Schwarzschild radius

Lore & Background

In 1916, Karl Schwarzschild obtained an exact solution to the Einstein field equations for the gravitational field outside a non-rotating, spherically symmetric body with mass. The solution contained terms of the form which have singularities at certain radii.

The parameter has come to be known as the Schwarzschild radius. The physical significance of these singularities was debated for decades; it was found that one singularity is a coordinate singularity, an artifact of the particular coordinate system used, while the other is a spacetime singularity that cannot be removed. The Schwarzschild radius is nonetheless a physically relevant quantity.

Any object whose radius is smaller than its Schwarzschild radius is called a black hole. The surface at the Schwarzschild radius acts as an event horizon in a non-rotating body (a rotating black hole operates slightly differently).

Neither light nor particles can escape through this surface from the region inside. Black holes can be classified based on their Schwarzschild radius, or equivalently, by their density. Small black holes are much more dense than large ones; the volume enclosed in the event horizon of the most massive black holes has an average density lower than main sequence stars.

The Schwarzschild radius of an object is proportional to its mass. The Sun has a Schwarzschild radius of approximately 3.0 km, Earth's is approximately 9 mm, and the Moon's is approximately 0.1 mm. The Schwarzschild radius of the supermassive black hole at the Galactic Center of the Milky Way is approximately 12 million kilometres.

Reader's Guide

The Schwarzschild radius is a fundamental concept in general relativity and black hole physics. It defines the event horizon for a non-rotating black hole, marking the boundary beyond which no light or matter can escape. This radius is directly proportional to mass, meaning that any object compressed within its Schwarzschild radius becomes a black hole. The concept allows classification of black holes by mass: supermassive black holes (with masses of hundreds of thousands to billions of solar masses) have low average densities, sometimes less than water, while stellar black holes have much higher densities.

Micro black holes, if they exist, would have extremely small Schwarzschild radii and densities so high that no known mechanism could form them; they might have formed in the early universe after the Big Bang. The Schwarzschild radius also appears in gravitational time dilation approximations and intersects with the Compton wavelength at the Planck scale, indicating where quantum gravity corrections become necessary. The term 'gravitational radius' is sometimes used as a synonym, though it can also refer to a constant half as large, leading to its avoidance in educational settings.