Extensions of Lorentzian spacetime geometry

Transcription

1 Extensions of Lorentzian spacetime geometry From Finsler to Cartan and vice versa Manuel Hohmann Teoreetilise Füüsika Labor Füüsika Instituut Tartu Ülikool 13. November 2013 Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

2 Outline 1 Introduction 2 Cartan geometry on observer space 3 Finsler spacetimes 4 From Finsler geometry to Cartan geometry 5 From Cartan geometry to Finsler geometry 6 Closing the circle 7 Finsler-Cartan-Gravity 8 Conclusion Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

3 Outline 1 Introduction 2 Cartan geometry on observer space 3 Finsler spacetimes 4 From Finsler geometry to Cartan geometry 5 From Cartan geometry to Finsler geometry 6 Closing the circle 7 Finsler-Cartan-Gravity 8 Conclusion Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

4 References This work: MH, Extensions of Lorentzian spacetime geometry: from Finsler to Cartan and vice versa, Phys. Rev. D 87 (2013) [arxiv: [gr-qc]]. Cartan geometry of observer space: S. Gielen and D. K. Wise, Lifting General Relativity to Observer Space, J. Math. Phys. 54 (2013) [arxiv: [gr-qc]]. Finsler spacetimes: C. Pfeifer and M. N. R. Wohlfarth, Causal structure and electrodynamics on Finsler spacetimes, Phys. Rev. D 84 (2011) [arxiv: [gr-qc]]. C. Pfeifer and M. N. R. Wohlfarth, Finsler geometric extension of Einstein gravity, Phys. Rev. D 85 (2012) [arxiv: [gr-qc]]. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

5 Physical motivation A simple experiment: light propagation in spacetime (M, g). A supernova occurs at some beacon event x 0 M. Light from the supernova follows a null geodesic γ in M. An astronomer observes the light at another event x M. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

6 Physical motivation A simple experiment: light propagation in spacetime (M, g). A supernova occurs at some beacon event x 0 M. Light from the supernova follows a null geodesic γ in M. An astronomer observes the light at another event x M. The measured data: Light intensity: photon rate measured with local clock. Spectrum: photon frequency measured with local clock. Location of the source: direction of incoming light in local frame. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

7 Physical motivation A simple experiment: light propagation in spacetime (M, g). A supernova occurs at some beacon event x 0 M. Light from the supernova follows a null geodesic γ in M. An astronomer observes the light at another event x M. The measured data: Light intensity: photon rate measured with local clock. Spectrum: photon frequency measured with local clock. Location of the source: direction of incoming light in local frame. Mathematical description of the measured data: General covariance: Physical quantities are tensors. Tensor components are measured with respect to local frame. No measurement without a frame. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

8 Physical motivation A simple experiment: light propagation in spacetime (M, g). A supernova occurs at some beacon event x 0 M. Light from the supernova follows a null geodesic γ in M. An astronomer observes the light at another event x M. The measured data: Light intensity: photon rate measured with local clock. Spectrum: photon frequency measured with local clock. Location of the source: direction of incoming light in local frame. Mathematical description of the measured data: General covariance: Physical quantities are tensors. Tensor components are measured with respect to local frame. No measurement without a frame. Consider observer frames as more fundamental than spacetime. Spacetime emerges from equivalence classes of observer frames. Geometric theory based on this assumption? Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

9 Why Cartan geometry on observer space? Quantum gravity may suggest breaking of general covariance: Loop quantum gravity Spin foam models Causal dynamical triangulations Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

10 Why Cartan geometry on observer space? Quantum gravity may suggest breaking of general covariance: Loop quantum gravity Spin foam models Causal dynamical triangulations Possible implications: Existence of preferred observers / timelike vector fields.... preferred spatial foliations of spacetime. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

11 Why Cartan geometry on observer space? Quantum gravity may suggest breaking of general covariance: Loop quantum gravity Spin foam models Causal dynamical triangulations Possible implications: Existence of preferred observers / timelike vector fields.... preferred spatial foliations of spacetime. Problems: Breaking of Copernican principle. No observation of (strongly) broken symmetry. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

12 Why Cartan geometry on observer space? Quantum gravity may suggest breaking of general covariance: Loop quantum gravity Spin foam models Causal dynamical triangulations Possible implications: Existence of preferred observers / timelike vector fields.... preferred spatial foliations of spacetime. Problems: Breaking of Copernican principle. No observation of (strongly) broken symmetry. Solution: Consider space O of all allowed observers. Describe experiments on observer space instead of spacetime. Observer dependence of physical quantities follows naturally. No preferred observers. Geometry of observer space modeled by Cartan geometry. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

13 Why Finsler geometry of spacetimes? Finsler geometry of space widely used in physics: Approaches to quantum gravity Electrodynamics in anisotropic media Modeling of astronomical data Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

14 Why Finsler geometry of spacetimes? Finsler geometry of space widely used in physics: Approaches to quantum gravity Electrodynamics in anisotropic media Modeling of astronomical data Finsler geometry generalizes Riemannian geometry: Clock postulate: proper time equals arc length along trajectories. Geometry described by Finsler metric. Well-defined notions of connections, curvature, parallel transport... Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

15 Why Finsler geometry of spacetimes? Finsler geometry of space widely used in physics: Approaches to quantum gravity Electrodynamics in anisotropic media Modeling of astronomical data Finsler geometry generalizes Riemannian geometry: Clock postulate: proper time equals arc length along trajectories. Geometry described by Finsler metric. Well-defined notions of connections, curvature, parallel transport... Finsler spacetimes are suitable backgrounds for: Gravity Electrodynamics Other matter field theories Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

16 Why Finsler geometry of spacetimes? Finsler geometry of space widely used in physics: Approaches to quantum gravity Electrodynamics in anisotropic media Modeling of astronomical data Finsler geometry generalizes Riemannian geometry: Clock postulate: proper time equals arc length along trajectories. Geometry described by Finsler metric. Well-defined notions of connections, curvature, parallel transport... Finsler spacetimes are suitable backgrounds for: Gravity Electrodynamics Other matter field theories Possible explanations of yet unexplained phenomena: Fly-by anomaly Galaxy rotation curves Accelerating expansion of the universe Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

17 Outline 1 Introduction 2 Cartan geometry on observer space 3 Finsler spacetimes 4 From Finsler geometry to Cartan geometry 5 From Cartan geometry to Finsler geometry 6 Closing the circle 7 Finsler-Cartan-Gravity 8 Conclusion Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

18 Frame bundles vs. homogeneous spaces Consider Lorentzian manifold (M, g). Orthonormal frame bundle π : P M. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

19 Frame bundles vs. homogeneous spaces Consider Lorentzian manifold (M, g). Orthonormal frame bundle π : P M. Split of the tangent spaces T p P: Local infinitesimal Lorentz transforms: Only change local frame. Leave base point x = π(p) invariant. Tangent vectors to fibers π 1 (x). Infinitesimal translations: Infinitesimal change of base point x. Frame is unchanged parallely transported. Orthogonal to infinitesimal Lorentz transforms. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

20 Frame bundles vs. homogeneous spaces Consider Lorentzian manifold (M, g). Orthonormal frame bundle π : P M. Split of the tangent spaces T p P = g: Local infinitesimal Lorentz transforms h: Only change local frame. Leave base point x = π(p) invariant. Tangent vectors to fibers π 1 (x). Infinitesimal translations z = g/h: Infinitesimal change of base point x. Frame is unchanged parallely transported. Orthogonal to infinitesimal Lorentz transforms. Description using Lie algebras: Poincaré algebra g. Lorentz algebra h g. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

21 Frame bundles vs. homogeneous spaces Consider Lorentzian manifold (M, g). Orthonormal frame bundle π : P M is principal H-bundle. Split of the tangent spaces T p P = g: Local infinitesimal Lorentz transforms h: Only change local frame. Leave base point x = π(p) invariant. Tangent vectors to fibers π 1 (x). Infinitesimal translations z = g/h: Infinitesimal change of base point x. Frame is unchanged parallely transported. Orthogonal to infinitesimal Lorentz transforms. Description using Lie algebras: Poincaré algebra g. Lorentz algebra h g. Corresponding Lie groups: Frame bundle P locally looks like Poincaré group G. Fibers π 1 (x) look like Lorentz group H. Spacetime M locally looks like homogeneous space G/H. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

22 Frame bundles vs. homogeneous spaces Consider Lorentzian manifold (M, g). Orthonormal frame bundle π : P M is principal H-bundle. Split of the tangent spaces T p P = g: Local infinitesimal Lorentz transforms h: Only change local frame. Leave base point x = π(p) invariant. Tangent vectors to fibers π 1 (x). Infinitesimal translations z = g/h: Infinitesimal change of base point x. Frame is unchanged parallely transported. Orthogonal to infinitesimal Lorentz transforms. Description using Lie algebras: Poincaré algebra g. Lorentz algebra h g. Corresponding Lie groups: Frame bundle P locally looks like Poincaré group G. Fibers π 1 (x) look like Lorentz group H. Spacetime M locally looks like homogeneous space G/H. Geometry encoded in mapping between T p P and g. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

23 The hamster ball Consider a hamster ball on a two-dimensional surface: Two-dimensional Riemannian manifold (M, g). Orthonormal frame bundle π : P M is principal SO(2)-bundle. Hamster position and orientation marks frame p P. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

24 The hamster ball Consider a hamster ball on a two-dimensional surface: Two-dimensional Riemannian manifold (M, g). Orthonormal frame bundle π : P M is principal SO(2)-bundle. Hamster position and orientation marks frame p P. Degrees of freedom of the hamster T p P: Rotations around its position x = π(p). Rolling without slippling over M. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

25 The hamster ball Consider a hamster ball on a two-dimensional surface: Two-dimensional Riemannian manifold (M, g). Orthonormal frame bundle π : P M is principal SO(2)-bundle. Hamster position and orientation marks frame p P. Degrees of freedom of the hamster T p P ball motions so(3): Rotations around its position x = π(p): subalgebra so(2). Rolling without slippling over M: quotient space so(3)/so(2). Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

26 The hamster ball Consider a hamster ball on a two-dimensional surface: Two-dimensional Riemannian manifold (M, g). Orthonormal frame bundle π : P M is principal SO(2)-bundle. Hamster position and orientation marks frame p P. Degrees of freedom of the hamster T p P ball motions so(3): Rotations around its position x = π(p): subalgebra so(2). Rolling without slippling over M: quotient space so(3)/so(2). Surface M modeled by homogeneous space SO(3)/ SO(2). Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

27 The hamster ball Consider a hamster ball on a two-dimensional surface: Two-dimensional Riemannian manifold (M, g). Orthonormal frame bundle π : P M is principal SO(2)-bundle. Hamster position and orientation marks frame p P. Degrees of freedom of the hamster T p P ball motions so(3): Rotations around its position x = π(p): subalgebra so(2). Rolling without slippling over M: quotient space so(3)/so(2). Surface M modeled by homogeneous space SO(3)/ SO(2) = S 2. Geometry of M encoded in Hamster ball motion. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

28 Cartan geometry Choose a Lie group G with a closed subgroup H G. Choose a principal H-bundle π : P M with dim P = dim G. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

29 Cartan geometry Choose a Lie group G with a closed subgroup H G. Choose a principal H-bundle π : P M with dim P = dim G. For each point p P with π(p) = x identify: T p P g T p π 1 (x) h Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

30 Cartan geometry Choose a Lie group G with a closed subgroup H G. Choose a principal H-bundle π : P M with dim P = dim G. For each point p P with π(p) = x identify: T p P A p g T p π 1 (x) h A p Cartan connection A Ω 1 (P, g): 1-form on P with values in g. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

31 Cartan geometry Choose a Lie group G with a closed subgroup H G. Choose a principal H-bundle π : P M with dim P = dim G. For each point p P with π(p) = x identify: T p P A p g T p π 1 (x) h A p Cartan connection A Ω 1 (P, g): 1-form on P with values in g. Fundamental vector fields A : g Γ(TP) as inverse of A. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

32 Cartan geometry Choose a Lie group G with a closed subgroup H G. Choose a principal H-bundle π : P M with dim P = dim G. For each point p P with π(p) = x identify: T p P A p g T p π 1 (x) h A p Cartan connection A Ω 1 (P, g): 1-form on P with values in g. Fundamental vector fields A : g Γ(TP) as inverse of A. Geometry of M encoded in A resp. A. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

33 Example: Cartan geometry of spacetime Choose Lie groups: Let SO 0 (4, 1) Λ > 0 G = ISO 0 (3, 1) Λ = 0, H = SO 0 (3, 1). SO 0 (3, 2) Λ < 0 Coset spaces G/H are the maximally symmetric spacetimes. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

34 Example: Cartan geometry of spacetime Choose Lie groups: Let SO 0 (4, 1) Λ > 0 G = ISO 0 (3, 1) Λ = 0, H = SO 0 (3, 1). SO 0 (3, 2) Λ < 0 Coset spaces G/H are the maximally symmetric spacetimes. Choose principal H-bundle: Let (M, g) be a Lorentzian manifold. Let P be the oriented time-oriented orthonormal frames on M. π : P M is a principal H-bundle. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

35 Example: Cartan geometry of spacetime Choose Lie groups: Let SO 0 (4, 1) Λ > 0 G = ISO 0 (3, 1) Λ = 0, H = SO 0 (3, 1). SO 0 (3, 2) Λ < 0 Coset spaces G/H are the maximally symmetric spacetimes. Choose principal H-bundle: Let (M, g) be a Lorentzian manifold. Let P be the oriented time-oriented orthonormal frames on M. π : P M is a principal H-bundle. Choose Cartan connection: g = h z splits into direct sum. Let e Ω 1 (P, z) be the solder form of π : P M. Let ω Ω 1 (P, h) be the Levi-Civita connection. A = ω + e Ω 1 (P, g) is a Cartan connection. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

36 Example: Cartan geometry of spacetime Choose Lie groups: Let SO 0 (4, 1) Λ > 0 G = ISO 0 (3, 1) Λ = 0, H = SO 0 (3, 1). SO 0 (3, 2) Λ < 0 Coset spaces G/H are the maximally symmetric spacetimes. Choose principal H-bundle: Let (M, g) be a Lorentzian manifold. Let P be the oriented time-oriented orthonormal frames on M. π : P M is a principal H-bundle. Choose Cartan connection: g = h z splits into direct sum. Let e Ω 1 (P, z) be the solder form of π : P M. Let ω Ω 1 (P, h) be the Levi-Civita connection. A = ω + e Ω 1 (P, g) is a Cartan connection. Metric g can be reconstructed from Cartan geometry. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

37 Example: Cartan geometry of observer space Choose Lie groups: [S. Gielen, D. Wise 12] Let SO 0 (4, 1) Λ > 0 G = ISO 0 (3, 1) Λ = 0, K = SO(3). SO 0 (3, 2) Λ < 0 Coset spaces G/K are the maximally symmetric observer spaces. Choose principal K -bundle: Let (M, g) be a Lorentzian manifold. Let O be the future unit timelike vectors on M. Let P be the oriented time-oriented orthonormal frames on M. π : P O is a principal K -bundle. Choose Cartan connection: g = h z splits into direct sum. Let e Ω 1 (P, z) be the solder form of π : P M. Let ω Ω 1 (P, h) be the Levi-Civita connection. A = ω + e Ω 1 (P, g) is a Cartan connection. Metric g can be reconstructed from Cartan geometry. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

38 Outline 1 Introduction 2 Cartan geometry on observer space 3 Finsler spacetimes 4 From Finsler geometry to Cartan geometry 5 From Cartan geometry to Finsler geometry 6 Closing the circle 7 Finsler-Cartan-Gravity 8 Conclusion Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

39 The clock postulate Proper time along a curve in Lorentzian spacetime: τ = t2 t 1 g ab (x(t))ẋ a (t)ẋ b (t)dt. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

40 The clock postulate Proper time along a curve in Lorentzian spacetime: τ = t2 t 1 g ab (x(t))ẋ a (t)ẋ b (t)dt. Finsler geometry: use a more general length functional: τ = t2 Finsler function F : TM R +. t 1 F(x(t), ẋ(t))dt. Parametrization invariance requires homogeneity: F(x, λy) = λf(x, y) λ > 0. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

41 Definition of Finsler spacetimes Finsler geometries suitable for spacetimes exist. [C. Pfeifer, M. Wohlfarth 11] Finsler metric with Lorentz signature: gab F (x, y) = 1 2 y a y b F 2 (x, y). Notion of timelike, lightlike, spacelike tangent vectors. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

42 Definition of Finsler spacetimes Finsler geometries suitable for spacetimes exist. [C. Pfeifer, M. Wohlfarth 11] Finsler metric with Lorentz signature: gab F (x, y) = 1 2 y a y b F 2 (x, y). Notion of timelike, lightlike, spacelike tangent vectors. Unit vectors y T x M defined by F 2 (x, y) = g F ab (x, y)y a y b = 1. Set Ω x T x M of unit timelike vectors at x M. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

43 Definition of Finsler spacetimes Finsler geometries suitable for spacetimes exist. [C. Pfeifer, M. Wohlfarth 11] Finsler metric with Lorentz signature: gab F (x, y) = 1 2 y a y b F 2 (x, y). Notion of timelike, lightlike, spacelike tangent vectors. Unit vectors y T x M defined by F 2 (x, y) = g F ab (x, y)y a y b = 1. Set Ω x T x M of unit timelike vectors at x M. Ω x contains a closed connected component S x Ω x. Causality: S x corresponds to physical observers. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

44 Outline 1 Introduction 2 Cartan geometry on observer space 3 Finsler spacetimes 4 From Finsler geometry to Cartan geometry 5 From Cartan geometry to Finsler geometry 6 Closing the circle 7 Finsler-Cartan-Gravity 8 Conclusion Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

45 Observer space Recall from the definition of Finsler spacetimes: Set Ω x T x M of unit timelike vectors at x M. Physical observers correspond to S x Ω x. Definition of observer space: O = x M S x. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

46 Observer space Recall from the definition of Finsler spacetimes: Set Ω x T x M of unit timelike vectors at x M. Physical observers correspond to S x Ω x. Definition of observer space: O = x M S x. Tangent vectors y S x satisfy g F ab (x, y)y a y b = 1. Complete y = f 0 to a frame f i with gab F (x, y)f a i fj b = η ij. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

47 Observer space Recall from the definition of Finsler spacetimes: Set Ω x T x M of unit timelike vectors at x M. Physical observers correspond to S x Ω x. Definition of observer space: O = x M S x. Tangent vectors y S x satisfy g F ab (x, y)y a y b = 1. Complete y = f 0 to a frame f i with gab F (x, y)f a i fj b = η ij. Let P be the space of all observer frames. π : P O is a principal SO(3)-bundle. In general no principal SO 0 (3, 1)-bundle π : P M. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

48 Construction of the Cartan connection Need to construct A Ω 1 (P, g). Recall that g = h z A = ω + e Need to construct ω Ω 1 (P, h) and e Ω 1 (P, z). Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

49 Construction of the Cartan connection Need to construct A Ω 1 (P, g). Recall that g = h z A = ω + e Need to construct ω Ω 1 (P, h) and e Ω 1 (P, z). Definition of e: Use the solder form. Let w T (x,f ) P be a tangent vector. Differential of the projection π : P M yields π (w) T x M. View frame f as a linear isometry f : z T x M. Solder form given by e(w) = f 1 ( π (w)). Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

50 Construction of the Cartan connection Need to construct A Ω 1 (P, g). Recall that g = h z A = ω + e Need to construct ω Ω 1 (P, h) and e Ω 1 (P, z). Definition of e: Use the solder form. Let w T (x,f ) P be a tangent vector. Differential of the projection π : P M yields π (w) T x M. View frame f as a linear isometry f : z T x M. Solder form given by e(w) = f 1 ( π (w)). Definition of ω: Use the Cartan linear connection. Tangent vector w T (x,f ) P shifts frame f by small amount. Compare shifted frame with parallely transported frame. Both frames differ by Lorentz transform. [C. Pfeifer, M. Wohlfarth 11] Measure the difference using the original frame: f a i = ɛf a j ω j i(w). Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

51 Cartan connection in components Translational part e Ω 1 (P, z): e i = f 1i adx a. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

52 Cartan connection in components Translational part e Ω 1 (P, z): e i = f 1i adx a. Boost / rotational part ω Ω 1 (P, h): [ ( )] ω i j = f 1i a dfj a + fj b dx c F a bc + (dx d N c d + df0 c )Ca bc. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

53 Cartan connection in components Translational part e Ω 1 (P, z): e i = f 1i adx a. Boost / rotational part ω Ω 1 (P, h): [ ( )] ω i j = f 1i a dfj a + fj b dx c F a bc + (dx d N c d + df0 c )Ca bc Coefficients of Cartan linear connection: N a b = 1 4 b [g ( F aq y p p q F 2 q F 2)], F a bc = 1 ( ) 2 gf ap δ b gpc F + δ c gbp F δ pgbc F, C a bc = 1 ( ) 2 gf ap b gpc F + c gbp F p gbc F.. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

54 Cartan connection in components Translational part e Ω 1 (P, z): e i = f 1i adx a. Boost / rotational part ω Ω 1 (P, h): [ ( )] ω i j = f 1i a dfj a + fj b dx c F a bc + (dx d N c d + df0 c )Ca bc Coefficients of Cartan linear connection: N a b = 1 4 b [g ( F aq y p p q F 2 q F 2)], F a bc = 1 ( ) 2 gf ap δ b gpc F + δ c gbp F δ pgbc F, C a bc = 1 ( ) 2 gf ap b gpc F + c gbp F p gbc F. A = ω + e is a Cartan connection on π : P O. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36.

55 Fundamental vector fields Let a = z i Z i hi jh i j g. Define the vector field A(a) = z i f a i ( a f b j F c ab j c ) ( + h i jfi a h i 0fi b fj c C bc) a a j. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

56 Fundamental vector fields Let a = z i Z i hi jh i j g. Define the vector field A(a) = z i f a i ( a f b j F c ab j c ) ( + h i jfi a h i 0fi b fj c C bc) a a j. For all p P we find For all w T p P we find A(A(a)(p)) = a. A(A(w))(p) = w. A p : T p P g and A p : g T p P complement each other. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

57 Fundamental vector fields Let a = z i Z i hi jh i j g. Define the vector field A(a) = z i f a i ( a f b j F c ab j c ) ( + h i jfi a h i 0fi b fj c C bc) a a j. For all p P we find For all w T p P we find A(A(a)(p)) = a. A(A(w))(p) = w. A p : T p P g and A p : g T p P complement each other. Horizontal vector fields A(z): translations. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

58 Fundamental vector fields Let a = z i Z i hi jh i j g. Define the vector field A(a) = z i f a i ( a f b j F c ab j c ) ( + h i jfi a h i 0fi b fj c C bc) a a j. For all p P we find For all w T p P we find A(A(a)(p)) = a. A(A(w))(p) = w. A p : T p P g and A p : g T p P complement each other. Horizontal vector fields A(z): translations. Vertical vector fields A(h): Lorentz transforms. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

59 Time translation Consider the fundamental vector field of the time translator Z 0, t = A(Z 0 ) = f a 0 a f a j N b a j b ω i j(t) = 0, e i (t) = δ i 0. Integral curve Γ : R P, λ (x(λ), f (λ)) of t. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

60 Time translation Consider the fundamental vector field of the time translator Z 0, t = A(Z 0 ) = f a 0 a f a j N b a j b ω i j(t) = 0, e i (t) = δ i 0. Integral curve Γ : R P, λ (x(λ), f (λ)) of t. From e i (t) = δ i 0 follows: ẋ a = f a 0. (x, f 0 ) is the canonical lift of a curve from M to O. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

61 Time translation Consider the fundamental vector field of the time translator Z 0, t = A(Z 0 ) = f a 0 a f a j N b a j b ω i j(t) = 0, e i (t) = δ i 0. Integral curve Γ : R P, λ (x(λ), f (λ)) of t. From e i (t) = δ i 0 follows: ẋ a = f a 0. (x, f 0 ) is the canonical lift of a curve from M to O. From ω i 0(t) = 0 follows: (x, f 0 ) is a Finsler geodesic. 0 = ḟ a 0 + Na bẋ b = ẍ a + N a bẋ b. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

62 Time translation Consider the fundamental vector field of the time translator Z 0, t = A(Z 0 ) = f a 0 a f a j N b a j b ω i j(t) = 0, e i (t) = δ i 0. Integral curve Γ : R P, λ (x(λ), f (λ)) of t. From e i (t) = δ i 0 follows: ẋ a = f a 0. (x, f 0 ) is the canonical lift of a curve from M to O. From ω i 0(t) = 0 follows: 0 = ḟ a 0 + Na bẋ b = ẍ a + N a bẋ b. (x, f 0 ) is a Finsler geodesic. From ω α β(t) = 0 follows: ( ) 0 = ḟ α a + fα b ẋ c F a bc + (ẋ d N c d + ḟ 0 c )Ca bc = (ẋ, ḟ 0 ) f α a. Frame f is parallely transported. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

63 Time translation Consider the fundamental vector field of the time translator Z 0, t = A(Z 0 ) = f a 0 a f a j N b a j b ω i j(t) = 0, e i (t) = δ i 0. Integral curve Γ : R P, λ (x(λ), f (λ)) of t. From e i (t) = δ i 0 follows: ẋ a = f a 0. (x, f 0 ) is the canonical lift of a curve from M to O. From ω i 0(t) = 0 follows: 0 = ḟ a 0 + Na bẋ b = ẍ a + N a bẋ b. (x, f 0 ) is a Finsler geodesic. From ω α β(t) = 0 follows: ( ) 0 = ḟ α a + fα b ẋ c F a bc + (ẋ d N c d + ḟ 0 c )Ca bc = (ẋ, ḟ 0 ) f α a. Frame f is parallely transported. Integral curves of t define freely falling observers. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

64 Curvature of the Cartan connection Curvature F = F h + F z Ω 2 (P, g) defined by F = da + 1 [A, A]. 2 Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

65 Curvature of the Cartan connection Curvature F = F h + F z Ω 2 (P, g) defined by F = da + 1 [A, A]. 2 Translational part F z Ω 2 (P, z) ( torsion ): F z = de i + ω i j e j = f 1i ac a bcdx b δf c 0 with δf c 0 = Nc ddx d + df c 0. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

66 Curvature of the Cartan connection Curvature F = F h + F z Ω 2 (P, g) defined by F = da + 1 [A, A]. 2 Translational part F z Ω 2 (P, z) ( torsion ): F z = de i + ω i j e j = f 1i ac a bcdx b δf c 0 with δf0 c = Nc ddx d + df0 c. Boost / rotational part F h Ω 2 (P, h): F h = dω i j + ω i k ω k j = 1 ( 2 f 1i d f j c R d cabdx a dx b + 2P d cabdx a δf b 0 + Sd cabδf a 0 δf b 0 ). Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

67 Curvature of the Cartan connection Curvature F = F h + F z Ω 2 (P, g) defined by F = da + 1 [A, A]. 2 Translational part F z Ω 2 (P, z) ( torsion ): F z = de i + ω i j e j = f 1i ac a bcdx b δf c 0 with δf0 c = Nc ddx d + df0 c. Boost / rotational part F h Ω 2 (P, h): F h = dω i j + ω i k ω k j = 1 ( 2 f 1i d f j c R d cabdx a dx b + 2P d cabdx a δf b 0 + Sd cabδf a 0 δf b 0 R d cab, P d cab, S d cab: curvature of Cartan linear connection. Cartan geometry reproduces well-known Finsler objects. ). Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

68 Outline 1 Introduction 2 Cartan geometry on observer space 3 Finsler spacetimes 4 From Finsler geometry to Cartan geometry 5 From Cartan geometry to Finsler geometry 6 Closing the circle 7 Finsler-Cartan-Gravity 8 Conclusion Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

69 Spacetime Condition 1: boost distribution A(h) must be integrable. A(h) can be integrated to a foliation F with leaf space M. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

70 Spacetime Condition 1: boost distribution A(h) must be integrable. A(h) can be integrated to a foliation F with leaf space M. Condition 2: foliation F must be strictly simple. Leaf space M is a smooth manifold. Canonical projection π : P M is a submersion. Canonical projections π = π π: P π O π M π Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

71 Observer trajectories Four-velocity of an observer? Embedding of observer space O into TM? Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

72 Observer trajectories Four-velocity of an observer? Embedding of observer space O into TM? Fundamental vector field t = A(Z 0 ) Γ(TP) of time translation. t transforms trivially under spatial rotations. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

73 Observer trajectories Four-velocity of an observer? Embedding of observer space O into TM? Fundamental vector field t = A(Z 0 ) Γ(TP) of time translation. t transforms trivially under spatial rotations. Unique vector field r Γ(TO) such that: P π O t r TP π TO Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

74 Observer trajectories Four-velocity of an observer? Embedding of observer space O into TM? Fundamental vector field t = A(Z 0 ) Γ(TP) of time translation. t transforms trivially under spatial rotations. Unique vector field r Γ(TO) such that: P π O t r TP π TO π TM σ Integral curves λ o(λ) O of r must be canonical lifts: σ(o(λ)) = d dλ π (o(λ)) = π (ȯ(λ)) = π (r(o(λ))). Uniquely defined map σ = π r. Condition 3: σ must be an embedding. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

75 Finsler geometry Finsler function must be positively homogeneous of degree one: F(x, λy) = λ F(x, y) Unit timelike condition: F(σ(o)) = 1 for all observers o O. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

76 Finsler geometry Finsler function must be positively homogeneous of degree one: F(x, λy) = λ F(x, y) Unit timelike condition: F(σ(o)) = 1 for all observers o O. Define F(λσ(o)) = λ on double cone Rσ(O). Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

77 Finsler geometry Finsler function must be positively homogeneous of degree one: F(x, λy) = λ F(x, y) Unit timelike condition: F(σ(o)) = 1 for all observers o O. Define F(λσ(o)) = λ on double cone Rσ(O). Condition 4: σ(o) must intersect each line (x, Ry) at most once. Condition 5: Finsler metric gab F must have Lorentz signature: gab F = 1 2 a b F 2 Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

78 Finsler geometry Finsler function must be positively homogeneous of degree one: F(x, λy) = λ F(x, y) Unit timelike condition: F(σ(o)) = 1 for all observers o O. Define F(λσ(o)) = λ on double cone Rσ(O). Condition 4: σ(o) must intersect each line (x, Ry) at most once. Condition 5: Finsler metric gab F must have Lorentz signature: gab F = 1 2 a b F 2 Finsler spacetime geometry on Rσ(O). No Finsler geometry on TM \ Rσ(O). Cartan geometry describes only geometry visible to observers. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

79 Outline 1 Introduction 2 Cartan geometry on observer space 3 Finsler spacetimes 4 From Finsler geometry to Cartan geometry 5 From Cartan geometry to Finsler geometry 6 Closing the circle 7 Finsler-Cartan-Gravity 8 Conclusion Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

80 Reconstruction of a given Finsler spacetime Idea: Start from a Finsler spacetime (M, F ). Construct a Cartan observer space (π : P O, A). Construct a new Finsler spacetime ( ˆM, ˆF ). Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

81 Reconstruction of a given Finsler spacetime Idea: Start from a Finsler spacetime (M, F ). Construct a Cartan observer space (π : P O, A). Construct a new Finsler spacetime ( ˆM, ˆF ). Equivalence of Finsler spacetimes (M, F) and ( ˆM, ˆF)? Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

82 Reconstruction of a given Finsler spacetime Idea: Start from a Finsler spacetime (M, F ). Construct a Cartan observer space (π : P O, A). Construct a new Finsler spacetime ( ˆM, ˆF ). Equivalence of Finsler spacetimes (M, F) and ( ˆM, ˆF)? There exists a diffeomorphism µ: TM M π µ ˆπ T ˆM TO σ r O π µ ˆπ ˆM Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

83 Reconstruction of a given Finsler spacetime Idea: Start from a Finsler spacetime (M, F ). Construct a Cartan observer space (π : P O, A). Construct a new Finsler spacetime ( ˆM, ˆF ). Equivalence of Finsler spacetimes (M, F) and ( ˆM, ˆF)? There exists a diffeomorphism µ: TM M π µ ˆπ T ˆM TO σ r O π µ ˆπ ˆM µ preserves the Finsler function on timelike vectors. Reconstruction of the original Finsler geometry. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

84 Reconstruction of a given Cartan observer space Idea: Start from a Cartan observer space (π : P O, A). Construct a Finsler spacetime (M, F). Construct a new Cartan observer space (ˆπ : ˆP Ô, Â). Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

85 Reconstruction of a given Cartan observer space Idea: Start from a Cartan observer space (π : P O, A). Construct a Finsler spacetime (M, F). Construct a new Cartan observer space (ˆπ : ˆP Ô, Â). Equivalence of (π : P O, A) and (ˆπ : ˆP Ô, Â)? Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

86 Reconstruction of a given Cartan observer space Idea: Start from a Cartan observer space (π : P O, A). Construct a Finsler spacetime (M, F). Construct a new Cartan observer space (ˆπ : ˆP Ô, Â). Equivalence of (π : P O, A) and (ˆπ : ˆP Ô, Â)? Only if a Cartan morphism χ exists: M π ˆπ O Ô σ π ˆπ χ P ˆP A(a) TP χ T ˆP Â(a) Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

87 Reconstruction of a given Cartan observer space Idea: Start from a Cartan observer space (π : P O, A). Construct a Finsler spacetime (M, F). Construct a new Cartan observer space (ˆπ : ˆP Ô, Â). Equivalence of (π : P O, A) and (ˆπ : ˆP Ô, Â)? Only if a Cartan morphism χ exists: M π ˆπ O Ô σ π ˆπ χ P ˆP A(a) TP χ π TO σ π T ˆP T Ô ˆπ Â(a) ˆπ Every Cartan morphism χ = (x, f ) takes the form x(p) = π (π(p)), f i (p) = π (π (A(Z i )(p))) Simple test for equivalence of (π : P O, A) and (ˆπ : ˆP Ô, Â). Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36 TM

88 Outline 1 Introduction 2 Cartan geometry on observer space 3 Finsler spacetimes 4 From Finsler geometry to Cartan geometry 5 From Cartan geometry to Finsler geometry 6 Closing the circle 7 Finsler-Cartan-Gravity 8 Conclusion Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

89 Gravity from Cartan to Finsler MacDowell-Mansouri gravity on observer space: [S. Gielen, D. Wise 12] S G = ɛ αβγ tr h (F h F h ) b α b β b γ O Hodge operator on h. Non-degenerate H-invariant inner product tr h on h. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

90 Gravity from Cartan to Finsler MacDowell-Mansouri gravity on observer space: [S. Gielen, D. Wise 12] S G = ɛ αβγ tr h (F h F h ) b α b β b γ O Hodge operator on h. Non-degenerate H-invariant inner product tr h on h. Translate terms into Finsler language (with R = dω [ω, ω]): Curvature scalar: Cosmological constant: Gauss-Bonnet term: [e, e] R g F ab R c acb dv. [e, e] [e, e] dv. R R ɛ abcd ɛ efgh R abef R cdgh dv. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

91 Gravity from Cartan to Finsler MacDowell-Mansouri gravity on observer space: [S. Gielen, D. Wise 12] S G = ɛ αβγ tr h (F h F h ) b α b β b γ O Hodge operator on h. Non-degenerate H-invariant inner product tr h on h. Translate terms into Finsler language (with R = dω [ω, ω]): Curvature scalar: Cosmological constant: Gauss-Bonnet term: [e, e] R g F ab R c acb dv. [e, e] [e, e] dv. R R ɛ abcd ɛ efgh R abef R cdgh dv. Gravity theory on Finsler spacetime. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

92 Gravity from Finsler to Cartan Finsler gravity action: [C. Pfeifer, M. Wohlfarth 11] S G = O Sasaki metric G on O. Non-linear curvature R a ab. d 4 x d 3 y GR a aby b. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

93 Gravity from Finsler to Cartan Finsler gravity action: [C. Pfeifer, M. Wohlfarth 11] S G = O Sasaki metric G on O. Non-linear curvature R a ab. d 4 x d 3 y GR a aby b. Translate terms into Cartan language: d 4 x d 3 y G = ɛ ijkl ɛ αβγ e i e j e k e l b α b β b γ, R a aby b = b α [A(Z α ), A(Z 0 )]. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

94 Gravity from Finsler to Cartan Finsler gravity action: [C. Pfeifer, M. Wohlfarth 11] S G = O Sasaki metric G on O. Non-linear curvature R a ab. d 4 x d 3 y GR a aby b. Translate terms into Cartan language: d 4 x d 3 y G = ɛ ijkl ɛ αβγ e i e j e k e l b α b β b γ, R a aby b = b α [A(Z α ), A(Z 0 )]. Gravity theory on observer space. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

95 Outline 1 Introduction 2 Cartan geometry on observer space 3 Finsler spacetimes 4 From Finsler geometry to Cartan geometry 5 From Cartan geometry to Finsler geometry 6 Closing the circle 7 Finsler-Cartan-Gravity 8 Conclusion Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

96 Summary Observer space: Lift physics from spacetime to the space of observers. Describe observer space geometry using Cartan geometry. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

97 Summary Observer space: Lift physics from spacetime to the space of observers. Describe observer space geometry using Cartan geometry. Finsler spacetime: Based on generalized length functional. Finsler metric is observer dependent. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

98 Summary Observer space: Lift physics from spacetime to the space of observers. Describe observer space geometry using Cartan geometry. Finsler spacetime: Based on generalized length functional. Finsler metric is observer dependent. From Finsler to Cartan: Cartan geometry on observer space derived from Finsler geometry. Connection calculated from Cartan linear connection. Parallely transported observer frames given by the flow of time. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

99 Summary Observer space: Lift physics from spacetime to the space of observers. Describe observer space geometry using Cartan geometry. Finsler spacetime: Based on generalized length functional. Finsler metric is observer dependent. From Finsler to Cartan: Cartan geometry on observer space derived from Finsler geometry. Connection calculated from Cartan linear connection. Parallely transported observer frames given by the flow of time. From Cartan to Finsler: Spacetime can (sometimes) be constructed from Cartan geometry. Observer dependent Finsler metric from Cartan connection. Observers have timelike four-velocities in TM. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

100 Summary Observer space: Lift physics from spacetime to the space of observers. Describe observer space geometry using Cartan geometry. Finsler spacetime: Based on generalized length functional. Finsler metric is observer dependent. From Finsler to Cartan: Cartan geometry on observer space derived from Finsler geometry. Connection calculated from Cartan linear connection. Parallely transported observer frames given by the flow of time. From Cartan to Finsler: Spacetime can (sometimes) be constructed from Cartan geometry. Observer dependent Finsler metric from Cartan connection. Observers have timelike four-velocities in TM. Both constructions complement each other. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

101 Summary Observer space: Lift physics from spacetime to the space of observers. Describe observer space geometry using Cartan geometry. Finsler spacetime: Based on generalized length functional. Finsler metric is observer dependent. From Finsler to Cartan: Cartan geometry on observer space derived from Finsler geometry. Connection calculated from Cartan linear connection. Parallely transported observer frames given by the flow of time. From Cartan to Finsler: Spacetime can (sometimes) be constructed from Cartan geometry. Observer dependent Finsler metric from Cartan connection. Observers have timelike four-velocities in TM. Both constructions complement each other. Gravity: MacDowell-Mansouri gravity from Cartan to Finsler. Finsler gravity from Finsler to Cartan. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

102 Outlook Current projects: Derive gravitational equations of motion. Translate more terms between both languages. Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36

103 Outlook Current projects: Derive gravitational equations of motion. Translate more terms between both languages. Future projects: Consistent matter coupling. Study of exact solutions. Effects of deviations from metric geometry? Geometrodynamics of Finsler spacetimes.... Manuel Hohmann (Tartu Ülikool) Finsler vs. Cartan 13. November / 36