Thermodynamics of Swirling Black Holes and Kerr-Taub-NUT

We establish the first law of black hole thermodynamics for the recently discovered Kerr swirling (KS) and accelerated Kerr swirling (AKS) black hole solutions. We also revisit the first law for the Kerr-Taub-NUT (KTN) spacetime. Guided by Noether's original insight that global symmetries give rise to conservation laws, we employ and adapt the surface charge method for computing the associated charges (mass and angular momentum) in the presence of spacetime string-like defects (conical singularities) along the axes of both KS and KTN black holes. We obtain finite integrated charges in a quasi-local manner, without relying on the non-flat asymptotic regions characteristic of these spacetimes. We then establish a first law without introducing independent stringy terms unrelated to symmetries. For both KS and KTN, the quantities that play the role of the black hole entropy differ from the simple geometrical rule of one-fourth of the horizon area. This mismatch arises because we keep first-law charges associated with conservation laws. It indicates that standard formulas may not be naively used for black holes with string-like defects; they have a different topology and suggest the need for a new geometrical framework. In stark contrast, for the AKS spacetime, which can be regularized outside the singularity and lacks string-like defects, we find that the charges are consistent with a first law having the usual one-fourth of the area entropy. We first determine the mass using the Christodoulou-Ruffini formula and then check its agreement with the result obtained from the surface charge method.