The Controversial Nature of Black Hole Horizons: Beyond the Schwarzschild Limit

For many years, the concept of a black hole has been a mythical subject in the field of astrophysics. One of the most intriguing propositions is the possibility that a black hole may not be as compact as traditionally thought. This article delves into the possibility that a black hole could instead have a star residing at its center, with a radius smaller than the Schwarzschild limit. We explore the definitions and properties of black holes, and analyze various scenarios where their horizons may differ from the well-known Schwarzschild radius.

Black Hole Radius and Definitions

In layman's terms, a black hole is defined as any compact set of spacetime events that are hidden behind an event horizon. An event horizon is the boundary dividing the black hole from its surroundings. The term "radius" in this context refers to the r-coordinate of the event horizon in specific coordinate systems.

Black Hole Horizon Comparisons

The radius of the event horizon can be less than, equal to, or even greater than the Schwarzschild radius, depending on the type of black hole.

Less than Schwarzschild Radius

Kerr Black Holes: These rotating black holes have an angular momentum characterized by the Kerr parameter. The outer bound of the horizon, or the outer Kerr horizon, is given by the formula r_ m sqrt{m^2 - a^2}. When the angular momentum a 0, the horizon reduces to the Schwarzschild radius.

Reissner-Nordstr?m (RN) Black Holes: Similar to Kerr black holes, these are electrically charged black holes. The formula for the outer horizon in RN black holes is r_ m sqrt{m^2 - Q^2}, where Q is the electric charge. Again, when Q 0, the horizon reverts to the Schwarzschild radius.

Equal to Schwarzschild Radius

Schwarzschild Black Holes: The classical, non-rotating and uncharged black hole. The Schwarzschild radius is the boundary where the escape velocity equals the speed of light, given by r_s 2m.

Greater than Schwarzschild Radius

Schwarzschild de Sitter Black Holes: These black holes incorporate a cosmological constant and are described by the Kottler spacetime. The metric coefficients can be more complex, but generally, the horizon is constrained by 2m . Here, the cosmological constant moves the event horizon outward.

Technical Notes on Schwarzschild Black Hole

The Schwarzschild black hole is the simplest case of a black hole, characterized by the metric g^{rr} g_{tt} 1 - 2m/r. Solving for the r-coordinate gives the Schwarzschild radius r_s 2m. The mass parameter in this case is the geometric mass defined as m GM/c^2, and the Schwarzschild radius is equal to itself under these conditions.

Conclusion

The possibility that a black hole might have a star residing at its center, or a radius smaller than the Schwarzschild limit, is a fascinating theoretical point of discussion. While the current understanding of black holes is based on the classical Schwarzschild radius, the exploration of alternative geometries like Kerr and RN black holes, as well as the more complex Schwarzschild de Sitter black holes, provides a deeper understanding of the potential properties of these cosmic phenomena. As our understanding of cosmology and astrophysics continues to evolve, the nature of black holes will remain a subject of intense research and debate.