The asymptotically Schwarzschild-like metric solutions

TL;DR

This work analyzes asymptotically Schwarzschild-like metrics as non-vacuum extensions of General Relativity that reproduce Schwarzschild behavior at large distances but differ markedly in the strong-field regime, notably by lacking an event horizon and potentially representing traversable wormholes or black holes surrounded by anisotropic matter. It develops a static, spherically symmetric framework with a single function $\alpha(r)$, derives the corresponding geodesic structure, and computes photon spheres, ISCOs, Regge–Wheeler potentials for spin-0 and spin-1 perturbations, and black-hole shadows, all within two specific exponential-inspired parametrizations of $\alpha(r)$. The results show mass- and parameter-dependent shifts in the photon-sphere and ISCO radii, modify perturbation potentials, and yield shadow sizes that can depart from the Schwarzschild prediction; they also reveal regions where energy conditions are violated or satisfied, highlighting the exotic matter content required to source these geometries. While solar-system tests disfavor these metrics as alternatives to GR in standard astrophysical contexts, their formal structure and connection to screened (Yukawa) potentials make them relevant for beyond-GR theories, regular black hole models, and explorations of how strong-field observables encode departures from vacuum GR.

Abstract

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.

Figures (6)

• Figure 1: The behavior of the function $\alpha(r)$ in the Schwarzschild ($\bf{A}$) and asymptotically Schwarzschild-like ($\bf{B}$) metrics. In the Schwarzschild metric, the function $\alpha(r)$ tends to $-\infty$ as $r \to r_g$, meaning there is an event horizon within the interval $r \in (0, +\infty)$. In the asymptotically Schwarzschild-like metric the function $\alpha(r)$ tends to $-\infty$ as $r \to -l$, meaning the event horizon lies outside the interval $r \in (0, +\infty)$, and this metric has no coordinate singularity within this interval.
• Figure 2: The potential function for $M=1$ and $l=0.1$. The graph shows the spin-zero Regge–Wheeler potential for $\tilde{l} = 0$. While these potentials are similar at large $r$, they differ radically at small $r$. In the Schwarzschild case, however, the potential at small $r$ is only formal, as this region lies behind the event horizon and cannot interact with the domain of outer communication.
• Figure 3: The potential function for $M =1$ and $l = 0.1$. The graph shows the spin one Regge–Wheeler potential for $\tilde{l} = 1$. While these potentials are similar at large $r$, they differ radically at small $r$. In the Schwarzschild case, however, the potential at small $r$ is only formal, as this region lies behind the event horizon and cannot interact with the domain of outer communication.
• Figure 4: The behavior of the function $\alpha(r)$ in the Schwarzschild ($\bf{A}$) and asymptotically Schwarzschild-like ($\bf{B}$) metrics. In the Schwarzschild metric, the function $\alpha(r)$ tends to $-\infty$ as $r \to r_g$, meaning there is an event horizon within the interval $r \in (0, +\infty)$. The asymptotically Schwarzschild-like metric has no any event horizon in the interval $r \in (0, +\infty)$.
• Figure 5: The potential function for $M = 1$ and $l = 0.5$. The graph shows the spin zero Regge–Wheeler potential for $\tilde{l} = 0$. As it is shown in the graph they are different in all radial coordinate $r$. In the Schwarzschild case, however, the potential at small $r$ is only formal, as this region lies behind the event horizon and cannot interact with the domain of outer communication.