Remarks on conserved quantities and entropy of BTZ black hole solutions. II. BCEA theory

The BTZ black hole solution for $\mathrm{}(2+1)\mathrm{}$ spacetime is considered as a solution of a triad-affine theory (BCEA) in which topological matter is introduced to replace the cosmological constant in the model. Conserved quantities and entropy are calculated via the Nöther theorem, reproducing in a geometrical and global framework earlier results found in the literature using local formalisms. Ambiguities in global definitions of conserved quantities are considered in detail. A dual and covariant Legendre transformation is performed to reformulate BCEA theory as a purely metric (natural) theory (BCG) coupled to topological matter. No ambiguities in the definition of mass and angular momentum arise in BCG theory. Moreover, gravitational and matter contributions to conserved quantities and entropy are isolated. Finally, a comparison of BCEA and BCG theories is carried out by relying on the results obtained in both theories.