Predicted Gravitational Radius and the Event Horizon of Black Holes with Constant c
Introduction
Karl Schwarzschild (1873–1916) was a German physicist and astronomer who investigated the mathematics that eventually led to the prediction of the existence of black holes.
Early Life and Education
Schwarzschild attended a Jewish primary school in Frankfurt up to the age of eleven, then entered the Gymnasium there. He became interested in astronomy at this stage, saving his pocket money to buy materials, such as lenses, to construct a telescope.
Karl’s father was friends with Professor J Epstein, who was a professor at the Philanthropin Academy and had his own private observatory. Their friendship arose through a shared interest in music. Professor Epstein had a son, Paul Epstein, who was two years older than Karl, and the two boys became good friends. They shared an interest in astronomy, and Karl learned how to use a telescope and some advanced mathematics from Paul Epstein.
Schwarzschild studied at the University of Strasbourg from 1891-93, where he learned a great deal of practical astronomy, and then at the University of Munich, where he obtained his doctorate. His dissertation, on an application of Poincaré’s theory of stable configurations of rotating bodies to tidal deformation of moons and to Laplace’s origin of the solar system, was supervised by Hugo von Seeliger. Schwarzschild found great inspiration in Seeliger’s teaching, which influenced him throughout his life.
Contributions
Schwarzschild’s key contributions include:
- The Schwarzschild solution, which uses Schwarzschild coordinates and the Schwarzschild metric, leads to a derivation of the Schwarzschild radius, the size of the event horizon of a non-rotating black hole.
- Schwarzschild Radius: Sometimes referred to as “Gravitational radius,” defined as r = 2Gm/c2. It is the radius of a sphere such that, if all the mass of an object were compressed into that sphere, the escape velocity from the surface of that sphere would equal the speed of light.
- Schwarzschild Metric: In Einstein’s theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild vacuum or Schwarzschild solution) is the solution to the Einstein field equations that describes the gravitational field outside a spherical mass, assuming the electric charge, angular momentum, and universal cosmological constant of the mass are all zero. The solution is a useful approximation for describing slowly rotating astronomical objects like many stars and planets, including Earth and the Sun.
- The Schwarzschild black hole is characterized by a surrounding spherical surface, called the event horizon, located at the Schwarzschild radius. Any non-rotating and non-charged mass smaller than its Schwarzschild radius forms a black hole. The solution to the Einstein field equations is valid for any mass M, so in principle (according to general relativity theory), a Schwarzschild black hole of any mass could exist if conditions were sufficiently favorable.
Black Holes
- Black holes were long considered a mathematical curiosity. In the 1960s, theoretical work showed they were a generic prediction of general relativity. The discovery of neutron stars sparked interest in gravitationally collapsed compact objects as a possible astrophysical reality.
- Karl Schwarzschild found the first modern solution of general relativity that would characterize a black hole in 1916, although its interpretation as a region of space from which nothing can escape was first published by David Finkelstein in 1958.
- It is postulated that stellar-mass black holes are expected to form when very massive stars collapse at the end of their life cycle. After a black hole has formed, it can continue to grow by absorbing mass from its surroundings. By absorbing other stars and merging with other black holes, supermassive black holes of millions of solar masses may form. There is general consensus that supermassive black holes exist in the centers of most galaxies.
Conjecture
While these are curious phenomena, some questions remain:
- Where do edges (asymptotic to a plane) end?
- If light is electromagnetic and can’t escape, how can gravity, if it is electromagnetic?
- How can they exist in a Big Bang universe if they have an infinite lifespan?
- There is also a problem with the divide-by-zero singularity.
Vision
Schwarzschild’s work provided a crucial early understanding of the extreme gravitational phenomena predicted by general relativity, paving the way for the modern study of black holes.
Legacy
Schwarzschild’s solution to Einstein’s field equations remains a cornerstone of general relativity and our understanding of black holes. His contributions continue to shape astrophysical research.