Schwarzschild Metric
The Schwarzschild metric is the first exact solution to the Einstein field equations, describing the curved spacetime geometry surrounding a spherically symmetric, non-rotating, uncharged mass — the simplest and most important solution in general relativity, underlying predictions from GPS corrections to black holes.
Curious
Einstein's field equations, published in November 1915, are notoriously difficult — ten coupled, nonlinear differential equations that resist exact solution in almost any realistic situation. Remarkably, within about a month of Einstein's publication, German physicist and astronomer Karl Schwarzschild — while serving on the German front in World War I — found an exact, complete solution for the simplest possible case: a single, non-rotating, spherically symmetric mass sitting alone in otherwise empty space.
This single solution turned out to describe, to excellent approximation, the spacetime around the Sun, the Earth, and most stars and planets, making it immediately useful for testing general relativity's predictions against real astronomical observations, and it later turned out to also describe black holes when the mass is sufficiently compact.
Exploring
| Symbol | Name | SI Unit |
|---|---|---|
| r_s | Schwarzschild radius | metre (m) |
| G | Newtonian gravitational constant | m³kg⁻¹s⁻² |
| M | Mass of the object | kilogram (kg) |
Every mass has a corresponding Schwarzschild radius — the radius that mass would need to be compressed within for its escape velocity to reach the speed of light, at which point (according to the metric) the object becomes a black hole. For the Sun, the Schwarzschild radius is about 3 km — far smaller than the Sun's actual radius of about 700,000 km, which is exactly why the Sun is not a black hole; for the Earth, it's less than a centimetre.
Worked example: Find the Schwarzschild radius of the Sun (M = 1.989×10³⁰ kg). r_s = 2GM/c² = 2(6.674×10⁻¹¹)(1.989×10³⁰)/(3×10⁸)² = 2.655×10²⁰/9×10¹⁶ ≈ 2,950 m ≈ 2.95 km.
Deep Dive
The full Schwarzschild metric, in standard spherical coordinates, is:
where dΩ² represents the ordinary angular part of spherical coordinates. Notice that the metric becomes singular (mathematically undefined) at r = r_s — this is the event horizon, and for ordinary astronomical objects like stars and planets, this radius lies deep inside the object where the vacuum solution doesn't apply anyway, so no physical singularity is encountered. Only for a sufficiently compact object (a black hole) does r_s lie outside all the mass, making the event horizon physically real and significant. There is also a genuine spacetime curvature singularity at r = 0, where the theory's predictions are believed to break down and require a yet-undiscovered theory of quantum gravity to resolve.
Schwarzschild died in 1916, only months after publishing this result, from a skin disease contracted while serving on the Russian front during World War I — meaning he never lived to see the enormous significance his solution would come to have for black hole physics, a field that only fully developed decades later.
Concept Relationship Map
Requires: curved spacetime
Enables: black holes, event horizon [PENDING FULL ENTRY], gravitational time dilation
Commonly confused with: assuming every object has a physically real Schwarzschild-radius surface (only compact objects like black holes do)
Sources
- Schwarzschild, K. (1916). Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften, 189–196.
- Misner, C.W., Thorne, K.S., Wheeler, J.A. (1973). Gravitation. W.H. Freeman, Ch. 31.