Karl Schwarzschild

While at Vienna in 1897, Schwarzschild developed a formula, now known as the Schwarzschild law, to calculate the optical density of photographic material. It involved an exponent now known as the Schwarzschild exponent, which is the ${\displaystyle p}$ in the formula:

${\displaystyle i=f(I\cdot t^{p})}$

(where ${\displaystyle i}$ is optical density of exposed photographic emulsion, a function of ${\displaystyle I}$, the intensity of the source being observed, and ${\displaystyle t}$, the exposure time, with ${\displaystyle p}$ a constant). This formula was important for enabling more accurate photographic measurements of the intensities of faint astronomical sources.

According to Wolfgang Pauli (Theory of relativity), Schwarzschild is the first to introduce the correct Lagrangian formalism of the electromagnetic field [7] as

${\displaystyle S=(1/2)\int (H^{2}-E^{2})dV+\int \rho (\phi -{\vec {A}}{\vec {u}})dV}$

where ${\displaystyle {\vec {E}},{\vec {H}}}$ are the electric and magnetic field, ${\displaystyle {\vec {A}}}$ is the vector potential and ${\displaystyle \phi }$ is the electric potential.

He also introduced a field free variational formulation of electrodynamics (also known as "action at distance" or "direct interparticle action") based only on the world line of particles as [8]

${\displaystyle S=\sum _{i}m_{i}\int _{C_{i}}ds_{i}+{\frac {1}{2}}\sum _{i,j}\iint _{C_{i},C_{j}}q_{i}q_{j}\delta \left(\left\Vert P_{i}P_{j}\right\Vert \right)d\mathbf {s} _{i}d\mathbf {s} _{j}}$

where ${\displaystyle C_{\alpha }}$ are the world lines of the particle, ${\displaystyle d\mathbf {s} _{\alpha }}$ the (vectorial) arc element along the world line. Two points on two world lines contribute to the Lagrangian (are coupled) only if they are a zero Minkowskian distance (connected by a light ray), hence the term ${\displaystyle \delta \left(\left\Vert P_{i}P_{j}\right\Vert \right)}$. The idea was further developed by Tetrode [9] and Fokker [10] in the 1920s and Wheeler and Feynman in the 1940s [11] and constitutes an alternative/equivalent formulation of electrodynamics.

The Kepler problem in general relativity, using the Schwarzschild metric

Einstein himself was pleasantly surprised to learn that the field equations admitted exact solutions, because of their prima facie complexity, and because he himself had only produced an approximate solution. Einstein's approximate solution was given in his famous 1915 article on the advance of the perihelion of Mercury. There, Einstein used rectangular coordinates to approximate the gravitational field around a spherically symmetric, non-rotating, non-charged mass. Schwarzschild, in contrast, chose a more elegant "polar-like" coordinate system and was able to produce an exact solution which he first set down in a letter to Einstein of 22 December 1915, written while Schwarzschild was serving in the war stationed on the Russian front. Schwarzschild concluded the letter by writing: "As you see, the war treated me kindly enough, in spite of the heavy gunfire, to allow me to get away from it all and take this walk in the land of your ideas."[12] In 1916, Einstein wrote to Schwarzschild on this result:

I have read your paper with the utmost interest. I had not expected that one could formulate the exact solution of the problem in such a simple way. I liked very much your mathematical treatment of the subject. Next Thursday I shall present the work to the Academy with a few words of explanation.

— Albert Einstein, [6]

Boundary region of Schwarzschild interior and exterior solution

Schwarzschild's second paper, which gives what is now known as the "Inner Schwarzschild solution" (in German: "innere Schwarzschild-Lösung"), is valid within a sphere of homogeneous and isotropic distributed molecules within a shell of radius r=R. It is applicable to solids; incompressible fluids; the sun and stars viewed as a quasi-isotropic heated gas; and any homogeneous and isotropic distributed gas.

Schwarzschild's first (spherically symmetric) solution does not contain a coordinate singularity on a surface that is now named after him. In Schwarzschild coordinates, this singularity lies on the sphere of points at a particular radius, called the Schwarzschild radius:

${\displaystyle R_{s}={\frac {2GM}{c^{2}}}}$

where G is the gravitational constant, M is the mass of the central body, and c is the speed of light in a vacuum.[13] In cases where the radius of the central body is less than the Schwarzschild radius, ${\displaystyle R_{s}}$ represents the radius within which all massive bodies, and even photons, must inevitably fall into the central body (ignoring quantum tunnelling effects near the boundary). When the mass density of this central body exceeds a particular limit, it triggers a gravitational collapse which, if it occurs with spherical symmetry, produces what is known as a Schwarzschild black hole. This occurs, for example, when the mass of a neutron star exceeds the Tolman-Oppenheimer-Volkoff limit (about three solar masses).