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Karl Schwarzschild

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Karl Schwarzschild (Template:IPA; 9 October 1873 – 11 May 1916) was a German physicist and astronomer.

Schwarzschild provided the first exact solution to the Einstein field equations of general relativity, for the limited case of a single spherical non-rotating mass, which he accomplished in 1915, the same year that Einstein first introduced general relativity. The Schwarzschild solution, which makes use of Schwarzschild coordinates and the Schwarzschild metric, leads to a derivation of the Schwarzschild radius, which is the size of the event horizon of a non-rotating black hole.

Schwarzschild accomplished this while serving in the German army during World War I. He died the following year, possibly from the autoimmune disease pemphigus, which he developed while at the Russian front.<ref>Template:Cite book</ref><ref>Template:Cite book</ref>

Asteroid 837 Schwarzschilda is named in his honour, as is the large crater Schwarzschild, on the far side of the Moon.<ref>Template:GPN</ref>

Life

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Karl Schwarzschild was born on 9 October 1873 in Frankfurt on Main, the eldest of six boys and one girl,<ref name=":0">Template:Cite web</ref><ref>Template:Cite web</ref> to Jewish parents. His father was active in the business community of the city, and the family had ancestors in Frankfurt from the sixteenth century onwards.<ref>Template:Cite book</ref> The family owned two fabric stores in Frankfurt. His brother Alfred became a painter.<ref>Template:Citation</ref> The young Schwarzschild attended a Jewish primary school until 11 years of age<ref>Template:Cite journal</ref> and then the Lessing-Gymnasium (secondary school). He received an all-encompassing education, including subjects like Latin, Ancient Greek, music and art, but developed a special interest in astronomy early on.<ref>Template:Cite book</ref> In fact he was something of a child prodigy, having two papers on binary orbits (celestial mechanics) published before the age of sixteen.<ref>Template:Cite journal</ref>

After graduation in 1890, he attended the University of Strasbourg to study astronomy. After two years he transferred to the Ludwig Maximilian University of Munich where he obtained his doctorate in 1896 for a work on Henri Poincaré's theories.

From 1897, he worked as assistant at the Kuffner Observatory in Vienna. His work here concentrated on the photometry of star clusters and laid the foundations for a formula linking the intensity of the starlight, exposure time, and the resulting contrast on a photographic plate. An integral part of that theory is the Schwarzschild exponent (astrophotography). In 1899, he returned to Munich to complete his Habilitation.

From 1901 until 1909, he was a professor at the prestigious Göttingen Observatory within the University of Göttingen,<ref>Template:Citation</ref> where he had the opportunity to work with some significant figures, including David Hilbert and Hermann Minkowski. Schwarzschild became the director of the observatory. He married Else Rosenbach, a great-granddaughter of Friedrich Wöhler and daughter of a professor of surgery at Göttingen, in 1909. Later that year they moved to Potsdam, where he took up the post of director of the Astrophysical Observatory. This was then the most prestigious post available for an astronomer in Germany.Template:Citation needed

Schwarzschild, third from left in the automobile; possibly during the Fifth Conference of the International Union for Co-operation in Solar Research, held in Bonn, Germany
Schwarzschild, third from left in the automobile; possibly during the Fifth Conference of the International Union for Co-operation in Solar Research, held in Bonn, Germany
File:Grave of Karl Schwarzschild at Stadtfriedhof Göttingen 2017 01.jpg
Karl Schwarzschild's grave at Stadtfriedhof (Göttingen)
File:Delegates to the Fourth Conference International Union for Cooperation in Solar Research at Mount Wilson Observatory.jpg
Schwarzschild at the Fourth Conference International Union for Cooperation in Solar Research at Mount Wilson Observatory, 1910

From 1912, Schwarzschild was a member of the Prussian Academy of Sciences.

At the outbreak of World War I in 1914, Schwarzschild volunteered for service in the German army despite being over 40 years old. He served on both the western and eastern fronts, specifically helping with ballistic calculations and rising to the rank of second lieutenant in the artillery.<ref name=":0" />

While serving on the front in Russia in 1915, he began to suffer from pemphigus, a rare and painful autoimmune skin-disease.<ref name=":1">Template:Cite web</ref> Nevertheless, he managed to write three outstanding papers, two on the theory of relativity and one on quantum theory. His papers on relativity produced the first exact solutions to the Einstein field equations, and a minor modification of these results gives the well-known solution that now bears his name — the Schwarzschild metric.<ref name="Levy">Template:Cite journal</ref>

In March 1916, Schwarzschild left military service because of his illness and returned to Göttingen. Two months later, on May 11, 1916, his struggle with pemphigus may have led to his death at the age of 42.<ref name=":1" />

He rests in his family grave at the Stadtfriedhof Göttingen.

With his wife Else he had three children:

Work

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Thousands of dissertations, articles, and books have since been devoted to the study of Schwarzschild's solutions to the Einstein field equations. However, although his best known work lies in the area of general relativity, his research interests were extremely broad, including work in celestial mechanics, observational stellar photometry, quantum mechanics, instrumental astronomy, stellar structure, stellar statistics, Halley's Comet, and spectroscopy.<ref name="Eisenstaedt, 1989">Eisenstaedt, “The Early Interpretation of the Schwarzschild Solution,” in D. Howard and J. Stachel (eds), Einstein and the History of General Relativity: Einstein Studies, Vol. 1, pp. 213-234. Boston: Birkhauser, 1989.</ref>

Some of his particular achievements include measurements of variable stars, using photography, and the improvement of optical systems, through the perturbative investigation of geometrical aberrations.

Physics of photography

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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 <math>p</math> in the formula:

<math>i = f ( I \,t^p )</math>

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

Electrodynamics

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According to Wolfgang Pauli,<ref>Pauli, W.. Theory of Relativity. United States, Dover Publications, 2013.</ref> Schwarzschild is the first to introduce the correct Lagrangian formalism of the electromagnetic field<ref name= "schwarzschil1903">K. Schwarzschild, Nachr. ges. Wiss. Gottingen (1903) 125</ref> as

<math> S = \frac12 \int (H^2-E^2) \mathrm dV + \int \rho(\phi - \mathbf{A} \cdot \mathbf{u}) \mathrm dV </math>

where <math> \mathbf{E},\mathbf{H} </math> are the electric and applied magnetic fields, <math>\mathbf{A}</math> is the vector potential and <math>\phi</math> 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<ref name= "schwarzschil1903b">K. Schwarzschild, Nachr. ges. Wiss. Gottingen (1903) 128,132</ref>

<math>

S=\sum_{i}m_{i}\int_{C_{i}}\mathrm 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)\mathrm d\mathbf{s}_{i}\mathrm d\mathbf{s}_{j} </math>

where <math> C_\alpha </math> are the world lines of the particle, <math> d\mathbf{s}_{\alpha} </math> 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 <math> \delta\left(\left\Vert P_{i}P_{j}\right\Vert \right) </math>. The idea was further developed by Hugo Tetrode<ref>H. Tetrode, Zeitschrift für Physik 10:137, 1922</ref> and Adriaan Fokker<ref>A. D. Fokker, Zeitschrift für Physik 58:386, 1929</ref> in the 1920s and John Archibald Wheeler and Richard Feynman in the 1940s<ref>Template:Cite journal</ref> and constitutes an alternative but equivalent formulation of electrodynamics.

Thermal radiation

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Template:Main In 1906, Schwarzchild developed the concept of radiative equilibrium between convection inside the Sun and thermal radiation at the surface. He developed an equation for radiative transfer and proved that, in accordance to data, the Sun’s photosphere is in radiative equilibrium.<ref name="mobley2019">Template:Cite web</ref>

Relativity

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File:Schwarzschild circular radii.svg
The Kepler problem in general relativity, using the Schwarzschild metric

Template:Main 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 produced only an approximate solution.<ref name="Levy"/> 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 he was serving in the war stationed on the Russian front. He concluded the letter by writing: "As you see, the war is kindly disposed toward me, allowing me, despite fierce gunfire at a decidedly terrestrial distance, to take this walk into this your land of ideas."<ref>Letter from K Schwarzschild to A Einstein dated 22 December 1915, in "The Collected Papers of Albert Einstein, Volume 8: The Berlin Years: Correspondence, 1914-1918 (English translation supplement)", Translated by Ann M. Hentschel, vol.8a, doc.#169.</ref> In 1916, Einstein wrote to Schwarzschild on this result:

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File:Schwarzschild interior.jpg
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 his coordinates, this singularity lies on the sphere of points at a particular radius, called the Schwarzschild radius:

<math>

R_{s} = \frac{2GM}{c^{2}} </math>

where G is the gravitational constant, M is the mass of the central body, and c is the speed of light in vacuum.<ref name="landau_1975" >Landau 1975.</ref> In cases where the radius of the central body is less than the Schwarzschild radius, <math>R_{s}</math> 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).

Cultural references

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Karl Schwarzschild appears as a character in the science fiction short story "Schwarzschild Radius" (1987) by Connie Willis.

Karl Schwarzchild appears as a fictionalized character in the story “Schwarzchild’s Singularity” in the collection "When We Cease to Understand the World" (2020) by Benjamín Labatut.

Works

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The entire scientific estate of Karl Schwarzschild is stored in a special collection of the Lower Saxony National- and University Library of Göttingen.

Relativity

Other papers

English translations

See also

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References

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