Vooludünaamika laiendatud geomeetrias

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1 Vooludünaamika laiendatud geomeetrias Manuel Hohmann Teoreetilise Füüsika Labor Füüsika Instituut Tartu Ülikool 28. oktoober 2014 Manuel Hohmann (Tartu Ülikool) Vooludünaamika 28. oktoober / 25

2 Fluid dynamics on generalized geometric backgrounds Manuel Hohmann Teoreetilise Füüsika Labor Füüsika Instituut Tartu Ülikool 28. oktoober 2014 Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

3 Outline 1 Motivation 2 Finsler geometry and observer space 3 Fluids on observer space 4 Conclusion Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

4 Outline 1 Motivation 2 Finsler geometry and observer space 3 Fluids on observer space 4 Conclusion Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

5 Fluids are everywhere Perfect fluid: No shear stress, no friction. Characterized by density ρ and pressure p. Dust, dark matter: p = 0. Radiation: p = 1 3 ρ. Dark energy: p < 1 3 ρ. Used in cosmology, parameterized post-newtonian formalism... Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

6 Fluids are everywhere Perfect fluid: No shear stress, no friction. Characterized by density ρ and pressure p. Dust, dark matter: p = 0. Radiation: p = 1 3 ρ. Dark energy: p < 1 3 ρ. Used in cosmology, parameterized post-newtonian formalism... Collisionless fluid: Model for dark matter. Used in structure formation... Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

7 Fluids are everywhere Perfect fluid: No shear stress, no friction. Characterized by density ρ and pressure p. Dust, dark matter: p = 0. Radiation: p = 1 3 ρ. Dark energy: p < 1 3 ρ. Used in cosmology, parameterized post-newtonian formalism... Collisionless fluid: Model for dark matter. Used in structure formation... Maxwell-Boltzmann gas: Collisions described by Boltzmann equation. Used in structure formation, atmosphere dynamics... Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

8 Fluids are everywhere Perfect fluid: No shear stress, no friction. Characterized by density ρ and pressure p. Dust, dark matter: p = 0. Radiation: p = 1 3 ρ. Dark energy: p < 1 3 ρ. Used in cosmology, parameterized post-newtonian formalism... Collisionless fluid: Model for dark matter. Used in structure formation... Maxwell-Boltzmann gas: Collisions described by Boltzmann equation. Used in structure formation, atmosphere dynamics... Charged, multi-component gas: Plasma, interacting gas including recombination / ionization. Used in stellar dynamics, pre-cmb era models... Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

9 From spacetime to observer space Fluid dynamics naturally lift to tangent bundle: Fluids conveniently modeled by particle dynamics (SPH... ). Physical fluids constituted by particles. Particle trajectories lift to tangent bundle: γ (γ, γ). Dynamics on the tangent bundle described by first order ODE. Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

10 From spacetime to observer space Fluid dynamics naturally lift to tangent bundle: Fluids conveniently modeled by particle dynamics (SPH... ). Physical fluids constituted by particles. Particle trajectories lift to tangent bundle: γ (γ, γ). Dynamics on the tangent bundle described by first order ODE. Velocity dependence of physical measurements: Physical observables are tensor components. Measured tensor components depend on observer velocity. Physical observer velocities are future unit timelike vectors. Observer space is space of physical velocities. Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

11 From spacetime to observer space Fluid dynamics naturally lift to tangent bundle: Fluids conveniently modeled by particle dynamics (SPH... ). Physical fluids constituted by particles. Particle trajectories lift to tangent bundle: γ (γ, γ). Dynamics on the tangent bundle described by first order ODE. Velocity dependence of physical measurements: Physical observables are tensor components. Measured tensor components depend on observer velocity. Physical observer velocities are future unit timelike vectors. Observer space is space of physical velocities. Quantum gravity: possible non-tensorial observer dependence. Modified gravity theories may have more general observer spaces. Physical observables become functions on observer space! Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

12 From spacetime to observer space Fluid dynamics naturally lift to tangent bundle: Fluids conveniently modeled by particle dynamics (SPH... ). Physical fluids constituted by particles. Particle trajectories lift to tangent bundle: γ (γ, γ). Dynamics on the tangent bundle described by first order ODE. Velocity dependence of physical measurements: Physical observables are tensor components. Measured tensor components depend on observer velocity. Physical observer velocities are future unit timelike vectors. Observer space is space of physical velocities. Quantum gravity: possible non-tensorial observer dependence. Modified gravity theories may have more general observer spaces. Physical observables become functions on observer space! Space of observers corresponds to particle tangent vectors. Consider fluid dynamics on observer space! Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

13 Finsler 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) Fluid dynamics 28. oktoober / 25

14 Finsler 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: Geometry described by Finsler function on the tangent bundle. Finsler function measures length of tangent vectors. Well-defined notions of connections, curvature, parallel transport... Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

15 Finsler 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: Geometry described by Finsler function on the tangent bundle. Finsler function measures length of tangent vectors. Well-defined notions of connections, curvature, parallel transport... Finsler spacetimes are suitable backgrounds for: Gravity Electrodynamics Other matter field theories Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

16 Finsler 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: Geometry described by Finsler function on the tangent bundle. Finsler function measures length of tangent vectors. 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) Fluid dynamics 28. oktoober / 25

17 Outline 1 Motivation 2 Finsler geometry and observer space 3 Fluids on observer space 4 Conclusion Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

18 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) Fluid dynamics 28. oktoober / 25

19 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) Fluid dynamics 28. oktoober / 25

20 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) Fluid dynamics 28. oktoober / 25

21 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) Fluid dynamics 28. oktoober / 25

22 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) Fluid dynamics 28. oktoober / 25

23 Geometry on the tangent bundle Cartan non-linear connection: N a b = 1 4 ] b [g F ac (y d d c F 2 c F 2 ) Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

24 Geometry on the tangent bundle Cartan non-linear connection: N a b = 1 4 ] b [g F ac (y d d c F 2 c F 2 ) Split of the tangent and cotangent bundles: Tangent bundle: TTM = HTM VTM δ a = a N b a b, a Cotangent bundle: T TM = H TM V TM dx a, δy a = dy a + N a bdx b Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

25 Geometry on the tangent bundle Cartan non-linear connection: N a b = 1 4 ] b [g F ac (y d d c F 2 c F 2 ) Split of the tangent and cotangent bundles: Tangent bundle: TTM = HTM VTM δ a = a N b a b, a Cotangent bundle: T TM = H TM V TM dx a, δy a = dy a + N a bdx b Sasaki metric: G = g F ab dx a dx b gf ab F 2 δy a δy b Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

26 Geometry on the tangent bundle Cartan non-linear connection: N a b = 1 4 ] b [g F ac (y d d c F 2 c F 2 ) Split of the tangent and cotangent bundles: Tangent bundle: TTM = HTM VTM δ a = a N b a b, a Cotangent bundle: T TM = H TM V TM dx a, δy a = dy a + N a bdx b Sasaki metric: G = g F ab dx a dx b gf ab F 2 δy a δy b Geodesic spray: S = y a δ a Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

27 Geometry on 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 TM. Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

28 Geometry on 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 TM. Sasaki metric G on O given by pullback of G to O. Volume form Σ of Sasaki metric G. Geodesic spray S restricts to Reeb vector field r on O. Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

29 Geometry on 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 TM. Sasaki metric G on O given by pullback of G to O. Volume form Σ of Sasaki metric G. Geodesic spray S restricts to Reeb vector field r on O. Geodesic hypersurface measure ω = ι r Σ. Note that L r Σ = 0 and dω = 0. Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

30 From metric to Finsler geometry Tangent bundle geometry: Finsler function: F (x, y) = g ab (x)y a y b Finsler metric: { gab(x, F g ab (x) y) = g ab (x) y timelike y spacelike Cartan non-linear connection: N a b(x, y) = Γ a bc(x)y c Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

31 From metric to Finsler geometry Tangent bundle geometry: Finsler function: F (x, y) = g ab (x)y a y b Finsler metric: { gab(x, F g ab (x) y) = g ab (x) y timelike y spacelike Cartan non-linear connection: N a b(x, y) = Γ a bc(x)y c Observer space: Space Ω x of unit timelike vectors at x M. Space S x of future unit timelike vectors at x M. Observer space O: union of shells S x. Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

32 Outline 1 Motivation 2 Finsler geometry and observer space 3 Fluids on observer space 4 Conclusion Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

33 Definition of fluids Single-component fluid: Constituted by classical, relativistic particles. Particles have equal properties (mass, charge,... ). Particles follow piecewise geodesic curves. Endpoints of geodesics are interactions with other particles. Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

34 Definition of fluids Single-component fluid: Constituted by classical, relativistic particles. Particles have equal properties (mass, charge,... ). Particles follow piecewise geodesic curves. Endpoints of geodesics are interactions with other particles. Collisionless fluid: Particles do not interact with other particles. Particles follow geodesics. Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

35 Definition of fluids Single-component fluid: Constituted by classical, relativistic particles. Particles have equal properties (mass, charge,... ). Particles follow piecewise geodesic curves. Endpoints of geodesics are interactions with other particles. Collisionless fluid: Particles do not interact with other particles. Particles follow geodesics. Multi-component fluid: multiple types of particles. Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

36 Geodesics on observer space Dynamics of fluids depends on geodesic equation. Geodesic equation for curve x(τ) on spacetime M: ẍ a + N a b(x, ẋ)ẋ b = 0. Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

37 Geodesics on observer space Dynamics of fluids depends on geodesic equation. Geodesic equation for curve x(τ) on spacetime M: ẍ a + N a b(x, ẋ)ẋ b = 0. Canonical lift of curve to tangent bundle TM: Lift of geodesic equation: x, y = ẋ. ẋ a = y a, ẏ a = N a b(x, y)y b. Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

38 Geodesics on observer space Dynamics of fluids depends on geodesic equation. Geodesic equation for curve x(τ) on spacetime M: ẍ a + N a b(x, ẋ)ẋ b = 0. Canonical lift of curve to tangent bundle TM: Lift of geodesic equation: x, y = ẋ. ẋ a = y a, ẏ a = N a b(x, y)y b. Solutions are integral curves of vector field: y a a y b N a b a = y a δ a = S. Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

39 Geodesics on observer space Dynamics of fluids depends on geodesic equation. Geodesic equation for curve x(τ) on spacetime M: ẍ a + N a b(x, ẋ)ẋ b = 0. Canonical lift of curve to tangent bundle TM: Lift of geodesic equation: x, y = ẋ. ẋ a = y a, ẏ a = N a b(x, y)y b. Solutions are integral curves of vector field: y a a y b N a b a = y a δ a = S. Tangent vectors are future unit timelike: (x, y) O. Particle trajectories are piecewise integral curves of r on O. Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

40 One-particle distribution function Recall: ω = ι r Σ Ω 6 (O) unique 6-form such that: ω non-degenerate on every hypersurface not tangent to r. dω = 0. Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

41 One-particle distribution function Recall: ω = ι r Σ Ω 6 (O) unique 6-form such that: ω non-degenerate on every hypersurface not tangent to r. dω = 0. Define one-particle distribution function φ : O R + such that: For every hypersurface σ O, N[σ] = φω # of particle trajectories through σ. σ Counting of particle trajectories respects hypersurface orientation. Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

42 One-particle distribution function Recall: ω = ι r Σ Ω 6 (O) unique 6-form such that: ω non-degenerate on every hypersurface not tangent to r. dω = 0. Define one-particle distribution function φ : O R + such that: For every hypersurface σ O, N[σ] = φω # of particle trajectories through σ. σ Counting of particle trajectories respects hypersurface orientation. For multi-component fluids: φ i for each component i. Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

43 Collisions & the Liouville equation Collision in spacetime interruption in observer space. Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

44 Collisions & the Liouville equation Collision in spacetime interruption in observer space. For any open set V O, φω = d(φω) = V V V L r φσ # of outbound trajectories - # of inbound trajectories. Collision density measured by L r φ. Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

45 Collisions & the Liouville equation Collision in spacetime interruption in observer space. For any open set V O, φω = d(φω) = V V V L r φσ # of outbound trajectories - # of inbound trajectories. Collision density measured by L r φ. Collisionless fluid: trajectories have no endpoints, L r φ = 0. Simple, first order equation of motion for collisionless fluid. φ is constant along integral curves of r. Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

46 Examples of fluids Geodesic dust fluid: φ(x, y) δ(y u(x)). Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

47 Examples of fluids Geodesic dust fluid: φ(x, y) δ(y u(x)). Jenkka Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

48 Examples of fluids Geodesic dust fluid: φ(x, y) δ(y u(x)). Collisionless fluid: L r φ = 0. Jenkka Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

49 Examples of fluids Geodesic dust fluid: φ(x, y) δ(y u(x)). Collisionless fluid: L r φ = 0. Jenkka Polkka Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

50 Examples of fluids Geodesic dust fluid: φ(x, y) δ(y u(x)). Collisionless fluid: L r φ = 0. Interacting fluid: L r φ 0. Jenkka Polkka Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

51 Examples of fluids Geodesic dust fluid: φ(x, y) δ(y u(x)). Collisionless fluid: L r φ = 0. Interacting fluid: L r φ 0. Jenkka Polkka Humppa Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

52 Averaged quantities Volume form Π x on unit timelike shells S x induced by G. Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

53 Averaged quantities Volume form Π x on unit timelike shells S x induced by G. How many particles pass hypersurface in spacetime? Averaged particle current density: J a (x) = φy a Π x S x Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

54 Averaged quantities Volume form Π x on unit timelike shells S x induced by G. How many particles pass hypersurface in spacetime? Averaged particle current density: J a (x) = φy a Π x S x How much energy-momentum passes hypersurface in spacetime? Averaged particle energy momentum tensor: T ab (x) = m φy a y b Π x S x Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

55 Averaged quantities Volume form Π x on unit timelike shells S x induced by G. How many particles pass hypersurface in spacetime? Averaged particle current density: J a (x) = φy a Π x S x How much energy-momentum passes hypersurface in spacetime? Averaged particle energy momentum tensor: T ab (x) = m φy a y b Π x S x Connection to well-known spacetime observables. Connection to measurements. Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

56 Symmetric solutions Infinitesimal diffeomorphism described by vector field ξ on M. Canonical lift of ξ to vector field on TM: ˆξ = ξ a a + y a a ξ b b Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

57 Symmetric solutions Infinitesimal diffeomorphism described by vector field ξ on M. Canonical lift of ξ to vector field on TM: ˆξ = ξ a a + y a a ξ b b Killing vector field ξ: Lˆξ F = 0 ˆξ is tangent to observer space O TM. Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

58 Symmetric solutions Infinitesimal diffeomorphism described by vector field ξ on M. Canonical lift of ξ to vector field on TM: ˆξ = ξ a a + y a a ξ b b Killing vector field ξ: Lˆξ F = 0 ˆξ is tangent to observer space O TM. Symmetric fluid solution: Lˆξ φ = 0 Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

59 Symmetric solutions Infinitesimal diffeomorphism described by vector field ξ on M. Canonical lift of ξ to vector field on TM: ˆξ = ξ a a + y a a ξ b b Killing vector field ξ: Lˆξ F = 0 ˆξ is tangent to observer space O TM. Symmetric fluid solution: Lˆξ φ = 0 Symmetry provides simplification of 7-dimensional O: Spherical symmetry: 4 dimensions remain. Static spherical symmetry: 3 dimensions remain. Cosmological symmetry: 2 dimensions remain. Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

60 Outline 1 Motivation 2 Finsler geometry and observer space 3 Fluids on observer space 4 Conclusion Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

61 Summary Basic idea: Model fluids by particle trajectories. Lift trajectories from spacetime to observer space. Describe geometry of observer space using Finsler geometry. Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

62 Summary Basic idea: Model fluids by particle trajectories. Lift trajectories from spacetime to observer space. Describe geometry of observer space using Finsler geometry. Finsler geometry suitable for generalized gravity theories. Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

63 Summary Basic idea: Model fluids by particle trajectories. Lift trajectories from spacetime to observer space. Describe geometry of observer space using Finsler geometry. Finsler geometry suitable for generalized gravity theories. Classical fluid variables obtained via averaging: Particle flux density. Energy-momentum tensor Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

64 Summary Basic idea: Model fluids by particle trajectories. Lift trajectories from spacetime to observer space. Describe geometry of observer space using Finsler geometry. Finsler geometry suitable for generalized gravity theories. Classical fluid variables obtained via averaging: Particle flux density. Energy-momentum tensor Symmetries defined as usual by Killing vector fields. Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

65 Outlook Coupling of fluids to non-metric gravity theories. Cosmological solutions with non-metric geometry. Extension of parameterized post-newtonian formalism.... Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25

66 References Kinetic theory on the tangent bundle: J. Ehlers, in: General Relativity and Cosmology, pp 1 70, Academic Press, New York / London, O. Sarbach and T. Zannias, AIP Conf. Proc (2013) 134 [arxiv: [gr-qc]]. O. Sarbach and T. Zannias, Class. Quant. Grav. 31 (2014) [arxiv: [gr-qc]]. Finsler spacetimes: C. Pfeifer and M. N. R. Wohlfarth, Phys. Rev. D 84 (2011) [arxiv: [gr-qc]]. C. Pfeifer and M. N. R. Wohlfarth, Phys. Rev. D 85 (2012) [arxiv: [gr-qc]]. MH, in: Mathematical structures of the Universe, pp 13 55, Copernicus Center Press, Krakow, Manuel Hohmann (Tartu Ülikool) Fluid dynamics 28. oktoober / 25