black-hole spacetimes are characterized by near-horizon co-rotating circular geodesics of radius rcircular\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_{\text {circular}}$$\end{document} with the property rcircular→rH+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_{\text {circular}}\rightarrow r^+_{\text {H}}$$\end{document}, where rH\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_{\text {H}}$$\end{document} is the horizon radius of the extremal black hole. Based on the famous Thorne hoop conjecture, in the present compact paper we provide evidence for the existence of a non-trivial lower bound rcircular-rHrH≳(μ/M)1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{r_{\text {circular}}-r_{\text {H}}}\over {r_{\text {H}}}}\gtrsim (\mu /M)^{1/2}$$\end{document} on the radii of circular orbits in the extremal Kerr black-hole spacetime, where μ/M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu /M$$\end{document} is the dimensionless mass ratio which characterizes the composed black-hole-orbiting-particle system.


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.
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 circular > r min > r 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 min = r min (μ/M) is a e-mail: shaharhod@gmail.com determined by the dimensionless mass ratio μ/M [13] which characterizes the composed extremal-black-hole-orbitingparticle system.

Circular orbits in the maximally-spinning Kerr black-hole spacetime
We consider a particle of proper mass μ which orbits around a maximally-spinning (extremal) Kerr black-hole of mass M (with μ/M 1) and angular momentum J = M 2 . The extremal curved black-hole spacetime is characterized by the line element [1], where (t, r, θ, φ) are the familiar Boyer-Lindquist spacetime coordinates and the metric functions in (1) are defined by the relations, The degenerate horizon of the extremal black-hole spacetime is defined by the radial functional relation (r = r H ) = 0, which yields the simple relation The dimensionless energy ratio E(r )/μ (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], 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], where the dimensionless parameter x (with the near-horizon property x 1) is defined by the relation,

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 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) 4π E total . That is, Thorne's famous hoop conjecture asserts that [12] C(E total ) ≤ 4π E total ⇒ Black-hole horizon exists. (8) As we shall now show, the hoop relation (8) provides compelling evidence for the existence of a lower bound (with r circular > r min > r 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] 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, Taking cognizance of Eqs. (5), (7), (8), and (10), one obtains the dimensionless lower bound 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 circular ≤ x min are expected to be engulfed by a larger horizon with r horizon ≥ r circular [see Eqs. (8) and (11)].

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 circular /r H → 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 horizon ≥ r circular ≥ r 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)]