Regular Black Strings and $(2+1)$ Black Hole in Unimodular Gravity Supported by Maxwell Fields

Abstract

In this work, we obtain regular solutions with a dynamical cosmological function in unimodular gravity. This alternative theory to General Relativity imposes an additional condition on the spacetime volume element by fixing the metric volume density to a prescribed non-dynamical quantity, which can be represented by a constant in unimodular coordinates. In this procedure, the cosmological constant does not appear directly in the action, but rather as an integration constant of the field equations. By using the non-conservation of the energy-momentum tensor, we show that the integration constant becomes a function $Λ(x)=Λ_0+δΛ(x)$, where $Λ_0$ is the vacuum cosmological constant and $δΛ(r)$ is a matter-induced cosmological contribution determined by the geometry and the electromagnetic sector. From the definition of the geometric function $H(r)$, we verify the validity of Maxwell electrodynamics as a source for the solutions. We further show that linear combinations of the mass functions associated with the individual solutions can also be consistently supported by the Maxwell field, a property that is not generally shared by nonlinear electrodynamics. This result allows us to construct a broader family of regular solutions and investigate their physical properties. In particular, we analyze the energy conditions, curvature invariants, and thermodynamic properties of the generated solutions.