Optimal entropy bound and the self-energy of test objects in the vicinity of a black hole

Recently Bekenstein and Mayo conjectured an entropy bound for charged rotating objects. On the basis of the no-hair principle for black holes, they speculate that this bound cannot be improved generically based on knowledge of other “quantum numbers,” e.g., baryon number, which may be borne by the object. Here we take a first step in the proof of this conjecture. The proof makes use of a gedanken experiment in which a massive object endowed with a scalar charge is lowered adiabatically towards a Schwarzschild’s black hole and then dropped into the black hole from some proper distance above the horizon. Central to the proof is the intriguing fact that the self-energy of the particle receives no contribution from the scalar charge. Thus the energy with which the object is assimilated consists of its gravitational energy alone. This of course agrees with the no-scalar-hair principle for black holes: after the object is assimilated into the black hole, any knowledge of the scalar field properties is lost. Using the generalized second law, we reach the conclusion that the original entropy bound was not improved by the knowledge of the scalar charge. At the end we speculate on whether or not massive vector fields may serve in the tightening of the entropy bound.