The Einstein Challenge for Differential Geometry - Cartan's Spaces

3. Cartan’s spaces Cartan’s first publication on “non-holonomous spaces” – 1921: Manuscripts. – 1922: Series of notes in the C.R. “Sur une généralisation de la notion de courbure de Riemann et les espaces à torsion” C.R. Feb. 27, 1922, . . . . – 1922: Connections described by Pfaffian (differential) forms “Sur les équations de la gravitation d’ Einstein” JdM 1922 (still without torsion). – 1923/24: “Sur les variétés a connexion affine et la théorie de la relativité généralisée I, II”, Ann. sci. ENS. 40/41 (I: Connections in affine group and in euclidean group, with torsion, Newtonian mechanics and GR in Cartanian terms chap. I “La dynamique des milieux continus et la notion de connexion affine de l’espace-temps” English in Renn/Schemmel (eds.) 2007 vol. 3 a bit of John’s “postmature Newstein”(?) II: generalization of GR to Cosserat type media . . . .) Cartan’s first publication on “non-holonomous spaces” – 1921: Manuscripts. – 1922: Series of notes in the C.R. “Sur une généralisation de la notion de courbure de Riemann et les espaces à torsion” C.R. Feb. 27, 1922, . . . . – 1922: Connections described by Pfaffian (differential) forms “Sur les équations de la gravitation d’ Einstein” JdM 1922 (still without torsion). – 1923/24: “Sur les variétés a connexion affine et la théorie de la relativité généralisée I, II”, Ann. sci. ENS. 40/41 (I: Connections in affine group and in euclidean group, with torsion, Newtonian mechanics and GR in Cartanian terms chap. I “La dynamique des milieux continus et la notion de connexion affine de l’espace-temps” English in Renn/Schemmel (eds.) 2007 vol. 3 a bit of John’s “postmature Newstein”(?) II: generalization of GR to Cosserat type media . . . .) Motivation for Cartan’s program – Toronto ICM 1924: Cartan discussed the gap between Klein’s conception of a geometry characterized by the principle of homogeneity (Klein spaces S = G /H) and Riemannian geometry which is only isotropic (not homogeneous) in infinitesimal neighbourhoods. – This gap had been filled only partially, by Levi-Civita’s introduction of parallel discplacement, due to the challenge of GR: Or, c’est le développement même de la théorie de la relativité, liée par l’obligation paradoxale d’interpréter dans et par un Univers non homogène les résultats de nombreuses expériences faites par des observateurs qui croyaient à l’homogénéié de cet Univers, qui permit de combler en partie le fossé qui séparait les espaces de Riemann de l’espace euclidien. Le premier pas dans cette voie fut l’oeuvre de M. Levi-Civita, par l’introduction de sa notion de parallélisme. (Cartan 1924 (ICM Toronto), 86) Motivation for Cartan’s program – Toronto ICM 1924: Cartan discussed the gap between Klein’s conception of a geometry characterized by the principle of homogeneity (Klein spaces S = G /H) and Riemannian geometry which is only isotropic (not homogeneous) in infinitesimal neighbourhoods. – This gap had been filled only partially, by Levi-Civita’s introduction of parallel discplacement, due to the challenge of GR: Or, c’est le développement même de la théorie de la relativité, liée par l’obligation paradoxale d’interpréter dans et par un Univers non homogène les résultats de nombreuses expériences faites par des observateurs qui croyaient à l’homogénéié de cet Univers, qui permit de combler en partie le fossé qui séparait les espaces de Riemann de l’espace euclidien. Le premier pas dans cette voie fut l’oeuvre de M. Levi-Civita, par l’introduction de sa notion de parallélisme. (Cartan 1924 (ICM Toronto), 86) The theory of relativity, facing the paradoxical task of interpreting in a non-homogeneous universe, and by means of it, the multiple experiences made by observers who used to believe (“qui croyaient”) in the homogeneity of this universe, allowed to partially fill the gap between Riemannian geometry and euclidean geometry [in the Kleinian sense, ES]. M. Levi-Civita made the first steps in this direction by introducing the concept of parallelism. ” (Cartan 1924, O.C. 594) Cartan’s program: infinitesimal homogeneity – Cartan proposed infinitesimally homogeneous, but “non-holonomous”, spaces as a second step for bridging the gap. – Infinitesimally homogeneous: Cartan parametrized infinitesimal neighbourhoods of points p in a manifold (space) S by a homogeneous space in Klein’s sense (of correct dimension). This was expressed by point dependent “répères”/frames (later Cartan gauge (Sharpe)). Result in modernized notation: Tp S ∼ = G /H – For any infinitesimal displacement δx (modernized, ξ ∈ Tp M) Cartan specified an infinitesimal element of the group by a set of differential (Pfaffian) forms. He called the whole set a “connection” (“affine, projective . . . ”). – Modernized: a differential form with values in g with specific transformation rules, now Cartan connection. Pointwise: ξ 7→ g (ξ) ∈ g = Lie(G ) , ξ ∈ Tp M . Cartan’s program: infinitesimal homogeneity – Cartan proposed infinitesimally homogeneous, but “non-holonomous”, spaces as a second step for bridging the gap. – Infinitesimally homogeneous: Cartan parametrized infinitesimal neighbourhoods of points p in a manifold (space) S by a homogeneous space in Klein’s sense (of correct dimension). This was expressed by point dependent “répères”/frames (later Cartan gauge (Sharpe)). Result in modernized notation: Tp S ∼ = G /H – For any infinitesimal displacement δx (modernized, ξ ∈ Tp M) Cartan specified an infinitesimal element of the group by a set of differential (Pfaffian) forms. He called the whole set a “connection” (“affine, projective . . . ”). – Modernized: a differential form with values in g with specific transformation rules, now Cartan connection. Pointwise: ξ 7→ g (ξ) ∈ g = Lie(G ) , ξ ∈ Tp M . Cartan’s program: infinitesimal homogeneity – Cartan proposed infinitesimally homogeneous, but “non-holonomous”, spaces as a second step for bridging the gap. – Infinitesimally homogeneous: Cartan parametrized infinitesimal neighbourhoods of points p in a manifold (space) S by a homogeneous space in Klein’s sense (of correct dimension). This was expressed by point dependent “répères”/frames (later Cartan gauge (Sharpe)). Result in modernized notation: Tp S ∼ = G /H – For any infinitesimal displacement δx (modernized, ξ ∈ Tp M) Cartan specified an infinitesimal element of the group by a set of differential (Pfaffian) forms. He called the whole set a “connection” (“affine, projective . . . ”). – Modernized: a differential form with values in g with specific transformation rules, now Cartan connection. Pointwise: ξ 7→ g (ξ) ∈ g = Lie(G ) , ξ ∈ Tp M . Cartan’s program: infinitesimal homogeneity – Cartan proposed infinitesimally homogeneous, but “non-holonomous”, spaces as a second step for bridging the gap. – Infinitesimally homogeneous: Cartan parametrized infinitesimal neighbourhoods of points p in a manifold (space) S by a homogeneous space in Klein’s sense (of correct dimension). This was expressed by point dependent “répères”/frames (later Cartan gauge (Sharpe)). Result in modernized notation: Tp S ∼ = G /H – For any infinitesimal displacement δx (modernized, ξ ∈ Tp M) Cartan specified an infinitesimal element of the group by a set of differential (Pfaffian) forms. He called the whole set a “connection” (“affine, projective . . . ”). – Modernized: a differential form with values in g with specific transformation rules, now Cartan connection. Pointwise: ξ 7→ g (ξ) ∈ g = Lie(G ) , ξ ∈ Tp M . Curvature and torsion – By “non-holonomous” Cartan meant that, after traversing an infinitesimal closed loop (in a surface element), the total “homogeneity operation” was different from the one in the corresponding Klein geometry. – This difference could be expressed by a collection of Pfaffian 2-forms; some of them characterized the deviation I in h = Lie(H) (isotropic or “rotational” curvature), I and a deviation in a transversal Liealgebra l (torsion), a generalized translational curvature. (g = l ⊕ h with specific properties – “reductivity”). – Riemannian geometry, with its Levi-Civita connection (“first step of bridging the gap”), was a special case of “nonholonomous space” with group G = GL(, R) or SO(n), depending on the “répères” and with vanishing torsion, Curvature and torsion – By “non-holonomous” Cartan meant that, after traversing an infinitesimal closed loop (in a surface element), the total “homogeneity operation” was different from the one in the corresponding Klein geometry. – This difference could be expressed by a collection of Pfaffian 2-forms; some of them characterized the deviation I in h = Lie(H) (isotropic or “rotational” curvature), I and a deviation in a transversal Liealgebra l (torsion), a generalized translational curvature. (g = l ⊕ h with specific properties – “reductivity”). – Riemannian geometry, with its Levi-Civita connection (“first step of bridging the gap”), was a special case of “nonholonomous space” with group G = GL(, R) or SO(n), depending on the “répères” and with vanishing torsion, Curvature and torsion – By “non-holonomous” Cartan meant that, after traversing an infinitesimal closed loop (in a surface element), the total “homogeneity operation” was different from the one in the corresponding Klein geometry. – This difference could be expressed by a collection of Pfaffian 2-forms; some of them characterized the deviation I in h = Lie(H) (isotropic or “rotational” curvature), I and a deviation in a transversal Liealgebra l (torsion), a generalized translational curvature. (g = l ⊕ h with specific properties – “reductivity”). – Riemannian geometry, with its Levi-Civita connection (“first step of bridging the gap”), was a special case of “nonholonomous space” with group G = GL(, R) or SO(n), depending on the “répères” and with vanishing torsion, Physics and mathematics From the very beginning of his presentations (C.R. Feb. 1922 etc.), Cartan discussed the relation of his new spaces to physical theories: – Einstein’s theory of general relativity, – the “beaux travaux de MM. E. et F. Cosserat sur, l’action euclidienne”: a generalized elasticity theory of media with non-symmetric tensor of tension (E. & M. Cosserat 1909). – and Weyl’s generalized spaces (which fitted in Cartan’s framework and were again generalized – by including torsion). Background info: Cosserat media (1909) – In 1909 Eugène and François Cosserat demanded to consider action principles of mechanics, invariant under the group of full Euclidean motions (“action euclidienne”). – In particular they studied a variational principle for elastic media with an action density dependent on an infinitesimal “trièdre trirectangulaire” (orthonormal frame) and invariant under infinitesimal Euclidean motions. – The resulting stress tensor was no longer necessarily symmetric. – An “infinitesimal” surface element (on the boundary of an elastic body under external influences) could be subject to an elastic torque in addition to an elastic force. In the language of the Ecole Polytechnique tradtion described by a “couple” of forces (parallel, inversely oriented, different lines of action). Background info: Cosserat media (1909) – In 1909 Eugène and François Cosserat demanded to consider action principles of mechanics, invariant under the group of full Euclidean motions (“action euclidienne”). – In particular they studied a variational principle for elastic media with an action density dependent on an infinitesimal “trièdre trirectangulaire” (orthonormal frame) and invariant under infinitesimal Euclidean motions. – The resulting stress tensor was no longer necessarily symmetric. – An “infinitesimal” surface element (on the boundary of an elastic body under external influences) could be subject to an elastic torque in addition to an elastic force. In the language of the Ecole Polytechnique tradtion described by a “couple” of forces (parallel, inversely oriented, different lines of action). Background info: Cosserat media (1909) – In 1909 Eugène and François Cosserat demanded to consider action principles of mechanics, invariant under the group of full Euclidean motions (“action euclidienne”). – In particular they studied a variational principle for elastic media with an action density dependent on an infinitesimal “trièdre trirectangulaire” (orthonormal frame) and invariant under infinitesimal Euclidean motions. – The resulting stress tensor was no longer necessarily symmetric. – An “infinitesimal” surface element (on the boundary of an elastic body under external influences) could be subject to an elastic torque in addition to an elastic force. In the language of the Ecole Polytechnique tradtion described by a “couple” of forces (parallel, inversely oriented, different lines of action). Elastic media in the light of the Einstein equation (1922) In his paper on “les équations d’Einstein” (1922) Cartan analyzed which quadratic differential forms Gij dx i dx j , satisfying a “conservation law” like the Einstein tensor, can be formed from up to second partial derivatives of the metric gij dx i dx j . He showed: – Gij is determined up to three arbitrary constants. – Gij can be expressed in terms of infinitesimal rotations of the Levi-Civita connection (Ricci coefficients). – The classical stress tensor of elastic media in Euclidean space can be re-written, by differential 2-forms Ωξ , depending on a direction vector ξ (Ωξ gives the norm of the force in direction ξ applied on a surface element dx ∧ dy .“Conservation law”: dΩξ = 0 – In Einstein gravity, Cartan expressed the coefficients of Ωξ by the Einstein tensor Gij – Einstein equation! – and thus, indirectly, by the Ricci coefficients. Elastic media in the light of the Einstein equation (1922) In his paper on “les équations d’Einstein” (1922) Cartan analyzed which quadratic differential forms Gij dx i dx j , satisfying a “conservation law” like the Einstein tensor, can be formed from up to second partial derivatives of the metric gij dx i dx j . He showed: – Gij is determined up to three arbitrary constants. – Gij can be expressed in terms of infinitesimal rotations of the Levi-Civita connection (Ricci coefficients). – The classical stress tensor of elastic media in Euclidean space can be re-written, by differential 2-forms Ωξ , depending on a direction vector ξ (Ωξ gives the norm of the force in direction ξ applied on a surface element dx ∧ dy .“Conservation law”: dΩξ = 0 – In Einstein gravity, Cartan expressed the coefficients of Ωξ by the Einstein tensor Gij – Einstein equation! – and thus, indirectly, by the Ricci coefficients. Elastic media in the light of the Einstein equation (1922) In his paper on “les équations d’Einstein” (1922) Cartan analyzed which quadratic differential forms Gij dx i dx j , satisfying a “conservation law” like the Einstein tensor, can be formed from up to second partial derivatives of the metric gij dx i dx j . He showed: – Gij is determined up to three arbitrary constants. – Gij can be expressed in terms of infinitesimal rotations of the Levi-Civita connection (Ricci coefficients). – The classical stress tensor of elastic media in Euclidean space can be re-written, by differential 2-forms Ωξ , depending on a direction vector ξ (Ωξ gives the norm of the force in direction ξ applied on a surface element dx ∧ dy .“Conservation law”: dΩξ = 0 – In Einstein gravity, Cartan expressed the coefficients of Ωξ by the Einstein tensor Gij – Einstein equation! – and thus, indirectly, by the Ricci coefficients. Elastic media in the light of the Einstein equation (1922) In his paper on “les équations d’Einstein” (1922) Cartan analyzed which quadratic differential forms Gij dx i dx j , satisfying a “conservation law” like the Einstein tensor, can be formed from up to second partial derivatives of the metric gij dx i dx j . He showed: – Gij is determined up to three arbitrary constants. – Gij can be expressed in terms of infinitesimal rotations of the Levi-Civita connection (Ricci coefficients). – The classical stress tensor of elastic media in Euclidean space can be re-written, by differential 2-forms Ωξ , depending on a direction vector ξ (Ωξ gives the norm of the force in direction ξ applied on a surface element dx ∧ dy .“Conservation law”: dΩξ = 0 – In Einstein gravity, Cartan expressed the coefficients of Ωξ by the Einstein tensor Gij – Einstein equation! – and thus, indirectly, by the Ricci coefficients. Origin of the terminology “torsion” (1921/22) – By formal analogy, Cartan expressed the stress tensor of elastic media in Euclidean space in terms of Ricci coefficients of an auxiliary metric (reading Einstein’s equ. in 3 dim. from right to left). – Generalizing the approach to “non-holonomous” spaces, a translational component appears in the connection, in addition to the rotational one (announced in CR notes, explained in 1923/24 paper. part II) – The corresponding stress tensor becomes asymmetric, if the second curvature, mentioned above, does not vanish. The asymmetry corresponds to a torque in addition to a force (on any surface element) like in the Cosserats’ analysis. – For stress, the second curvature leads to a torque like the rotational curvature (Ricci coefficients) corresponds to surface force. Thus Cartan’s surprising terminological choice “torsion” for a translational (!) curvature component of the infinitesimal group. Origin of the terminology “torsion” (1921/22) – By formal analogy, Cartan expressed the stress tensor of elastic media in Euclidean space in terms of Ricci coefficients of an auxiliary metric (reading Einstein’s equ. in 3 dim. from right to left). – Generalizing the approach to “non-holonomous” spaces, a translational component appears in the connection, in addition to the rotational one (announced in CR notes, explained in 1923/24 paper. part II) – The corresponding stress tensor becomes asymmetric, if the second curvature, mentioned above, does not vanish. The asymmetry corresponds to a torque in addition to a force (on any surface element) like in the Cosserats’ analysis. – For stress, the second curvature leads to a torque like the rotational curvature (Ricci coefficients) corresponds to surface force. Thus Cartan’s surprising terminological choice “torsion” for a translational (!) curvature component of the infinitesimal group. Origin of the terminology “torsion” (1921/22) – By formal analogy, Cartan expressed the stress tensor of elastic media in Euclidean space in terms of Ricci coefficients of an auxiliary metric (reading Einstein’s equ. in 3 dim. from right to left). – Generalizing the approach to “non-holonomous” spaces, a translational component appears in the connection, in addition to the rotational one (announced in CR notes, explained in 1923/24 paper. part II) – The corresponding stress tensor becomes asymmetric, if the second curvature, mentioned above, does not vanish. The asymmetry corresponds to a torque in addition to a force (on any surface element) like in the Cosserats’ analysis. – For stress, the second curvature leads to a torque like the rotational curvature (Ricci coefficients) corresponds to surface force. Thus Cartan’s surprising terminological choice “torsion” for a translational (!) curvature component of the infinitesimal group. Origin of the terminology “torsion” (1921/22) – By formal analogy, Cartan expressed the stress tensor of elastic media in Euclidean space in terms of Ricci coefficients of an auxiliary metric (reading Einstein’s equ. in 3 dim. from right to left). – Generalizing the approach to “non-holonomous” spaces, a translational component appears in the connection, in addition to the rotational one (announced in CR notes, explained in 1923/24 paper. part II) – The corresponding stress tensor becomes asymmetric, if the second curvature, mentioned above, does not vanish. The asymmetry corresponds to a torque in addition to a force (on any surface element) like in the Cosserats’ analysis. – For stress, the second curvature leads to a torque like the rotational curvature (Ricci coefficients) corresponds to surface force. Thus Cartan’s surprising terminological choice “torsion” for a translational (!) curvature component of the infinitesimal group.