The self-force on a non-minimally coupled static scalar charge outside a Schwarzschild black hole

Details The self-force on a non-minimally coupled static scalar charge outside a Schwarzschild black hole The self-force on a non-minimally coupled static scalar charge outside a Schwarzschild black hole

Mar 2007

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2007cqgra..24.1035c&link_type=abstract

Classical and Quantum Gravity, Volume 24, Issue 5, pp. 1035-1048 (2007).

Computer Science

4

Scientific paper

The finite part of the self-force on a static, non-minimally coupled scalar test charge outside a Schwarzschild black hole is zero. This result is determined from the work required to slowly raise or lower the charge through an infinitesimal distance. Unlike similar force calculations for minimally-coupled scalar charges or electric charges, we find that we must account for a flux of field energy that passes through the horizon and changes the mass and area of the black hole when the charge is displaced. This occurs even for an arbitrarily slow displacement of the non-minimally coupled scalar charge. For a positive coupling constant, the area of the hole increases when the charge is lowered and decreases when the charge is raised. The fact that the self-force vanishes for a static, non-minimally coupled scalar charge in Schwarzschild spacetime agrees with a simple prediction of the Quinn Wald axioms. However, Zel'nikov and Frolov computed a non-vanishing self-force for a non-minimally coupled charge. Our method of calculation closely parallels the derivation of Zel'nikov and Frolov, and we show that their omission of this unusual flux is responsible for their (incorrect) result. When the flux is accounted for, the self-force vanishes. This correction eliminates a potential counter example to the Quinn Wald axioms. The fact that the area of the black hole changes when the charge is displaced brings up two interesting questions that did not arise in similar calculations for static electric charges and minimally coupled scalar charges. (1) How can we reconcile a decrease in the area of the black hole horizon with the area theorem which concludes that δAreahorizon >= 0? The key hypothesis of the area theorem is that the stress energy tensor must satisfy a null-energy condition Tαβlαlβ >= 0 for any null vector lα. We explicitly show that the stress energy associated with a non-minimally coupled field does not satisfy this condition, and this violation of the hypothesis leads directly to the decreasing area. (2) Since the entropy of a Schwarzschild black hole is proportional to the area of the horizon, and the area of the horizon will change while we slowly raise or lower the charge, we must ask: does this simple process conserve entropy? The process does conserve entropy; however, the appropriate entropy for a gravitational theory with a non-minimally coupled scalar field is the Iyer Wald generalized entropy which—in addition to the area of the black hole—includes a contribution from the scalar field evaluated on the horizon. We explicitly calculate the generalized entropy of a Schwarzschild black hole bathed in the field of a static, non-minimally coupled scalar test charge, and show that it is conserved when the charge is slowly raised or lowered.

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