GEODESIC ORBITS AND LYAPUNOV EXPONENTS OF FROLOV'S BLACK HOLE

Abstract

Binary black holes maintain unstable orbits at very close distances. In the simplest case of geodesics around a Schwarzschild black hole, the orbits, although unstable, are regular and depend only on the mass. In more complex cases, geodesics may depend on charge, rotation, and other parameters. When perturbed, unstable orbits can become a source of chaos. All unstable orbits, whether regular or chaotic, can be quantified by their Lyapunov exponents. Exponents are important for observations because the phase of gravitational waves can decohere in Lyapunov time. If the time scale of dissipation due to gravitational waves is shorter than the Lyapunov time, the chaos will be damped and practically unobservable. These two time scales can be compared. Lyapunov exponents should be used with caution for several reasons: they are relative and dependent on the coordinate system used, they vary from orbit to orbit, and finally, they can be deceptively diluted by transitional behavior for orbits that pass in and out of unstable regions. The stability of circular geodesic orbits of Frolov's black hole space-time is studied in this work. The influence of the black hole charge and the scale parameter on the stability of geodesic orbits and the Lyapunov exponent is analyzed. It is shown that the region of stable circular orbits increases with the black hole charge Q and the scale parameter ℓ . The largest region of stable circular orbits of Frolov's black hole is reached at Q = M and ℓ = 0.75M.