The Einstein Challenge for Differential Geometry

The Einstein challenge for differential geometry The case of Weyl and Cartan gewahlt ist, da% die g,, die in (4) gegebenen speziellen konstanten Wertefur haben. Relativitatstheorie einen Teil des vierdiinensionalen Raumes, so wiihlbarvon undeinem so in welchem das Koordinatensystem Betrachten wir eben dieseKO Bewegung beliebig gewahlt ist, da% die g,, die in (4) gegebenen kon-er sich von K , aus, speziellen so bewegt gewiihlten Koordinatensystem stanten Werte haben. nach den Uberlegungen des Kr, &us, beurteilt 2 in eineiii Betrachten wir eben diese Bewegung von einem beliebig Gravitationsfelde. Das Bewegungsgesetz auf h', K , aus, so bewegt ermit sichBezug von gewiihlten Koordinatensystem ergibt sich leicht Uberlegung. Mit Bezug auf Kr, &us, beurteilt nach aus den folgender Uberlegungen des 0 2 in eineiii KO ist das Bewegungsgesetz einemitvierdimensionale Gravitationsfelde. Das Bewegungsgesetz Bezug auf h', Geradc. ergibt aus folgender Uberlegung. auf also sich eineleicht geodiitische Linie. Da nunMitdieBezug geodiitische Link KO ist das Bewegungsgesetz eine vierdimensionale unabhiingig vom Bezugssystem defiriiert ist,Geradc. wird ihre Gleialso eine geodiitische Linie. Da nun die geodiitische Link chung auch die Bewegungsgleichung des materiellen Punktes unabhiingig vom Bezugssystem defiriiert ist, wird ihre Gleiin bezug auf Bewegungsgleichung K , sein. Setzen des wir materiellen Punktes chung auch die in bezug auf K , sein. Setzen wir Einstein in 1916 0 about Christoffel symbols: components of the gravitational field. r;] r;] r;, = - (45) r;, = - (45) so lautet also die Gleichuig der Punktbewegung inbezug auf K , so lautet also die Gleichuig der Punktbewegung inbezug auf K , Wir machen die naheliegende sehr naheliegende Wir machen nun nun die sehr Annahme,Annahme, daB dieses daB dieses allgemein kovariante Gleichungssystem die Bewegung des allgemein kovariante Gleichungssystem die Bewegung des Punktes im Gravitationsfeld auch in dern Falle bestimmt: Punktes im Gravitationsfeld auch in dern Falle bestimmt: KOexistiert, bezuglichbezuglich dessen in dessen end da% kein Bezugssystem KOexistiert, in end da% kein Bezugssystem lichen Riiumen die spezielle Relativitiitstheorie gilt. Zu diesel lichen Riiumen die spezielle Relativitiitstheorie gilt. Zu diesel Annahme sind wir urn so berechtigter, als (46) nur erste AbAnnahme wir urn so berechtigter, als (46) nur erste Ableitungen der sind g, enthiilt, zwischen denen auch im Spezialleitungen der von g, KO enthiilt, zwischen denen auch im Spezialfalle der Existenz keine Beziehungen bestehen.l) falle der Existenz keinesich Beziehungen bestehen.l) bewegt der Punkt geradVerschwinden die von soKO linig und gleichformig; die diese GroBen bedingen sich also der die Abso bewegt Punkt geradVerschwinden weichung der Bewegung von der Gleichformigkeit. Sie linig und gleichformig; diese GroBen bedingen sind also die AbGravitationsfelcies. dieweichung Komponenten der des Bewegung von der Gleichformigkeit. Sie sind ... r,zy, r,zy, des der Gravitationsfelcies. Sdie 14. Komponenten Die Feldgleichungen Qravitation bei Abwesenheit von Materie. SWir14.unterscheiden Die Feldgleichungen derzwischen Qravitation bei Abwesenheit im folgenden ,,GravitationsMaterie. feld" und ,,Materie", in demvon Sinne, da% alles auBer d e n Gravitationsfeld als ,,Materie" im bezeichnet wird,zwischen also nicht ,,Gravitationsnur Wir unterscheiden folgenden feld" und ,,Materie", in dem Sinne, da% alles auBer d e n nicht nur Bier 1) Erst zwischen den zweiten (und ersten) Ableitungen bestelicn Gravitationsfeld als ,,Materie" bezeichnet wird, also gemiiB f 12 die Beziehungen = 0. 1) Erst zwischen den zweiten (und ersten) Ableitungen bestelicn gemiiB f 12 die Beziehungen = 0. Bier x 1916. 7. ANNALEN DER PHYSIK. VIERTE FOME. BAND49. .- 1. D i e GrurtdZage dsr a l l g e r n d n m ReZat4&t&t8theor&; von A. EinsteBn. Die im nachfolgenden dargelegte Theorie bildet die denkbar weitgehenclste Verallgemeherung der heute allgemein als .,Relativita ts t heorie ' ' bezeichne ten Theorie ; die letztere nenne ich im folgenden zur Unterscheidnng von der ersteren ,,spezielle RelativitHtstheorie" und setze sie als bekannt voraus. Die Versllgemeinerung der Relativitatstheorie wurde sehr erleichtert durch die Gestalt, welche der speziellen Relativitgtstheorie durch Min ko ws k i gegeben wurde, welcher Mathematiker zuerst die formale Gleichwertigkeit der raumlichen Koordinaten und der Zeitkoordinate klar erkannte und fiir den Aufbau der Theorie nutzbar machte. Die fiir die allgemeine Relativitatstheorie notigen mathematischen Hilfsmittel lagen fertig bereit in dem ,,absoluten Differentialkalkiil", welcher auf den Forschungen von Gauss, R i e m a n n und Chris toff el uber nichteuklidische Mannigfaltigkeiten ruht und von R i c c i und L e v i - C i v i t a in ein System gebracht und bereits auf Probleme der theoretischen Physik angewendet wurde. Ich habe im Abschnitt B der vorliegenden Abhandlung alle fiir uns notigen, bei dem Physiker nicht als bekannt vorauszusetzenden mathematischen Rilfsmittel in moglichst einfacher uncl durchsichtiger Weise entwickelt, so daS ein Studium nisthematischer Literatur fiir das Verstandnis der vorliegenden Abhandlung nicht erforderlich ist . Endlich sei an dieser Stelle dankbar meines Freundes, des Mathematikers G r o s s m a n n , gedacht, der rnir durch seine Hilfe nicht nur clas Studium der einschliigigen mathematischen Literatur ersparte, sondern mich auch beim Snchen nach den Feldgleichungen der Gravitat,ion unt'erstutzte. Annalen der Phyaik. I V . Folge. 49. 50 Response of mathematicians – Levi-Civita 1917: Geometrical analysis/interpretation of Christoffel symbols as infinitesimal parallel displacement for manifolds embedded in euclidean Rn+ – Levi-Civita 1917, generalization to any Riemannian manifold: affine connection Γ (derived from metric) – Weyl 1918: affine connection as abstract structure in diff’ble manifold (not necessarily metric) – Weyl 1918, physical interpretation: gravito-inertial guiding field (compare Einstein’s 1916 remark), important for his view of GR. – . . . a little later taken up by Eddington, Einstein etc. Response of mathematicians – Levi-Civita 1917: Geometrical analysis/interpretation of Christoffel symbols as infinitesimal parallel displacement for manifolds embedded in euclidean Rn+ – Levi-Civita 1917, generalization to any Riemannian manifold: affine connection Γ (derived from metric) – Weyl 1918: affine connection as abstract structure in diff’ble manifold (not necessarily metric) – Weyl 1918, physical interpretation: gravito-inertial guiding field (compare Einstein’s 1916 remark), important for his view of GR. – . . . a little later taken up by Eddington, Einstein etc. Response of mathematicians – Levi-Civita 1917: Geometrical analysis/interpretation of Christoffel symbols as infinitesimal parallel displacement for manifolds embedded in euclidean Rn+ – Levi-Civita 1917, generalization to any Riemannian manifold: affine connection Γ (derived from metric) – Weyl 1918: affine connection as abstract structure in diff’ble manifold (not necessarily metric) – Weyl 1918, physical interpretation: gravito-inertial guiding field (compare Einstein’s 1916 remark), important for his view of GR. – . . . a little later taken up by Eddington, Einstein etc. Response of mathematicians – Levi-Civita 1917: Geometrical analysis/interpretation of Christoffel symbols as infinitesimal parallel displacement for manifolds embedded in euclidean Rn+ – Levi-Civita 1917, generalization to any Riemannian manifold: affine connection Γ (derived from metric) – Weyl 1918: affine connection as abstract structure in diff’ble manifold (not necessarily metric) – Weyl 1918, physical interpretation: gravito-inertial guiding field (compare Einstein’s 1916 remark), important for his view of GR. – . . . a little later taken up by Eddington, Einstein etc. “Postmature” development (if any) – but not only – John Stachel 2007 (“Newstein . . . ”): postmature development “ ‘affine connection’ is a postmature concept, the absence of which during the course of development of the general theory of relativity had a crucial negative influence on its development and subsequent interpretation.”, p. 1043) – . . . interesting point of view from philosophy of science, – From a historian’s point of view it was, perhaps even more, a starting point for a blossoming study of diverse differential geometric structures. – This shall be the main subject of my talk . . . “Postmature” development (if any) – but not only – John Stachel 2007 (“Newstein . . . ”): postmature development “ ‘affine connection’ is a postmature concept, the absence of which during the course of development of the general theory of relativity had a crucial negative influence on its development and subsequent interpretation.”, p. 1043) – . . . interesting point of view from philosophy of science, – From a historian’s point of view it was, perhaps even more, a starting point for a blossoming study of diverse differential geometric structures. – This shall be the main subject of my talk . . . “Postmature” development (if any) – but not only – John Stachel 2007 (“Newstein . . . ”): postmature development “ ‘affine connection’ is a postmature concept, the absence of which during the course of development of the general theory of relativity had a crucial negative influence on its development and subsequent interpretation.”, p. 1043) – . . . interesting point of view from philosophy of science, – From a historian’s point of view it was, perhaps even more, a starting point for a blossoming study of diverse differential geometric structures. – This shall be the main subject of my talk . . . Outline 1. New diff geom structures, an overview 2. Weyl’s gauge geometry 3. Cartan’s spaces Outlook 1. New structures: (i) Gauge geometry and gauge field(s) – Weyl 1918: purely infinitesimal geometry without direct comparison of lengths (or physical quantities) at different points of “the world”. – Basic concepts: conformal structure plus scale gauge; both together imply a unique affine connection conformal curvature (Weyl tensor). – Electromagnetism as gauge field (scale !): geometrically unified field theory and attempt to improve Hilbert-Mie theory of matter. – Twisted history of reception and criticism by physicists and philosophico-mathematical “analysis of the problem of space” (PoS) by Weyl (1920–1923). – After rise of QM shift of gauge concept from scale to phase; for Weyl conceptually bound to general relativistic framework of Driac field (1929 Weyl and Fock). – Later (1940s) reflection on relationship and difference between mathematical and physical automorphisms. 1. New structures: (i) Gauge geometry and gauge field(s) – Weyl 1918: purely infinitesimal geometry without direct comparison of lengths (or physical quantities) at different points of “the world”. – Basic concepts: conformal structure plus scale gauge; both together imply a unique affine connection conformal curvature (Weyl tensor). – Electromagnetism as gauge field (scale !): geometrically unified field theory and attempt to improve Hilbert-Mie theory of matter. – Twisted history of reception and criticism by physicists and philosophico-mathematical “analysis of the problem of space” (PoS) by Weyl (1920–1923). – After rise of QM shift of gauge concept from scale to phase; for Weyl conceptually bound to general relativistic framework of Driac field (1929 Weyl and Fock). – Later (1940s) reflection on relationship and difference between mathematical and physical automorphisms. 1. New structures: (i) Gauge geometry and gauge field(s) – Weyl 1918: purely infinitesimal geometry without direct comparison of lengths (or physical quantities) at different points of “the world”. – Basic concepts: conformal structure plus scale gauge; both together imply a unique affine connection conformal curvature (Weyl tensor). – Electromagnetism as gauge field (scale !): geometrically unified field theory and attempt to improve Hilbert-Mie theory of matter. – Twisted history of reception and criticism by physicists and philosophico-mathematical “analysis of the problem of space” (PoS) by Weyl (1920–1923). – After rise of QM shift of gauge concept from scale to phase; for Weyl conceptually bound to general relativistic framework of Driac field (1929 Weyl and Fock). – Later (1940s) reflection on relationship and difference between mathematical and physical automorphisms. 1. New structures: (i) Gauge geometry and gauge field(s) – Weyl 1918: purely infinitesimal geometry without direct comparison of lengths (or physical quantities) at different points of “the world”. – Basic concepts: conformal structure plus scale gauge; both together imply a unique affine connection conformal curvature (Weyl tensor). – Electromagnetism as gauge field (scale !): geometrically unified field theory and attempt to improve Hilbert-Mie theory of matter. – Twisted history of reception and criticism by physicists and philosophico-mathematical “analysis of the problem of space” (PoS) by Weyl (1920–1923). – After rise of QM shift of gauge concept from scale to phase; for Weyl conceptually bound to general relativistic framework of Driac field (1929 Weyl and Fock). – Later (1940s) reflection on relationship and difference between mathematical and physical automorphisms. 1. New structures: (i) Gauge geometry and gauge field(s) – Weyl 1918: purely infinitesimal geometry without direct comparison of lengths (or physical quantities) at different points of “the world”. – Basic concepts: conformal structure plus scale gauge; both together imply a unique affine connection conformal curvature (Weyl tensor). – Electromagnetism as gauge field (scale !): geometrically unified field theory and attempt to improve Hilbert-Mie theory of matter. – Twisted history of reception and criticism by physicists and philosophico-mathematical “analysis of the problem of space” (PoS) by Weyl (1920–1923). – After rise of QM shift of gauge concept from scale to phase; for Weyl conceptually bound to general relativistic framework of Driac field (1929 Weyl and Fock). – Later (1940s) reflection on relationship and difference between mathematical and physical automorphisms. (ii) Conformal and projective structures – Schouten (1920/21): study of conformal differential geometry. – Weyl 1921: conformal and projective structure (“Beschaffenheit”), conceptual role in GR (causal and inertial structure), projective curvature tensor. – Cartan 1922ff.: conformal and projective Cartan spaces, part of wider program (below). – Veblen/Eisenhart 1922ff.: geometry of paths and projective structures in mathematics and physics (diss: Thomas, Church . . . ) All these strongly influenced by general relativity; often with interest in foundations for GR or for generalizations. (ii) Conformal and projective structures – Schouten (1920/21): study of conformal differential geometry. – Weyl 1921: conformal and projective structure (“Beschaffenheit”), conceptual role in GR (causal and inertial structure), projective curvature tensor. – Cartan 1922ff.: conformal and projective Cartan spaces, part of wider program (below). – Veblen/Eisenhart 1922ff.: geometry of paths and projective structures in mathematics and physics (diss: Thomas, Church . . . ) All these strongly influenced by general relativity; often with interest in foundations for GR or for generalizations. (ii) Conformal and projective structures – Schouten (1920/21): study of conformal differential geometry. – Weyl 1921: conformal and projective structure (“Beschaffenheit”), conceptual role in GR (causal and inertial structure), projective curvature tensor. – Cartan 1922ff.: conformal and projective Cartan spaces, part of wider program (below). – Veblen/Eisenhart 1922ff.: geometry of paths and projective structures in mathematics and physics (diss: Thomas, Church . . . ) All these strongly influenced by general relativity; often with interest in foundations for GR or for generalizations. (ii) Conformal and projective structures – Schouten (1920/21): study of conformal differential geometry. – Weyl 1921: conformal and projective structure (“Beschaffenheit”), conceptual role in GR (causal and inertial structure), projective curvature tensor. – Cartan 1922ff.: conformal and projective Cartan spaces, part of wider program (below). – Veblen/Eisenhart 1922ff.: geometry of paths and projective structures in mathematics and physics (diss: Thomas, Church . . . ) All these strongly influenced by general relativity; often with interest in foundations for GR or for generalizations. (ii) Conformal and projective structures – Schouten (1920/21): study of conformal differential geometry. – Weyl 1921: conformal and projective structure (“Beschaffenheit”), conceptual role in GR (causal and inertial structure), projective curvature tensor. – Cartan 1922ff.: conformal and projective Cartan spaces, part of wider program (below). – Veblen/Eisenhart 1922ff.: geometry of paths and projective structures in mathematics and physics (diss: Thomas, Church . . . ) All these strongly influenced by general relativity; often with interest in foundations for GR or for generalizations. (iii) Cartan spaces – Multiple backgrounds: Darboux’s diff. geometry (“trièdres”/3-frames) + Lie “groups” (Lie algebras)+ diff. forms + E. & F. Cosserat’s theory of generalized elastic media + Einstein’ theory. – Cartan 1921ff.: construction of generalized spaces from infinitesimal Klein spaces, using differential forms and infinitesimal Lie groups – Connections with values in Lie algebra (modernized language) two kinds of curvature: isotropy component (known before in different disguise) and a kind of translational curvature, called torsion by Cartan (role in geometrized Cosserat theory). – Huge reseach program with repercussions inside mathematics (differential geometry, partial diff equs.) and physics (general relativity, elasticity). – Technicalities; very difficult to understand in original form. Later restructured in terminology of principle fibre bundles and connections in these (Ehresmann 1950sf.) (iii) Cartan spaces – Multiple backgrounds: Darboux’s diff. geometry (“trièdres”/3-frames) + Lie “groups” (Lie algebras)+ diff. forms + E. & F. Cosserat’s theory of generalized elastic media + Einstein’ theory. – Cartan 1921ff.: construction of generalized spaces from infinitesimal Klein spaces, using differential forms and infinitesimal Lie groups – Connections with values in Lie algebra (modernized language) two kinds of curvature: isotropy component (known before in different disguise) and a kind of translational curvature, called torsion by Cartan (role in geometrized Cosserat theory). – Huge reseach program with repercussions inside mathematics (differential geometry, partial diff equs.) and physics (general relativity, elasticity). – Technicalities; very difficult to understand in original form. Later restructured in terminology of principle fibre bundles and connections in these (Ehresmann 1950sf.) (iii) Cartan spaces – Multiple backgrounds: Darboux’s diff. geometry (“trièdres”/3-frames) + Lie “groups” (Lie algebras)+ diff. forms + E. & F. Cosserat’s theory of generalized elastic media + Einstein’ theory. – Cartan 1921ff.: construction of generalized spaces from infinitesimal Klein spaces, using differential forms and infinitesimal Lie groups – Connections with values in Lie algebra (modernized language) two kinds of curvature: isotropy component (known before in different disguise) and a kind of translational curvature, called torsion by Cartan (role in geometrized Cosserat theory). – Huge reseach program with repercussions inside mathematics (differential geometry, partial diff equs.) and physics (general relativity, elasticity). – Technicalities; very difficult to understand in original form. Later restructured in terminology of principle fibre bundles and connections in these (Ehresmann 1950sf.) (iii) Cartan spaces – Multiple backgrounds: Darboux’s diff. geometry (“trièdres”/3-frames) + Lie “groups” (Lie algebras)+ diff. forms + E. & F. Cosserat’s theory of generalized elastic media + Einstein’ theory. – Cartan 1921ff.: construction of generalized spaces from infinitesimal Klein spaces, using differential forms and infinitesimal Lie groups – Connections with values in Lie algebra (modernized language) two kinds of curvature: isotropy component (known before in different disguise) and a kind of translational curvature, called torsion by Cartan (role in geometrized Cosserat theory). – Huge reseach program with repercussions inside mathematics (differential geometry, partial diff equs.) and physics (general relativity, elasticity). – Technicalities; very difficult to understand in original form. Later restructured in terminology of principle fibre bundles and connections in these (Ehresmann 1950sf.) (iii) Cartan spaces – Multiple backgrounds: Darboux’s diff. geometry (“trièdres”/3-frames) + Lie “groups” (Lie algebras)+ diff. forms + E. & F. Cosserat’s theory of generalized elastic media + Einstein’ theory. – Cartan 1921ff.: construction of generalized spaces from infinitesimal Klein spaces, using differential forms and infinitesimal Lie groups – Connections with values in Lie algebra (modernized language) two kinds of curvature: isotropy component (known before in different disguise) and a kind of translational curvature, called torsion by Cartan (role in geometrized Cosserat theory). – Huge reseach program with repercussions inside mathematics (differential geometry, partial diff equs.) and physics (general relativity, elasticity). – Technicalities; very difficult to understand in original form. Later restructured in terminology of principle fibre bundles and connections in these (Ehresmann 1950sf.)