Quasi-local mass at axially symmetric null infinity

We give a brief review of the definition of the Wang–Yau quasi-local mass and discuss the evaluation of this quantity on surfaces of unit size at null infinity of an axisymmetric spacetime in Bondi–van der Burg–Metzner coordinates.

Thermodynamic Volume Product in Spherically Symmetric and Axisymmetric Spacetime

We have examined the thermodynamic volume products for spherically symmetric and axisymmetric spacetime in the framework of extended phase space. Such volume products are usually formulated in terms of the outer horizon (H+) and the inner horizon (H-) of black hole (BH) spacetime. Besides volume product, the other thermodynamic formulations like volume sum, volume minus, and volume division are considered for a wide variety of spherically symmetric spacetime and axisymmetric spacetime. Like area (or entropy) product of multihorizons, the mass-independent (universal) features of volume products sometimes also fail. In particular, for a spherically symmetric AdS spacetime, the simple thermodynamic volume product of H± is not mass-independent. In this case, more complicated combinations of outer and inner horizon volume products are indeed mass-independent. For a particular class of spherically symmetric cases, i.e., Reissner Nordström BH of Einstein gravity and Kehagias-Sfetsos BH of Hořava Lifshitz gravity, the thermodynamic volume products of H± are indeed universal. For axisymmetric class of BH spacetime in Einstein gravity, all the combinations are mass-dependent. There has been no chance to formulate any combinations of volume product relation to be mass-independent. Interestingly, only the rotating BTZ black hole in 3D provides that the volume product formula is mass-independent, i.e., universal, and hence it is quantized.

Carter's constant and superintegrability

Carter's constant is a nontrivial conserved quantity of motion of a particle moving in stationary axisymmetric spacetime. In the version of the theorem originally given by Carter, due to the presence of two Killing vectors, the system effectively has two degrees of freedom. We propose an extension to the first version of Carter's theorem to a system having three degrees of freedom to find two functionally independent Carter-like integrals of motion. We further generalize the theorem to a dynamical system with [Formula: see text] degrees of freedom. We further study the implications of Carter's constant to superintegrability and present a different approach to probe a superintegrable system. Our formalism gives another viewpoint to a superintegrable system using the simple observation of separable Hamiltonian according to Carter's criteria. We then give some examples by constructing some two-dimensional superintegrable systems based on this idea and also show that all three-dimensional simple classical superintegrable potentials are also Carter separable.