Static Spherically Symmetric Black Holes Sourced by Exponentially Localized Anisotropic Matter Fields

We derive a static and spherically symmetric black-hole spacetime sourced by an anisotropic matter distribution. Starting from a Schwarzschild-like metric ansatz with a deformation function, the Einstein equations are used to express the energy density, radial pressure, and tangential pressure in terms of a single metric function. By imposing an anisotropic equation of state for the tangential pressure, we obtain a second-order differential equation whose solution leads to an exponentially localized matter profile. The resulting spacetime reduces to the Schwarzschild geometry in the absence of the anisotropic matter parameter and approaches a Reissner–Nordstr"om-type form in the weak-localization limit. The corresponding energy density and pressure components are analyzed, and the main energy conditions are discussed. It is shown that the null, weak, and strong energy conditions are satisfied for positive model parameters, while the dominant energy condition may be violated in the tangential direction due to the anisotropic nature of the source.

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