$$(2+1)$$ ( 2 + 1 ) -dimensional black holes of Einstein’s theory with Born–Infeld type electrodynamic sources

A new family of $$(2+1)$$ -dimensional black holes are investigated in the background of Born–Infeld type theories coupled to a Riemannian curved spacetime. We know that both the scale and dual invariances are violated for these nonlinear electromagnetic theories. In this set-up, first we consider a pure magnetic source in a model of exponential electrodynamics and find a magnetically charged $$(2+1)$$ -dimensional black hole solution in terms of magnetic charge q and nonlinearity parameter $$\beta $$ . In the second case we consider a pure electric source of gravity in the framework of arcsin electrodynamics and derive the associated $$(2+1)$$ -dimensional black hole solution in terms of electric charge Q and the parameter $$\beta $$ . The asymptotic behaviour of the solutions at infinity as well as at $$r\rightarrow 0$$ in both the frameworks is discussed. The asymptotic expressions of curvature invariants in the case of exponential electrodynamics shows that there exists a finite value of curvature at the origin, while in arcsin electrodynamics, the corresponding asymptotic behaviour shows that there is a true curvature singularity at the centre of the charged object. Furthermore, thermodynamics of the resulting charged black holes within the context of both the models is studied. It is shown that the thermodynamic quantities corresponding to these objects satisfy the first law of black hole thermodynamics.