Editing Karl Schwarzschild

{{Short description|German physicist (1873–1916)}}
{{Infobox scientist
| name              = Karl Schwarzschild
| image             = Karl schwarzschild.portrait.jpg
| image_size        = 190
| caption           =
| birth_date        = {{birth date|1873|10|9|df=y}}
| birth_place       = [[Frankfurt am Main]], [[German Empire]]
| death_date        = {{death date and age|1916|5|11|1873|10|9|df=y}}<ref>[http://zelmanov.ptep-online.com/papers/zj-2008-b3.pdf Biography of Karl Schwarzschild] by Indranu Suhendro, ''The Abraham Zelmanov Journal'', 2008, Volume 1.</ref>{{rp|xix}}
| death_place       = [[Potsdam]], German Empire
| fields             = [[Physics]]<br />[[Astronomy]]
|children = 3, including [[Agathe Thornton|Agathe]] and [[Martin Schwarzschild|Martin]]
| work_institutions = 
| alma_mater        = [[Ludwig Maximilian University of Munich]]<br />[[University of Strasbourg]]
| doctoral_advisor  = [[Hugo von Seeliger]]
| doctoral_students = 
| known_for         = [[Schwarzschild metric]]<br>[[Schwarzschild coordinates]]<br>[[Schwarzschild radius]]<br>[[Schwarzschild law]]<br>[[Interior Schwarzschild metric]]<br>[[Schwarzschild's equation for radiative transfer]]<br>[[Schwarzschild telescope]]<br>[[Conic constant]]<br>[[Lagrangian (field theory)|Langrangian of electromagnetism]]
| prizes            = 
| religion          = 
| footnotes         = 
| module            = {{Infobox military person
 |embed  = yes
 | allegiance = {{flag|German Empire}} 
 | branch = [[German Army (German Empire)|Imperial German Army]]
 | serviceyears = 1914–1916
 | rank = [[Leutnant|Lieutenant]] 
 | battles = [[World War I]]
 }} 
}}

'''Karl Schwarzschild''' ({{IPA|de|kaʁl ˈʃvaʁtsʃɪlt|lang|LL-Q188 (deu)-Shisma-Schwarzschild.wav}}; 9 October 1873 – 11 May 1916) was a German [[physicist]] and astronomer.

Schwarzschild provided the first [[Exact solutions in general relativity|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 [[Autoimmunity|autoimmune disease]] [[pemphigus]], which he developed while at the [[Eastern Front (World War I)|Russian front]].<ref>{{cite book |last1=Snygg |first1=John |title=A new approach to differential geometry using Clifford's geometric algebra |date=2012 |publisher=Springer Science |location=New York |isbn=978-0-8176-8283-5 |page=400 |doi=10.1007/978-0-8176-8283-5 |url=https://link.springer.com/book/10.1007/978-0-8176-8283-5}}</ref><ref>{{cite book |last1=Ahsan |first1=Zafar |title=Tensors : mathematics of differential geometry and relativity |date=2015 |publisher=Prentice Hall India |location=Delhi |isbn=9788120350885 |page=205 }}</ref>

Asteroid [[837 Schwarzschilda]] is named in his honour, as is the large crater [[Schwarzschild (crater)|''Schwarzschild'']], on the far side of the [[Moon]].<ref>{{GPN|5389|name=Crater Schwarzschild}}</ref>

==Life==
Karl Schwarzschild was born on 9 October 1873 in [[Frankfurt|Frankfurt on Main]], the eldest of six boys and one girl,<ref name=":0">{{Cite web |title=The mystery of the dark bodies |url=https://www.mpg.de/11225504/the-mystery-of-the-dark-bodies |access-date=2022-05-15 |website=www.mpg.de |language=en}}</ref><ref>{{Cite web |title=Alfred Schwarzschild Biography |url=http://alfredschwarzschild.com/Alfred_Schwarzschild_Biography.html |access-date=2022-05-15 |website=alfredschwarzschild.com}}</ref> to [[History of the Jews in Germany|Jewish]] parents. His father was active in the [[business sector|business community]] of the city, and the family had ancestors in Frankfurt from the sixteenth century onwards.<ref>{{Cite book|url= https://www.worldcat.org/oclc/1244019080|title= "Nachforschung der Wahrheit" von der alten Lateinschule zum Lessing-Gymnasium in Frankfurt am Main : Festschrift zum 500-jährigen Jubiläum der Schule|date= 2020 |others=Bernhard Mieles, Carolin Ritter, Christoph Wolf, Lessing-Gymnasium Frankfurt am Main, Frankfurter Societäts-Medien GmbH|isbn=978-3-95542-379-7|location=Frankfurt am Main|oclc=1244019080}}</ref> The family owned two fabric stores in Frankfurt. His brother Alfred became a painter.<ref>{{Citation|last=Schwarzschild|first=Karl|title=Karl Schwarzschild Lectures|date= 1992|url= http://dx.doi.org/10.1007/978-3-642-58086-4_2|work= Gesammelte Werke Collected Works|pages= 29–42|place=Berlin, Heidelberg|publisher=Springer Berlin Heidelberg|doi=10.1007/978-3-642-58086-4_2 |isbn=978-3-642-63467-3|access-date= 2021-05-18}}</ref> The young Schwarzschild attended a Jewish primary school until 11 years of age<ref>{{Cite journal|date=2016-01-18|title=MacTutor History of Mathematics Archive|url= http://dx.doi.org/10.1108/rr-08-2015-0205|journal=Reference Reviews|volume=30|issue=1|pages=27–28|doi= 10.1108/rr-08-2015-0205|issn=0950-4125}}</ref> and then the [[Lessing-Gymnasium, Frankfurt|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>{{Cite book|url= https://www.worldcat.org/oclc/981916699|title=Karl Schwarzschild (1873-1916) ein Pionier und Wegbereiter der Astrophysik|date=2017|others=Klaus Reinsch, Axel Wittmann, Universitätsverlag Göttingen|isbn=978-3-86395-295-2|location=Göttingen|oclc=981916699}}</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>{{Cite journal|last= Hertzsprung|first= Ejnar|date= June 1917|title= Karl Schwarzschild|journal=The Astrophysical Journal |language= en|volume=45|pages=285|doi=10.1086/142329|bibcode= 1917ApJ....45..285H|issn=0004-637X|doi-access=free}}</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 (astronomy)|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>{{Citation|last=Schwarzschild|first=Karl|title=Biography of Karl Schwarzschild (1873-1916)|date= 1992|url=http://dx.doi.org/10.1007/978-3-642-58086-4_1|work=Gesammelte Werke Collected Works|pages=1–28|place=Berlin, Heidelberg|publisher=Springer Berlin Heidelberg|doi=10.1007/978-3-642-58086-4_1 |isbn=978-3-642-63467-3|access-date=2021-05-18}}</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.{{citation needed|date= July 2021}}
[[File:Frederick Stratton, John William Nicholson, K. Schwarzschild, Frank Watson Dyson ride in automobile.jpg|alt=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|left|thumb|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|thumb|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|thumb|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 [[Imperial German Army|German army]] despite being over 40 years old. He served on both the western and eastern fronts, specifically helping with [[Ballistics|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">{{Cite web |title=Karl Schwarzschild - Important Scientists - The Physics of the Universe |url=https://www.physicsoftheuniverse.com/scientists_schwarzschild.html |access-date=2022-05-15 |website=www.physicsoftheuniverse.com}}</ref> Nevertheless, he managed to write three outstanding papers, two on the [[theory of relativity]] and one on [[Quantum mechanics|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">{{cite journal |last1=Levy |first1=Adam |title=How black holes morphed from theory to reality |journal=Knowable Magazine |date=January 11, 2021 |doi=10.1146/knowable-010921-1|s2cid=250662997 |doi-access=free |url=https://knowablemagazine.org/article/physical-world/2021/how-black-holes-morphed-theory-reality |access-date=25 March 2022}}</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)|Stadtfriedhof Göttingen]].

With his wife Else he had three children:

* [[Agathe Thornton]] (1910–2006) emigrated to Great Britain in 1933. In 1946, she moved to New Zealand, where she became a classics professor at the [[University of Otago]] in Dunedin.<ref>{{Cite web |last1=Graham |first1=Reg |last2=Taonga |first2=New Zealand Ministry for Culture and Heritage Te Manatu |title=Agathe Thornton |url=https://teara.govt.nz/mi/photograph/43514/agathe-thornton |access-date=2022-05-15 |website=teara.govt.nz |language=mi}}</ref>
* [[Martin Schwarzschild]] (1912–1997) became a professor of astronomy at [[Princeton University]].<ref>{{Cite web |title=Princeton - News - Princeton Astrophysicist Martin Schwarzschild Dies |url=https://pr.princeton.edu/news/97/q2/0411schw.html |access-date=2022-05-15 |website=pr.princeton.edu}}</ref>
* Alfred Schwarzschild (1914–1944) remained in Nazi Germany and was murdered during the [[The Holocaust|Holocaust]].<ref>{{Cite book |last1=Nicolini |first1=Piero |url=https://books.google.com/books?id=6B_UCgAAQBAJ&dq=alfred+schwarzschild+holocaust+son+of+karl&pg=PA10 |title=1st Karl Schwarzschild Meeting on Gravitational Physics |last2=Kaminski |first2=Matthias |last3=Mureika |first3=Jonas |last4=Bleicher |first4=Marcus |publisher=Springer |year=2015 |isbn=9783319200460 |pages=10}}</ref>

==Work==
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 (astronomy)|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 star]]s, using photography, and the improvement of optical systems, through the perturbative investigation of geometrical aberrations.

===Physics of photography===
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===

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 (field theory)|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>{{Cite journal |last1=Wheeler |first1=John Archibald |last2=Feynman |first2=Richard Phillips |date=1949-07-01 |title=Classical Electrodynamics in Terms of Direct Interparticle Action |journal=Reviews of Modern Physics |language=en |volume=21 |issue=3 |pages=425–433 |doi=10.1103/RevModPhys.21.425 |issn=0034-6861|doi-access=free |bibcode=1949RvMP...21..425W }}</ref> and constitutes an alternative but equivalent formulation of electrodynamics.

=== Thermal radiation ===
{{Main|Schwarzschild's equation for radiative transfer}}
In 1906, Schwarzchild developed the concept of radiative equilibrium between convection inside the Sun and thermal radiation at the surface. He developed an [[Schwarzschild's equation for radiative transfer|equation for radiative transfer]] and proved that, in accordance to data, the Sun’s [[photosphere]] is in radiative equilibrium.<ref name="mobley2019">{{cite web |last1=Mobley |first1=Curtis |title=Notes on The Evolution of Radiative Transfer Theory |url=https://www.oceanopticsbook.info/packages/iws_l2h/conversion/files/Mobley_EvolutionRTT_draft.pdf |access-date=19 October 2023 |website=Ocean Optics}}</ref>

===Relativity===
[[Image:Schwarzschild circular radii.svg|thumb|250px|The [[Kepler problem]] in general relativity, using the [[Schwarzschild metric]]]]
{{main|Deriving the Schwarzschild solution}}
Einstein himself was pleasantly surprised to learn that the [[Einstein field equations|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>[https://einsteinpapers.press.princeton.edu/vol8-trans/191?highlightText=schwarzschild 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:

{{Blockquote|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]]|<ref name="Eisenstaedt, 1989"/>}}

[[Image:Schwarzschild interior.jpg|thumb|250px|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 [[Mathematical singularity|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==
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==
The entire scientific estate of Karl Schwarzschild is stored in a special collection of the [https://www.sub.uni-goettingen.de/sub-aktuell/ Lower Saxony National- and University Library of Göttingen].

'''Relativity'''
* ''[[s:de:Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie|Über das Gravitationsfeld eines Massenpunktes nach der Einstein’schen Theorie.]]'' Reimer, Berlin 1916, S. 189 ff. (Sitzungsberichte der Königlich-Preussischen Akademie der Wissenschaften; 1916)
* ''Über das Gravitationsfeld einer Kugel aus inkompressibler Flüssigkeit.'' Reimer, Berlin 1916, S. 424-434 (Sitzungsberichte der Königlich-Preussischen Akademie der Wissenschaften; 1916)

'''Other papers'''
* ''Untersuchungen zur geometrischen Optik I. Einleitung in die Fehlertheorie optischer Instrumente auf Grund des Eikonalbegriffs'', 1906, [http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN251726223_0004 Abhandlungen der Gesellschaft der Wissenschaften in Göttingen, Band 4], Nummero 1, S. 1-31
* ''Untersuchungen zur geometrischen Optik II. Theorie der Spiegelteleskope'', 1906, [http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN251726223_0004 Abhandlungen der Gesellschaft der Wissenschaften in Göttingen, Band 4], Nummero 2, S. 1-28
* ''Untersuchungen zur geometrischen Optik III. Über die astrophotographischen Objektive'', 1906, [http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN251726223_0004 Abhandlungen der Gesellschaft der Wissenschaften in Göttingen, Band 4], Nummero 3, S. 1-54
* ''[http://dz-srv1.sub.uni-goettingen.de/contentserver/contentserver?command=docconvert&docid=D58230 Über Differenzformeln zur Durchrechnung optischer Systeme]{{dead link|date=December 2017 |bot=InternetArchiveBot |fix-attempted=yes }}'', [http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN252457811_1907 1907, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen], S. 551-570
* ''Aktinometrie der Sterne der B. D. bis zur Größe 7.5 in der Zone 0° bis +20° Deklination. Teil A. Unter Mitwirkung von Br. Meyermann, A. Kohlschütter und O. Birck'', 1910, [http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN251726223_0006 Abhandlungen der Gesellschaft der Wissenschaften in Göttingen, Band 6], Numero 6, S. 1-117
* ''[http://dz-srv1.sub.uni-goettingen.de/contentserver/contentserver?command=docconvert&docid=D58114 Über das Gleichgewicht der Sonnenatmosphäre]{{dead link|date=December 2017 |bot=InternetArchiveBot |fix-attempted=yes }}'', [http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN252457811_1906 1906, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen], S. 41-53
* ''[http://dz-srv1.sub.uni-goettingen.de/contentserver/contentserver?command=docconvert&docid=D36450 Die Beugung und Polarisation des Lichts durch einen Spalt. I.]{{dead link|date=December 2017 |bot=InternetArchiveBot |fix-attempted=yes }}'', 1902, Mathematische Annalen, [http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN235181684_0055 Band 55], S. 177-247
* ''[http://dz-srv1.sub.uni-goettingen.de/contentserver/contentserver?command=docconvert&docid=D57298 Zur Elektrodynamik. I. Zwei Formen des Princips der Action in der Elektronentheorie]{{dead link|date=December 2017 |bot=InternetArchiveBot |fix-attempted=yes }}'', [http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN252457811_1903 1903, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen], S. 126-131
* ''[http://dz-srv1.sub.uni-goettingen.de/contentserver/contentserver?command=docconvert&docid=D57299 Zur Elektrodynamik. II. Die elementare elektrodynamische Kraft]{{dead link|date=December 2017 |bot=InternetArchiveBot |fix-attempted=yes }}'', [http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN252457811_1903 1903, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen], S. 132-141
* ''[http://dz-srv1.sub.uni-goettingen.de/contentserver/contentserver?command=docconvert&docid=D57299 Zur Elektrodynamik. III. Ueber die Bewegung des Elektrons]{{dead link|date=December 2017 |bot=InternetArchiveBot |fix-attempted=yes }}'', [http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN252457811_1903 1903, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen], S. 245-278
* ''[http://dz-srv1.sub.uni-goettingen.de/contentserver/contentserver?command=docconvert&docid=D58241 Ueber die Eigenbewegungen der Fixsterne]{{dead link|date=December 2017 |bot=InternetArchiveBot |fix-attempted=yes }}'', [http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN252457811_1907 1907, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen], S. 614-632
* ''[http://dz-srv1.sub.uni-goettingen.de/contentserver/contentserver?command=docconvert&docid=D62941 Ueber die Bestimmung von Vertex und Apex nach der Ellipsoidhypothese aus einer geringeren Anzahl beobachteter Eigenbewegungen]{{dead link|date=December 2017 |bot=InternetArchiveBot |fix-attempted=yes }}'', [http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN252457811_1908 1908, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen], S. 191-200
* K. Schwarzschild, E. Kron: ''[http://dz-srv1.sub.uni-goettingen.de/contentserver/contentserver?command=docconvert&docid=D63058 Ueber die Helligkeitsverteilung im Schweif des Halley´schen Kometen]{{dead link|date=December 2017 |bot=InternetArchiveBot |fix-attempted=yes }}'', [http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN252457811_1911 1911, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen], S. 197-208
* ''[http://dz-srv1.sub.uni-goettingen.de/contentserver/contentserver?command=docconvert&docid=D244900 Die naturwissenschaftlichen Ergebnisse und Ziele der neueren Mechanik.]{{dead link|date=December 2017 |bot=InternetArchiveBot |fix-attempted=yes }}'', 1904, [http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN37721857X_0013 Jahresbericht der Deutschen Mathematiker-Vereinigung, Band 13], S. 145-156
* ''[http://dz-srv1.sub.uni-goettingen.de/contentserver/contentserver?command=docconvert&docid=D244507 Über die astronomische Ausbildung der Lehramtskandidaten.]{{dead link|date=December 2017 |bot=InternetArchiveBot |fix-attempted=yes }}'', 1907, [http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN37721857X_0016 Jahresbericht der Deutschen Mathematiker-Vereinigung, Band 16], S. 519-522

'''English translations'''
* ''On the Gravitational Field of a Point-Mass, According to Einstein's Theory'', [https://web.archive.org/web/20220331132918/http://zelmanov.ptep-online.com/papers/zj-2008-03.pdf The Abraham Zelmanov Journal, 2008, Volume 1, P. 10-19]
* ''On the Gravitational Field of a Sphere of Incompressible Liquid, According to Einstein's Theory'', [http://zelmanov.ptep-online.com/papers/zj-2008-04.pdf The Abraham Zelmanov Journal, 2008, Volume 1, P. 20-32]
* ''On the Permissible Numerical Value of the Curvature of Space'', [http://zelmanov.ptep-online.com/papers/zj-2008-06.pdf The Abraham Zelmanov Journal, Volume 1, 2008, pp. 64-73]

==See also==
*[[List of things named after Karl Schwarzschild]]

==References==
{{Reflist}}

==External links==
{{commons category}}
{{Wikisourcelang|de|Karl Schwarzschild|Works by Karl Schwarzschild}}
* {{MacTutor Biography|id=Schwarzschild}}
* Roberto B. Salgado [https://web.archive.org/web/20060617222412/http://www.phy.syr.edu/courses/modules/LIGHTCONE/schwarzschild.html The Light Cone: The Schwarzschild Black Hole]
* [http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1917ApJ....45..285H Obituary in the Astrophysical Journal], written by [[Ejnar Hertzsprung]]
* {{MathGenealogy|id=52349}}
* [http://zelmanov.ptep-online.com/papers/zj-2008-b3.pdf Biography of Karl Schwarzschild] {{Webarchive|url=https://web.archive.org/web/20210302153030/http://zelmanov.ptep-online.com/papers/zj-2008-b3.pdf |date=2021-03-02 }} by Indranu Suhendro, ''The Abraham Zelmanov Journal'', 2008, Volume 1.

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