The asymptotically Schwarzschild-like metric solutions

In this article we investigate the properties of the asymptotically Schwarzschild-like metric as an alternative to the Schwarzschild solution in General Relativity. While asymptotically flat and similar to Schwarzschild at large distances $r$, this metric exhibits a fundamentally different strong-field behavior: it lacks an event horizon and is best interpreted as a traversable wormhole or a black hole surrounded by a specific anisotropic fluid, rather than a true vacuum solution. We analyze key phenomenological features, demonstrating significant deviations from Schwarzschild in the radii of the photon sphere and the Innermost Stable Circular Orbit (ISCO). Furthermore, we derive the Regge-Wheeler equation for gravitational and electromagnetic perturbations and compute the black hole shadow, providing a direct comparison with the Schwarzschild metric. Despite being ruled out by solar system tests, the exponential metric remains relevant for theories beyond standard GR, regular black hole models, and its connection to screened Yukawa potentials.

K.K. Ernazarov · 2025