Negative Energy Condition and Black Holes on the Brane

We propose that for non-localizable energy distribution the relevant energy condition is determined by the gravitational field energy which is negative for positive non-gravitational energy. That is negativity of the non-localized energy is the ”positive” energy condition. This would have direct application and relevance for a black hole on the brane which would be sitting in a trace free stresses induced by the Weyl curvature of the bulk. PACS Numbers: 04.50.+h, 04.60., 04.70.-s, 98.80.Cq In view of the developments in the string theory, the study of gravity in dimensions higher than the usual four has been in vogue for quite a while. The recent results indicate that extra dimensions need not be compact, they could be large and even infinite [1,2]. In the Randall Sundrum (R S) model [2], the 4-D Universe is a 3-brane acting as a domain wall between the two semi infinite 5-D negative λ regions. That is, the actual Universe is embedded in the 5-D bulk satisfying the appropriate boundary conditions. All non-gravitational matter fields remain localized in the brane while free gravitational field and gravitational waves also propagate off the brane in the 5-D bulk. The free field in the bulk backreacts on the brane to induce a trace free stress tensor through the 5-D Weyl tensor evaluated at the brane. The Einstein field equation for the brane has been formulated by Shiromizu, Maeda and Sasaki [3]. In the absence of energy distribution on the brane, the stress tensor on the brane would have only the trace free Weyl term which need not a priori satisfy the usual positive energy conditions. The energy density of the induced stresses could as well be negative. The effective stress tensor on the brane could thus legitimately include negative energy. It turns out that for a static black hole on the brane, the induced energy from the free gravitational field in the bulk must be negative to contribute positively to the hole’s gravity (attractively). In this context we would like to argue that the proper energy condition should be that the induced energy must be negative to contribute