The Schwarzschild radius is the size a mass would need to be compressed to for its own gravity to trap light — the defining threshold of a black hole. Named after Karl Schwarzschild, who derived the exact solution just weeks after Einstein published general relativity in 1915, it depends on nothing but the object's mass.
How the Schwarzschild Radius Calculator works
The calculator uses rs = 2GM/c², the exact solution Karl Schwarzschild found to Einstein's field equations for the gravitational field outside a spherically symmetric, non-rotating mass. It can also be derived from simpler Newtonian escape-velocity reasoning — setting v = √(2GM/r) equal to c and solving for r — which happens to give the same answer, though the full general-relativity derivation is what makes it physically rigorous at these extreme scales.
G, the gravitational constant, is fixed at 6.674×10⁻¹¹ N·m²/kg², and c, the speed of light, is fixed at 2.998×10⁸ m/s (299,792,458 m/s exactly).
Inputs and what they mean
Mass (M) is the only required input — the total mass of the object, in kilograms or solar masses (M☉, where one solar mass is about 1.989×10³⁰ kg). The calculator accepts scientific notation directly, which is the natural way to enter astronomical values like a star's or black hole's mass.
The Schwarzschild radius scales linearly with mass: doubling the mass doubles rs. That's a very different relationship from ordinary density, which is why a supermassive black hole like Sagittarius A* — despite having a colossal mass — has an average density inside its event horizon far lower than a stellar-mass black hole's.
Limits and edge cases
This formula describes a non-rotating (Schwarzschild) black hole with no electric charge — real astrophysical black holes typically rotate, and their true event-horizon shape is described by the more complex Kerr metric, which gives a slightly smaller horizon for a given mass. The Schwarzschild radius remains an excellent first-order estimate and is the standard reference figure quoted for black hole sizes.
For everyday objects like the Sun or Earth, the Schwarzschild radius is a purely theoretical number — no known physical process can compress a star or planet that far without first blowing it apart. It only becomes physically relevant for compact objects like neutron star remnants above the Tolman–Oppenheimer–Volkoff limit, where gravitational collapse can actually proceed to a true black hole.