Spacetime and the cosmos

1916

The first black hole solution

Karl Schwarzschild

Could something be so dense that not even light gets out?

Serving in the German army during the First World War, Schwarzschild found an exact solution to Einstein’s equations for a single round mass. It contained a radius from which nothing, not even light, could escape. It took decades for physicists to accept such objects are real.

How they figured it out

In 1783 an English clergyman, John Michell, imagined “dark stars” whose escape speed exceeded the speed of light. The idea was forgotten.

Karl Schwarzschild, director of the Potsdam observatory, volunteered for the German army at 40. Serving on the Russian front, he found an exact solution of Einstein’s brand-new equations within weeks and posted it to Einstein, who was surprised one existed. Schwarzschild died in May 1916 of an illness caught at the front. The name “black hole” only caught on in 1967, popularised by John Wheeler.

The equation

rs=2GMc2r_s = \frac{2GM}{c^2}

Squeeze any mass inside this radius and it becomes a black hole. For the Sun, about 3 km.

Everyday echoSqueezing the Earth into a black hole would make it about the size of a marble.

See it in real life

Escape speed

Throw a ball up at 11.2 km/s and it never comes back: that is Earth’s escape speed. Squeeze the Earth smaller and escape speed rises. Squeeze it down to about 9 mm, the size of a marble, and escape speed reaches the speed of light.

Where engineers use it

  • Escape speed

    The same balance of energies gives rocket escape speeds: v=2GM/rv = \sqrt{2GM/r}.

Worked example: Squeeze the Earth

  1. rs=2GM/c2=2×3.986×1014/(3×108)2r_s = 2GM/c^2 = 2 \times 3.986\times10^{14} / (3\times10^8)^2
  2. ≈8.9\approx 8.9 mm: about the size of a marble.

Where it came from

What it led to