The fastest way to circle a black hole

📝 Abstract

Black-hole spacetimes with a “photonsphere”, a hypersurface on which massless particles can orbit the black hole on circular null geodesics, are studied. We prove that among all possible trajectories (both geodesic and non-geodesic) which circle the central black hole, the null circular geodesic is characterized by the {\it shortest} possible orbital period as measured by asymptotic observers. Thus, null circular geodesics provide the fastest way to circle black holes. In addition, we conjecture the existence of a universal lower bound for orbital periods around compact objects (as measured by flat-space asymptotic observers): $T_{\infty}\geq 4\pi M $, where $M$ is the mass of the central object. This bound is saturated by the null circular geodesic of the maximally rotating Kerr black hole.

💡 Analysis

Black-hole spacetimes with a “photonsphere”, a hypersurface on which massless particles can orbit the black hole on circular null geodesics, are studied. We prove that among all possible trajectories (both geodesic and non-geodesic) which circle the central black hole, the null circular geodesic is characterized by the {\it shortest} possible orbital period as measured by asymptotic observers. Thus, null circular geodesics provide the fastest way to circle black holes. In addition, we conjecture the existence of a universal lower bound for orbital periods around compact objects (as measured by flat-space asymptotic observers): $T_{\infty}\geq 4\pi M $, where $M$ is the mass of the central object. This bound is saturated by the null circular geodesic of the maximally rotating Kerr black hole.

📄 Content

arXiv:1201.0068v1 [gr-qc] 30 Dec 2011 The fastest way to circle a black hole Shahar Hod The Ruppin Academic Center, Emeq Hefer 40250, Israel and The Hadassah Institute, Jerusalem 91010, Israel (Dated: November 1, 2018) Black-hole spacetimes with a “photonsphere”, a hypersurface on which massless particles can orbit the black hole on circular null geodesics, are studied. We prove that among all possible trajectories (both geodesic and non-geodesic) which circle the central black hole, the null circular geodesic is characterized by the shortest possible orbital period as measured by asymptotic observers. Thus, null circular geodesics provide the fastest way to circle black holes. In addition, we conjecture the existence of a universal lower bound for orbital periods around compact objects (as measured by flat-space asymptotic observers): T∞≥4πM, where M is the mass of the central object. This bound is saturated by the null circular geodesic of the maximally rotating Kerr black hole. I. INTRODUCTION The motion of test particles in black-hole spacetimes has been extensively studied for more than four decades, see [1–4] are references therein. Of particular importance are geodesic motions which provide valuable information on the structure of the spacetime geometry. Circular null geodesics (also known as “photonspheres”) are especially interesting from both an astrophysical and theoretical points of view [5]. As pointed out in [4], the optical appearance to external observers of a star undergoing gravitational collapse is related to the properties of the unstable circular null geodesic [4, 6, 7]. Furthermore, null circular geodesics are closely related to the characteristic oscillation modes of black holes (see e.g. [8, 9] for detailed reviews). These quasinormal resonances can be interpreted in terms of null particles trapped at the unstable circular orbit and slowly leaking out [4, 10–14]. The real part of the complex quasinormal frequencies is related to the angular velocity at the unsta- ble null geodesic (as measured by asymptotic observers) while the imaginary part of the resonances is related to the instability timescale of the orbit [4, 10–14] (or the inverse Lyapunov exponent of the geodesic [4]). An important physical quantity for the analysis of circular orbits in black-hole spacetimes is the angular frequency Ω∞of the orbit as measured by asymptotic observers. In this paper we shall reveal an interesting property of null circular geodesics which is related to this important quantity: We shall show that null cir- cular geodesics provide the fastest way to circle black holes. More explicitly, we shall prove that the null cir- cular geodesic of a black-hole spacetime is characterized by the shortest possible orbital period (the largest orbital frequency) as measured by asymptotic observers. It is worth pointing out that the orbital period T around a spherical compact object must be bounded from below by the mass M of the central object: Suppose the compact object has radius R, then obviously T ≥2πR. (We shall use natural units in which G = c = 1). In addition, the central object must be larger than its grav- itational radius, R ≥2M. Thus, the orbital period must be bounded from below by T ≥4πM . (1) However, it should be realized that the above rea- soning is actually too naive – it does not take into ac- count the possible influence of the spacetime curvature (in the region near the surface of the compact object) on the orbital period. Due to the influence of the gravita- tional time dilation effect (redshift), the orbital period T∞as measured by asymptotic observers would actually be larger than 2πR. Moreover, we shall show below that due to the influence of the redshift factor, the circular orbit with the shortest orbital period (as measured by asymptotic observers) is distinct from the circular orbit with the smallest circumference (that is, rfast ̸= R in gen- eral, where rfast to be determined below is the radius of the circular trajectory with the shortest orbital period). II. SPHERICALLY SYMMETRIC SPACETIMES We shall first consider static spherically symmetric asymptotically flat black-hole spacetimes. The line el- ement may take the following form in Schwarzschild co- ordinates [4, 15] ds2 = −f(r)dt2 + 1 g(r)dr2 + r2(dθ2 + sin2 θdφ2) , (2) where the metric functions f(r) and g(r) depend only on the Schwarzschild areal coordinate r. These func- tions should be determined by solving the field equations. Since we do not specify the field equations, our results would be valid for all spherically symmetric asymptot- ically flat black holes. We note, in particular, that we do not assume g(r) = f(r) (a property which character- izes the familiar Schwarzschild and Reissner-Nordstr¨om spacetimes) and thus our results would be applicable to hairy black-hole configurations as well [in these space- times g(r) ̸= f(r), see [16–20] and references therein]. Asymptotic flatness requires that as r →∞, f(r) →1 and g(r) →1

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