Comment on “Generalized black diholes”

Abstract

We show that a recent solution published by Cabrera-Munguia et al. is physically inconsistent since the quantity σ it involves does not have a correct limit R → ∞.

Smarr formula for black holes endowed with both electric and magnetic charges

“…the formula (27). At this point, turning to the lower horizon, it might seem plausible to make use of the symmetry of the dyonic configuration under consideration and just conclude that the lower dyon verifies the generalized mass formula (13) too as it has the same physical characteristics (48) and (49) as the upper one (albeit the sign change in Ω H that does not affect (13) because J also changes its sign). However, in reality, coming to such an obvious physical conclusion is not straightforward at all, since Tomimatsu's formula (7) leads on the lower horizon to the equation…”

“…thus violating the generalized Smarr formula (13). Precisely for that reason the constant b 0 in (3) must be set equal to zero; however, in some more complex dyonic solutions the constant b 0 , as will be seen in the next two sections, must be assigned non-zero values to ensure consistent verification of the mass formula (13). It should be also noted that the values of the potentials Φ H and Φ H m of the KN dyon suggest that the last two (electromagnetic) terms in the generalized Smarr formula (13) can be combined in one expression.…”

“…thus being also consistent with the choice (34) for b 0 . Note that it is precisely the absence of the constant b 0 in the potential A ϕ of the paper [10] that led the authors of the latter to an unphysical redefinition J − QB of the angular momentum throughout the solution, thus making their results physically inconsistent (see [13] for details).…”

“…and it is straightforward to check that (36)-(40) satisfy (13) identically. Apparently, the lower dyon constituent satisfies (13) too, as the latter equation is invariant under the sign change J → −J,…”

“…Then a simple check indicates that the quantities (48) and (49) verify the generalized Smarr formula (13) or, by introducing…”