General relativistic formulas for mass and spin of a Kerr black hole in terms of redshifts and orbital parameters

Abstract

We derive closed formulas for the mass M and spin a parameters of a Kerr black hole in terms of a minimal quantity of observational data: the red-/blue-shifts of photons emitted by massive particles (stars) moving on geodesics around the black hole and their respective orbital radius. It turns out that for a single star orbiting the black hole, we need a minimal set of four observational measurements to analytically determine both parameters. Indeed, given a set of two (three) stars revolving around the black hole, these formulas involve just eight (twelve) observational data.

Mass and spin of Kerr black holes in terms of observational quantities: The dragging effect on the redshift

“…We would like to emphasize that the mass expression (15) is given in terms of the total Kerr frequency shifts (11)-( 12) which are directly measured quantities in real astrophysical systems. This elegant relation differs from previous attempts to express the mass parameter in terms of the kinematic frequency shift [9,13,14,23]. Indeed, since Eq.…”

“…Indeed, since Eq. ( 13) is a linear relation for M , it has no degeneracy in its values as in the case of higher order algebraic equations arising in [13,14] when expressing M in terms of the kinematic redshift for several spherically symmetric spacetimes and for configurations involving several photon sources [23]. Now, by substituting the relation ( 15) into ( 14), we obtain the rotation parameter versus the same quantities…”

“…By directly substituting ( 5)-( 7) and ( 21) into (23) [or, equivalently, by employing ( 5), (8), and (21) in relations ( 26)-( 27)], it is possible to write these expressions versus the black hole parameters and orbital radii…”