Comments on claimed risk from metastable black holes

In a recent note, arXiv:0808.1415, it was argued that a hypothetical metastable black hole scenario could pose collider risk not excluded by our previous study. We comment on inconsistency of this proposed scenario. CERN-PH-TH/2008-184 August 29, 2008 [email protected] [email protected] The recent paper [1] claims to produce a scenario for LHC risk that is not excluded by our previous work[2], or by the paper [3] supporting the same conclusions. This proposal is based on an idea, extrapolating claims in [4], that Hawking radiation is suppressed until a certain mass threshold, which [1] proposes occurs for black hole radii comparable to the scale where higher-dimensional gravity matches onto four-dimensional gravity. Ref. [1] claims that once the Hawking radiation turns on, the power output would be at dangerous levels like 10 W, and that moreover such a scenario is not excluded by astrophysical constraints such as those derived in [2] from white dwarfs and neutron stars. To assess this scenario, let us begin by observing that a universal relation for the energy output of Hawking radiation is of the form dE dt = ξ (D − 3) 3840π 1 R ∼ (D − 3) × 10 1 R , (1) where D is the spacetime dimension, R is the Schwarzschild radius, and ξ is an O(1) graybody factor parameterizing the deviations from a precise blackbody spectrum. This general formula follows from two facts: the Stefan-Boltzmann law, and the formula for the Hawking temperature of the black hole, T = D − 3 4π R . (2) Moreover, this formula agrees with eq. (2) of [1]. The proposal of [1] is that Hawking radiation is suppressed compared to this usual Hawking result (hence the black hole is “metastable”) until the black hole reaches the radius scale R ∼ L, where L is comparable to the scale of transition to four-dimensional behavior. (This could be anywhere in the range between the parameters RD and RC introduced in [2].) When it reaches this scale, the usual Hawking radiation is then claimed to switch on. Ref. [1] considers in particular scales near L ∼ 10cm, where one is claimed to find the large power output stated above. However, using these parameters one readily finds from the formula (1) a negligible power output of size dE dt ∼ .1μW , (3) differing by a factor of 10 from the claim of [1]. Where did [1] go wrong? The answer is in the inconsistent application of formula (2) of that paper. In the type of warped scenario that [1] considers, the black hole would evolve up to a radius R ∼ RD via higher-dimensional evolution, and then would experience a large mass gain in transitioning to a slightly higher radius R ∼ RC , as [1] acknowledges. Throughout this region, in the usual Hawking scenario, the temperature formula (2) should hold. Thus if the black hole radiance is suppressed compared to this, as the author of [1] proposes, it can’t exceed a value of size (3). However, [1] then applies the formula (1) written in terms of the mass using the four-dimensional relationship between radius and mass, but does this in a region where the four-dimensional relation between radius and mass is clearly wrong. Indeed,