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49-487: In Lorentzian geometry, a number of Ricci-flat metrics are known from works of Karl Schwarzschild , Roy Kerr , and Yvonne Choquet-Bruhat . In Riemannian geometry , Shing-Tung Yau 's resolution of the Calabi conjecture produced a number of Ricci-flat metrics on Kähler manifolds . A pseudo-Riemannian manifold is said to be Ricci-flat if its Ricci curvature is zero. It is direct to verify that, except in dimension two,

98-494: A closed manifold is flat. His work, using techniques of partial differential equations , established a comprehensive existence theory for Ricci-flat metrics in the special case of Kähler metrics on closed complex manifolds . Due to his analytical techniques, the metrics are non-explicit even in the simplest cases. Such Riemannian manifolds are often called Calabi–Yau manifolds , although various authors use this name in slightly different ways. Relative to harmonic coordinates ,

147-692: A Riemannian manifold may also be characterized (roughly speaking) by the existence of a 2-sphere of complex structures which are all parallel . This says in particular that every hyperkähler metric is Kähler; furthermore, via the Ambrose–Singer theorem , every such metric is Ricci-flat. The Calabi–Yau theorem specializes to this context, giving a general existence and uniqueness theorem for hyperkähler metrics on compact Kähler manifolds admitting holomorphically symplectic structures. Examples of hyperkähler metrics on noncompact spaces had earlier been obtained by Eugenio Calabi . The Eguchi–Hanson space , discovered at

196-686: 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 . 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 Harmonic coordinate The harmonic coordinate condition

245-570: A harmonic coordinate system is the closest approximation available in general relativity to an inertial frame of reference in special relativity. In general relativity, we have to use the covariant derivative instead of the partial derivative in d'Alembert's equation, so we get: Since the coordinate x is not actually a scalar, this is not a tensor equation. That is, it is not generally invariant. But coordinate conditions must not be generally invariant because they are supposed to pick out (only work for) certain coordinate systems and not others. Since

294-470: A metric is Ricci-flat if and only if its Einstein tensor is zero. Ricci-flat manifolds are one of three special types of Einstein manifold , arising as the special case of scalar curvature equaling zero. From the definition of the Weyl curvature tensor , it is direct to see that any Ricci-flat metric has Weyl curvature equal to Riemann curvature tensor . By taking traces , it is straightforward to see that

343-660: A minor modification of these results gives the well-known solution that now bears his name — the Schwarzschild metric . 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. He rests in his family grave at the Stadtfriedhof Göttingen . With his wife Else he had three children: Thousands of dissertations, articles, and books have since been devoted to

392-508: A nonlinear geometric analogue of the difference between the Laplace equation and the wave equation . Yau's existence theorem for Ricci-flat Kähler metrics established the precise topological condition under which such a metric exists on a given closed complex manifold : the first Chern class of the holomorphic tangent bundle must be zero. The necessity of this condition was previously known by Chern–Weil theory . Beyond Kähler geometry,

441-495: A painter. The young Schwarzschild attended a Jewish primary school until 11 years of age 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. In fact he was something of a child prodigy, having two papers on binary orbits ( celestial mechanics ) published before

490-424: 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

539-567: Is real-analytic . This also holds in the broader setting of Einstein Riemannian metrics. Analogously, relative to harmonic coordinates, Ricci-flatness of a Lorentzian metric can be interpreted as a system of hyperbolic partial differential equations . Based on this perspective, Yvonne Choquet-Bruhat developed the well-posedness of the Ricci-flatness condition. She reached a definitive result in collaboration with Robert Geroch in

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588-413: Is Ricci-flat if and only if the holonomy group is contained in the special unitary group . On a general Kähler manifold, the if direction still holds, but only the restricted holonomy group of a Ricci-flat Kähler metric is necessarily contained in the special unitary group. A hyperkähler manifold is a Riemannian manifold whose holonomy group is contained in the symplectic group . This condition on

637-500: Is a Riemannian manifold whose holonomy group is contained in the Lie groups Spin(7) or G 2 . The Ambrose–Singer theorem implies that any such manifold is Ricci-flat. The existence of closed manifolds of this type was established by Dominic Joyce in the 1990s. Marcel Berger commented that all known examples of irreducible Ricci-flat Riemannian metrics on simply-connected closed manifolds have special holonomy groups, according to

686-503: Is already indicated by the fundamental distinction between the geodesically complete metrics which are typical of Riemannian geometry and the maximal globally hyperbolic developments which arise from Choquet-Bruhat and Geroch's work. Moreover, the analyticity and corresponding unique continuation of a Ricci-flat Riemannian metric has a fundamentally different character than Ricci-flat Lorentzian metrics, which have finite speeds of propagation and fully localizable phenomena. This can be viewed as

735-693: Is not an invariant scalar, and so its covariant derivative is not the same as its ordinary derivative. Rather, − g ; ρ = 0 {\displaystyle {\sqrt {-g}}_{;\rho }=0\!} because g ; ρ μ ν = 0 {\displaystyle g_{;\rho }^{\mu \nu }=0\!} , while − g , ρ = − g Γ σ ρ σ . {\displaystyle {\sqrt {-g}}_{,\rho }={\sqrt {-g}}\Gamma _{\sigma \rho }^{\sigma }\,.} Contracting ν with ρ and applying

784-609: Is one of several coordinate conditions in general relativity , which make it possible to solve the Einstein field equations . A coordinate system is said to satisfy the harmonic coordinate condition if each of the coordinate functions x (regarded as scalar fields) satisfies d'Alembert's equation . The parallel notion of a harmonic coordinate system in Riemannian geometry is a coordinate system whose coordinate functions satisfy Laplace's equation . Since d'Alembert's equation

833-469: Is optical density of exposed photographic emulsion, a function of I {\displaystyle I} , the intensity of the source being observed, and t {\displaystyle t} , the exposure time, with p {\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 , Schwarzschild

882-619: Is the first to introduce the correct Lagrangian formalism of the electromagnetic field as S = ( 1 / 2 ) ∫ ( H 2 − E 2 ) d V + ∫ ρ ( ϕ − A → ⋅ u → ) d V {\displaystyle S=(1/2)\int (H^{2}-E^{2})dV+\int \rho (\phi -{\vec {A}}\cdot {\vec {u}})dV} where E → , H → {\displaystyle {\vec {E}},{\vec {H}}} are

931-518: Is the generalization of Laplace's equation to space-time, its solutions are also called "harmonic". The laws of physics can be expressed in a generally invariant form. In other words, the real world does not care about our coordinate systems. However, for us to be able to solve the equations, we must fix upon a particular coordinate system. A coordinate condition selects one (or a smaller set of) such coordinate system(s). The Cartesian coordinates used in special relativity satisfy d'Alembert's equation, so

980-582: Is the large crater Schwarzschild , on the far side of the Moon . Karl Schwarzschild was born on 9 October 1873 in Frankfurt on Main , the eldest of six boys and one girl, 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. The family owned two fabric stores in Frankfurt. His brother Alfred became

1029-531: 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 from the autoimmune disease pemphigus , which he developed while at the Russian front . Various forms of the disease particularly affect people of Ashkenazi Jewish origin. Asteroid 837 Schwarzschilda is named in his honour, as

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1078-436: Is usually considered as a topic unto itself. As such, the study of Ricci-flat metrics is only a distinct topic in dimension four and above. As noted above, any flat metric is Ricci-flat. However it is nontrivial to identify Ricci-flat manifolds whose full curvature is nonzero. In 1916, Karl Schwarzschild found the Schwarzschild metrics , which are Ricci-flat Lorentzian manifolds of nonzero curvature. Roy Kerr later found

1127-671: The Kerr metrics , a two-parameter family containing the Schwarzschild metrics as a special case. These metrics are fully explicit and are of fundamental interest in the mathematics and physics of black holes . More generally, in general relativity , Ricci-flat Lorentzian manifolds represent the vacuum solutions of Einstein's field equations with vanishing cosmological constant . Many pseudo-Riemannian manifolds are constructed as homogeneous spaces . However, these constructions are not directly helpful for Ricci-flat Riemannian metrics, in

1176-402: The Schwarzschild radius : where G is the gravitational constant , M is the mass of the central body, and c is the speed of light in vacuum. In cases where the radius of the central body is less than the Schwarzschild radius, R s {\displaystyle R_{s}} represents the radius within which all massive bodies, and even photons , must inevitably fall into

1225-449: The 1960s, establishing how a certain class of maximally extended Ricci-flat Lorentzian metrics are prescribed and constructed by certain Riemannian data. These are known as maximal globally hyperbolic developments . In general relativity, this is typically interpreted as an initial value formulation of Einstein's field equations for gravitation. The study of Ricci-flatness in the Riemannian and Lorentzian cases are quite distinct. This

1274-444: The above possibilities. It is not known whether this suggests an unknown general theorem or simply a limitation of known techniques. For this reason, Berger considered Ricci-flat manifolds to be "extremely mysterious." Notes. Sources. Karl Schwarzschild Karl Schwarzschild ( German: [kaʁl ˈʃvaʁtsʃɪlt] ; 9 October 1873 – 11 May 1916) was a German physicist and astronomer. Schwarzschild provided

1323-685: The age of sixteen. 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

1372-519: 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). Karl Schwarzschild appears as

1421-408: The condition of Ricci-flatness for a Riemannian metric can be interpreted as a system of elliptic partial differential equations . It is a straightforward consequence of standard elliptic regularity results that any Ricci-flat Riemannian metric on a smooth manifold is analytic, in the sense that harmonic coordinates define a compatible analytic structure , and the local representation of the metric

1470-501: The converse also holds. This may also be phrased as saying that Ricci-flatness is characterized by the vanishing of the two non-Weyl parts of the Ricci decomposition . Since the Weyl curvature vanishes in two or three dimensions, every Ricci-flat metric in these dimensions is flat . Conversely, it is automatic from the definitions that any flat metric is Ricci-flat. The study of flat metrics

1519-1159: The electric and applied magnetic fields, A → {\displaystyle {\vec {A}}} is the vector potential and ϕ {\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 S = ∑ i m i ∫ C i d s i + 1 2 ∑ i , j ∬ C i , C j q i q j δ ( ‖ P i P j ‖ ) d s i d s j {\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 C α {\displaystyle C_{\alpha }} are

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1568-538: 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. 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

1617-411: 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

1666-425: 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 , where he had

1715-476: The harmonic coordinate condition to the second term, we get: Thus, we get that an alternative way of expressing the harmonic coordinate condition is: If one expresses the Christoffel symbol in terms of the metric tensor, one gets Discarding the factor of g α δ {\displaystyle g^{\alpha \delta }\,} and rearranging some indices and terms, one gets In

1764-568: The improvement of optical systems, through the perturbative investigation of geometrical aberrations. 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 p {\displaystyle p} in the formula: i = f ( I ⋅ t p ) {\displaystyle i=f(I\cdot t^{p})} (where i {\displaystyle i}

1813-402: The notion of enlargeability of a closed manifold. The class of enlargeable manifolds is closed under homotopy equivalence , the taking of products, and under the connected sum with an arbitrary closed manifold. Every Ricci-flat Riemannian manifold in this class is flat, which is a corollary of Cheeger and Gromoll's splitting theorem . On a simply-connected Kähler manifold, a Kähler metric

1862-489: 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

1911-785: The partial derivative of a coordinate is the Kronecker delta , we get: And thus, dropping the minus sign, we get the harmonic coordinate condition (also known as the de Donder gauge after Théophile de Donder ): This condition is especially useful when working with gravitational waves. Consider the covariant derivative of the density of the reciprocal of the metric tensor: The last term − g μ ν Γ σ ρ σ − g {\displaystyle -g^{\mu \nu }\Gamma _{\sigma \rho }^{\sigma }{\sqrt {-g}}} emerges because − g {\displaystyle {\sqrt {-g}}}

1960-453: The rank of second lieutenant in the artillery. While serving on the front in Russia in 1915, he began to suffer from pemphigus , a rare and painful autoimmune skin-disease. 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

2009-505: The same time, is a special case of his construction. A quaternion-Kähler manifold is a Riemannian manifold whose holonomy group is contained in the Lie group Sp(n)·Sp(1) . Marcel Berger showed that any such metric must be Einstein. Furthermore, any Ricci-flat quaternion-Kähler manifold must be locally hyperkähler, meaning that the restricted holonomy group is contained in the symplectic group. A G 2 manifold or Spin(7) manifold

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2058-412: The sense that any homogeneous Riemannian manifold which is Ricci-flat must be flat. However, there are homogeneous (and even symmetric ) Lorentzian manifolds which are Ricci-flat but not flat, as follows from an explicit construction and computation of Lie algebras . Until Shing-Tung Yau 's resolution of the Calabi conjecture in the 1970s, it was not known whether every Ricci-flat Riemannian metric on

2107-469: The situation is not as well understood. A four-dimensional closed and oriented manifold supporting any Einstein Riemannian metric must satisfy the Hitchin–Thorpe inequality on its topological data. As particular cases of well-known theorems on Riemannian manifolds of nonnegative Ricci curvature, any manifold with a complete Ricci-flat Riemannian metric must: Mikhael Gromov and Blaine Lawson introduced

2156-505: 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 . Some of his particular achievements include measurements of variable stars , using photography, and

2205-598: The world lines of the particle, d s α {\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 δ ( ‖ P i P j ‖ ) {\displaystyle \delta \left(\left\Vert P_{i}P_{j}\right\Vert \right)} . The idea

2254-467: Was further developed by Hugo Tetrode and Adriaan Fokker in the 1920s and John Archibald Wheeler and Richard Feynman in the 1940s and constitutes an alternative but equivalent formulation of electrodynamics. 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. Einstein's approximate solution

2303-437: 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

2352-471: 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." 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

2401-578: Was then the most prestigious post available for an astronomer in Germany. 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

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