Isao Okamoto

Fluctuations in isothermal spheres

Isolated isothermal spheres of N gravitationally interacting points with equal mass are believed to be stable when density contrasts do not exceed 709. That stability limit does not, however, take into consideration fluctuations of temperature near the onset of instability. These are important when N is finite. Here we correlate global mean quadratic temperature fluctuations with the onset of instability. We show that such fluctuations trigger instability when the density contrast reaches a value near 709×exp(−3.3N−1/3). These lower values of limiting density contrasts are significantly smaller than 709 when N is not very large, and this suggests (i) that numerical calculations with small N may not reflect correctly the onset of core collapse in clusters with large N, and (ii) that a greater number of globular clusters than is normally believed may already be in an advanced stage of core collapse, because most of the observed globular clusters with parameters that fit quasi-isothermal configurations are close to marginal stability.

Thermodynamics of a black hole in a cavity

We present a unified thermodynamic description of configurations consisting of self-gravitating radiation with or without a black hole. We compute the thermal fluctuations and evaluate where they will induce a transition from metastable configurations towards stable ones. We show that the probability of finding such a transition is exponentially small. This indicates that, in a sequence of quasi-equilibrium configurations, the system will remain in metastable states until it approaches very closely the critical point beyond which no metastable configuration exists. Near that point, we relate the divergence of local temperature fluctuations to the approach of instability of the whole system, thereby generalizing the usual fluctuation analysis to cases where long-range forces are present. When angular momentum is added to the cavity, the above picture is slightly modified. Nevertheless, at high angular momentum, the black hole loses most of its mass before it reaches the critical point at which it evaporates completely.

Electromagnetic Extraction of Energy from Kerr Black Holes

The energy extraction process from Kerr black holes is elucidated compared with that from pulsars in force-free degenerate electrodynamics (FFDE). It is argued for the force-free magnetosphere of a Kerr black hole in a steady axisymmetric state that the function of unipolar inductors in the presence of global magnetic fluxes threading the horizon is equipped by a frame-dragging effect in the Kerr spacetime metric to the upper null surface, SN, with the axial distance, � N, where the local value of the angular velocity of the frame-dragging, ω, is equal to the rotational frequency of each field line, i.e., ω(� N )= ΩF, and the rotational velocity of each field line, vF, measured by “zeroangular-momentum-observers” changes in sign. There must be a pair-creation gap at SN, from which the paired winds, ingoing and outgoing, are initiated. The poloidal electric current, I, for each wind is determined by solving the eigenvalue problem with the “criticality condition” imposed at the fast surfaces nearly at the horizon and at infinity, and the “current-closure condition” at SN determines the ultimate eigenvalues of ΩF and I in terms of the hole’s angular velocity, ΩH = ω(rH), and the distribution of the magnetic flux at the horizon. It is not in the ordinary ergosphere, but in the effective ergosphere in the region of ΩH >ω> ΩF and vF < 0 that general-relativistic effects play the most significant role for electromagnetic extraction of the hole’s energy. The flow there must be regarded as an ingoing magneto-centrifugal wind, rather than an accretion flow. In a kind of extreme state that “force-free” field lines be frozen in “massless” particles in FFDE, the “membrane paradigm” combined with the DC circuit model is useful in black hole as well as pulsar magnetospheres.

A comment on fluctuations and stability limits with application to `superheated' black holes

We point out that, contrary to signs of heat capacities, thermodynamic fluctuations are simply and unequivocally related to the onset of instabilities that show up near critical points. Fluctuation theory is then applied to Schwarzschild black holes surrounded by radiation. This shows that slowly evolving black holes along quasi-equilibrium states in cavities greater than Planck lengths will not evaporate below the critical Hawking limit temperature despite the fact that pure radiation has a much higher entropy.