Finsler Steepest Descent with Applications to Piecewise-regular
Laboratory of Theoretical Physics Physics Institute...
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Fluid dynamics on Finsler spacetimes
Manuel Hohmann
Laboratory of Theoretical PhysicsPhysics Institute
University of Tartu
DPG-Tagung Berlin – Session GR 1519. März 2015
Manuel Hohmann (University of Tartu) Finsler fluid dynamics DPG Berlin 19. März 2015 1 / 11
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Motivation
Fluids are everywhere:Perfect fluid (radiation, dust, dark matter. . . ) - cosmology.Maxwell-Boltzmann gas - atmospheres.Plasma - stellar dynamics, primordial plasma.
Lift fluid dynamics to observer space:All measurements are performed by observers.Measurements depend on observer’s frame (velocity).Quantum gravity: possible non-tensorial velocity dependence.Observer space: space of all physical velocities.Fluids naturally modeled as densities on observer space.
Finsler spacetimes as observer space geometry:Finsler geometry of space widely used in physics.Finsler geometry generalizes Riemannian geometry.Finsler spacetimes are suitable backgrounds for physics.Possible explanations of yet unexplained phenomena.
Manuel Hohmann (University of Tartu) Finsler fluid dynamics DPG Berlin 19. März 2015 2 / 11
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Motivation
Fluids are everywhere:Perfect fluid (radiation, dust, dark matter. . . ) - cosmology.Maxwell-Boltzmann gas - atmospheres.Plasma - stellar dynamics, primordial plasma.
Lift fluid dynamics to observer space:All measurements are performed by observers.Measurements depend on observer’s frame (velocity).Quantum gravity: possible non-tensorial velocity dependence.Observer space: space of all physical velocities.Fluids naturally modeled as densities on observer space.
Finsler spacetimes as observer space geometry:Finsler geometry of space widely used in physics.Finsler geometry generalizes Riemannian geometry.Finsler spacetimes are suitable backgrounds for physics.Possible explanations of yet unexplained phenomena.
Manuel Hohmann (University of Tartu) Finsler fluid dynamics DPG Berlin 19. März 2015 2 / 11
![Page 4: Laboratory of Theoretical Physics Physics Institute ...kodu.ut.ee/~manuel/talks/finsler/2015_03_19_berlin2.pdf · Fluid dynamics on Finsler spacetimes Manuel Hohmann Laboratory of](https://reader035.fdocuments.in/reader035/viewer/2022063015/5fd1e8e45bb62a653420866c/html5/thumbnails/4.jpg)
Motivation
Fluids are everywhere:Perfect fluid (radiation, dust, dark matter. . . ) - cosmology.Maxwell-Boltzmann gas - atmospheres.Plasma - stellar dynamics, primordial plasma.
Lift fluid dynamics to observer space:All measurements are performed by observers.Measurements depend on observer’s frame (velocity).Quantum gravity: possible non-tensorial velocity dependence.Observer space: space of all physical velocities.Fluids naturally modeled as densities on observer space.
Finsler spacetimes as observer space geometry:Finsler geometry of space widely used in physics.Finsler geometry generalizes Riemannian geometry.Finsler spacetimes are suitable backgrounds for physics.Possible explanations of yet unexplained phenomena.
Manuel Hohmann (University of Tartu) Finsler fluid dynamics DPG Berlin 19. März 2015 2 / 11
![Page 5: Laboratory of Theoretical Physics Physics Institute ...kodu.ut.ee/~manuel/talks/finsler/2015_03_19_berlin2.pdf · Fluid dynamics on Finsler spacetimes Manuel Hohmann Laboratory of](https://reader035.fdocuments.in/reader035/viewer/2022063015/5fd1e8e45bb62a653420866c/html5/thumbnails/5.jpg)
Finsler spacetime geometry
Generalize metric length measure to Finsler function:
τ =
∫ t2
t1
√|gab(x(t))xa(t)xb(t)|dt
∫ t2
t1F (x(t), x(t))dt .
Finsler function F : TM → R+.Parametrization invariance requires homogeneity:
F (x , λy) = λF (x , y) ∀λ > 0 .
Finsler spacetime [C. Pfeifer, M. Wohlfarth ’11]:Length measure for tangent vectors.Notion of timelike, lightlike, spacelike tangent vectors.“Future unit timelike” vectors: physically allowed velocities.
⇒ Observer space O ⊂ TM of allowed velocities.
Manuel Hohmann (University of Tartu) Finsler fluid dynamics DPG Berlin 19. März 2015 3 / 11
![Page 6: Laboratory of Theoretical Physics Physics Institute ...kodu.ut.ee/~manuel/talks/finsler/2015_03_19_berlin2.pdf · Fluid dynamics on Finsler spacetimes Manuel Hohmann Laboratory of](https://reader035.fdocuments.in/reader035/viewer/2022063015/5fd1e8e45bb62a653420866c/html5/thumbnails/6.jpg)
Finsler spacetime geometry
Generalize metric length measure to Finsler function:
τ =
∫ t2
t1
√|gab(x(t))xa(t)xb(t)|dt
∫ t2
t1F (x(t), x(t))dt .
Finsler function F : TM → R+.Parametrization invariance requires homogeneity:
F (x , λy) = λF (x , y) ∀λ > 0 .
Finsler spacetime [C. Pfeifer, M. Wohlfarth ’11]:Length measure for tangent vectors.Notion of timelike, lightlike, spacelike tangent vectors.“Future unit timelike” vectors: physically allowed velocities.
⇒ Observer space O ⊂ TM of allowed velocities.
Manuel Hohmann (University of Tartu) Finsler fluid dynamics DPG Berlin 19. März 2015 3 / 11
![Page 7: Laboratory of Theoretical Physics Physics Institute ...kodu.ut.ee/~manuel/talks/finsler/2015_03_19_berlin2.pdf · Fluid dynamics on Finsler spacetimes Manuel Hohmann Laboratory of](https://reader035.fdocuments.in/reader035/viewer/2022063015/5fd1e8e45bb62a653420866c/html5/thumbnails/7.jpg)
Finsler spacetime geometry
Generalize metric length measure to Finsler function:
τ =
∫ t2
t1
√|gab(x(t))xa(t)xb(t)|dt
∫ t2
t1F (x(t), x(t))dt .
Finsler function F : TM → R+.Parametrization invariance requires homogeneity:
F (x , λy) = λF (x , y) ∀λ > 0 .
Finsler spacetime [C. Pfeifer, M. Wohlfarth ’11]:Length measure for tangent vectors.Notion of timelike, lightlike, spacelike tangent vectors.“Future unit timelike” vectors: physically allowed velocities.
⇒ Observer space O ⊂ TM of allowed velocities.
Manuel Hohmann (University of Tartu) Finsler fluid dynamics DPG Berlin 19. März 2015 3 / 11
![Page 8: Laboratory of Theoretical Physics Physics Institute ...kodu.ut.ee/~manuel/talks/finsler/2015_03_19_berlin2.pdf · Fluid dynamics on Finsler spacetimes Manuel Hohmann Laboratory of](https://reader035.fdocuments.in/reader035/viewer/2022063015/5fd1e8e45bb62a653420866c/html5/thumbnails/8.jpg)
Finsler spacetime geometry
Generalize metric length measure to Finsler function:
τ =
∫ t2
t1
√|gab(x(t))xa(t)xb(t)|dt
∫ t2
t1F (x(t), x(t))dt .
Finsler function F : TM → R+.Parametrization invariance requires homogeneity:
F (x , λy) = λF (x , y) ∀λ > 0 .
Finsler spacetime [C. Pfeifer, M. Wohlfarth ’11]:Length measure for tangent vectors.Notion of timelike, lightlike, spacelike tangent vectors.“Future unit timelike” vectors: physically allowed velocities.
⇒ Observer space O ⊂ TM of allowed velocities.
Manuel Hohmann (University of Tartu) Finsler fluid dynamics DPG Berlin 19. März 2015 3 / 11
![Page 9: Laboratory of Theoretical Physics Physics Institute ...kodu.ut.ee/~manuel/talks/finsler/2015_03_19_berlin2.pdf · Fluid dynamics on Finsler spacetimes Manuel Hohmann Laboratory of](https://reader035.fdocuments.in/reader035/viewer/2022063015/5fd1e8e45bb62a653420866c/html5/thumbnails/9.jpg)
Point mass dynamics on observer space
Point mass follows curve γ : R→ M on spacetime M.γ is extremal curve of Finsler length measure:
δ
∫F (γ(t), γ(t))dt = 0 .
Canonical lift Γ of curve to tangent bundle TM:
Γ = (γ, γ) .
Lift of geodesic equation to TM is first order differential equation:
Γ(t) = S(Γ(t)).
⇒ Solutions are integral curves of vector field S on TM.Physically allowed velocities: Γ(t) ∈ O.Restriction r = S|O: Reeb vector field.
⇒ Physical geodesics are integral curves of r on O.
Manuel Hohmann (University of Tartu) Finsler fluid dynamics DPG Berlin 19. März 2015 4 / 11
![Page 10: Laboratory of Theoretical Physics Physics Institute ...kodu.ut.ee/~manuel/talks/finsler/2015_03_19_berlin2.pdf · Fluid dynamics on Finsler spacetimes Manuel Hohmann Laboratory of](https://reader035.fdocuments.in/reader035/viewer/2022063015/5fd1e8e45bb62a653420866c/html5/thumbnails/10.jpg)
Point mass dynamics on observer space
Point mass follows curve γ : R→ M on spacetime M.γ is extremal curve of Finsler length measure:
δ
∫F (γ(t), γ(t))dt = 0 .
Canonical lift Γ of curve to tangent bundle TM:
Γ = (γ, γ) .
Lift of geodesic equation to TM is first order differential equation:
Γ(t) = S(Γ(t)).
⇒ Solutions are integral curves of vector field S on TM.
Physically allowed velocities: Γ(t) ∈ O.Restriction r = S|O: Reeb vector field.
⇒ Physical geodesics are integral curves of r on O.
Manuel Hohmann (University of Tartu) Finsler fluid dynamics DPG Berlin 19. März 2015 4 / 11
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Point mass dynamics on observer space
Point mass follows curve γ : R→ M on spacetime M.γ is extremal curve of Finsler length measure:
δ
∫F (γ(t), γ(t))dt = 0 .
Canonical lift Γ of curve to tangent bundle TM:
Γ = (γ, γ) .
Lift of geodesic equation to TM is first order differential equation:
Γ(t) = S(Γ(t)).
⇒ Solutions are integral curves of vector field S on TM.Physically allowed velocities: Γ(t) ∈ O.Restriction r = S|O: Reeb vector field.
⇒ Physical geodesics are integral curves of r on O.
Manuel Hohmann (University of Tartu) Finsler fluid dynamics DPG Berlin 19. März 2015 4 / 11
![Page 12: Laboratory of Theoretical Physics Physics Institute ...kodu.ut.ee/~manuel/talks/finsler/2015_03_19_berlin2.pdf · Fluid dynamics on Finsler spacetimes Manuel Hohmann Laboratory of](https://reader035.fdocuments.in/reader035/viewer/2022063015/5fd1e8e45bb62a653420866c/html5/thumbnails/12.jpg)
From particles to fluids
Model fluid by classical, relativistic particles:Particles follow piecewise geodesic curves.Endpoints of geodesics are interactions with other particles.
Particle measure: ω ∈ Ω6(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] =
∫σ
φω
# of particle trajectories through σ.
Counting of particle trajectories respects hypersurface orientation.
Manuel Hohmann (University of Tartu) Finsler fluid dynamics DPG Berlin 19. März 2015 5 / 11
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From particles to fluids
Model fluid by classical, relativistic particles:Particles follow piecewise geodesic curves.Endpoints of geodesics are interactions with other particles.
Particle measure: ω ∈ Ω6(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] =
∫σ
φω
# of particle trajectories through σ.
Counting of particle trajectories respects hypersurface orientation.
Manuel Hohmann (University of Tartu) Finsler fluid dynamics DPG Berlin 19. März 2015 5 / 11
![Page 14: Laboratory of Theoretical Physics Physics Institute ...kodu.ut.ee/~manuel/talks/finsler/2015_03_19_berlin2.pdf · Fluid dynamics on Finsler spacetimes Manuel Hohmann Laboratory of](https://reader035.fdocuments.in/reader035/viewer/2022063015/5fd1e8e45bb62a653420866c/html5/thumbnails/14.jpg)
From particles to fluids
Model fluid by classical, relativistic particles:Particles follow piecewise geodesic curves.Endpoints of geodesics are interactions with other particles.
Particle measure: ω ∈ Ω6(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] =
∫σ
φω
# of particle trajectories through σ.
Counting of particle trajectories respects hypersurface orientation.
Manuel Hohmann (University of Tartu) Finsler fluid dynamics DPG Berlin 19. März 2015 5 / 11
![Page 15: Laboratory of Theoretical Physics Physics Institute ...kodu.ut.ee/~manuel/talks/finsler/2015_03_19_berlin2.pdf · Fluid dynamics on Finsler spacetimes Manuel Hohmann Laboratory of](https://reader035.fdocuments.in/reader035/viewer/2022063015/5fd1e8e45bb62a653420866c/html5/thumbnails/15.jpg)
Collisions & the Liouville equation
Collision in spacetime! interruption in observer space.
!
For any open set V ∈ O,∫∂Vφω =
∫V
d(φω) =
∫VLrφΣ
# of outbound trajectories - # of inbound trajectories.⇒ Collision density measured by Lrφ.
Collisionless fluid: trajectories have no endpoints, Lrφ = 0.⇒ Simple, first order equation of motion for collisionless fluid.⇒ φ is constant along integral curves of r.
Manuel Hohmann (University of Tartu) Finsler fluid dynamics DPG Berlin 19. März 2015 6 / 11
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Collisions & the Liouville equation
Collision in spacetime! interruption in observer space.
!For any open set V ∈ O,∫
∂Vφω =
∫V
d(φω) =
∫VLrφΣ
# of outbound trajectories - # of inbound trajectories.⇒ Collision density measured by Lrφ.
Collisionless fluid: trajectories have no endpoints, Lrφ = 0.⇒ Simple, first order equation of motion for collisionless fluid.⇒ φ is constant along integral curves of r.
Manuel Hohmann (University of Tartu) Finsler fluid dynamics DPG Berlin 19. März 2015 6 / 11
![Page 17: Laboratory of Theoretical Physics Physics Institute ...kodu.ut.ee/~manuel/talks/finsler/2015_03_19_berlin2.pdf · Fluid dynamics on Finsler spacetimes Manuel Hohmann Laboratory of](https://reader035.fdocuments.in/reader035/viewer/2022063015/5fd1e8e45bb62a653420866c/html5/thumbnails/17.jpg)
Collisions & the Liouville equation
Collision in spacetime! interruption in observer space.
!For any open set V ∈ O,∫
∂Vφω =
∫V
d(φω) =
∫VLrφΣ
# of outbound trajectories - # of inbound trajectories.⇒ Collision density measured by Lrφ.
Collisionless fluid: trajectories have no endpoints, Lrφ = 0.⇒ Simple, first order equation of motion for collisionless fluid.⇒ φ is constant along integral curves of r.
Manuel Hohmann (University of Tartu) Finsler fluid dynamics DPG Berlin 19. März 2015 6 / 11
![Page 18: Laboratory of Theoretical Physics Physics Institute ...kodu.ut.ee/~manuel/talks/finsler/2015_03_19_berlin2.pdf · Fluid dynamics on Finsler spacetimes Manuel Hohmann Laboratory of](https://reader035.fdocuments.in/reader035/viewer/2022063015/5fd1e8e45bb62a653420866c/html5/thumbnails/18.jpg)
Examples of fluids
Geodesic dust fluid:φ(x , y) ∼ δ(y−u(x)) .
“Jenkka”
Collisionless fluid:Lrφ = 0 .
“Polkka”
Interacting fluid:Lrφ 6= 0 .
“Humppa”
Manuel Hohmann (University of Tartu) Finsler fluid dynamics DPG Berlin 19. März 2015 7 / 11
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Examples of fluids
Geodesic dust fluid:φ(x , y) ∼ δ(y−u(x)) .
“Jenkka”
Collisionless fluid:Lrφ = 0 .
“Polkka”
Interacting fluid:Lrφ 6= 0 .
“Humppa”
Manuel Hohmann (University of Tartu) Finsler fluid dynamics DPG Berlin 19. März 2015 7 / 11
![Page 20: Laboratory of Theoretical Physics Physics Institute ...kodu.ut.ee/~manuel/talks/finsler/2015_03_19_berlin2.pdf · Fluid dynamics on Finsler spacetimes Manuel Hohmann Laboratory of](https://reader035.fdocuments.in/reader035/viewer/2022063015/5fd1e8e45bb62a653420866c/html5/thumbnails/20.jpg)
Examples of fluids
Geodesic dust fluid:φ(x , y) ∼ δ(y−u(x)) .
“Jenkka”
Collisionless fluid:Lrφ = 0 .
“Polkka”
Interacting fluid:Lrφ 6= 0 .
“Humppa”
Manuel Hohmann (University of Tartu) Finsler fluid dynamics DPG Berlin 19. März 2015 7 / 11
![Page 21: Laboratory of Theoretical Physics Physics Institute ...kodu.ut.ee/~manuel/talks/finsler/2015_03_19_berlin2.pdf · Fluid dynamics on Finsler spacetimes Manuel Hohmann Laboratory of](https://reader035.fdocuments.in/reader035/viewer/2022063015/5fd1e8e45bb62a653420866c/html5/thumbnails/21.jpg)
Examples of fluids
Geodesic dust fluid:φ(x , y) ∼ δ(y−u(x)) .
“Jenkka”
Collisionless fluid:Lrφ = 0 .
“Polkka”
Interacting fluid:Lrφ 6= 0 .
“Humppa”
Manuel Hohmann (University of Tartu) Finsler fluid dynamics DPG Berlin 19. März 2015 7 / 11
![Page 22: Laboratory of Theoretical Physics Physics Institute ...kodu.ut.ee/~manuel/talks/finsler/2015_03_19_berlin2.pdf · Fluid dynamics on Finsler spacetimes Manuel Hohmann Laboratory of](https://reader035.fdocuments.in/reader035/viewer/2022063015/5fd1e8e45bb62a653420866c/html5/thumbnails/22.jpg)
Examples of fluids
Geodesic dust fluid:φ(x , y) ∼ δ(y−u(x)) .
“Jenkka”
Collisionless fluid:Lrφ = 0 .
“Polkka”
Interacting fluid:Lrφ 6= 0 .
“Humppa”
Manuel Hohmann (University of Tartu) Finsler fluid dynamics DPG Berlin 19. März 2015 7 / 11
![Page 23: Laboratory of Theoretical Physics Physics Institute ...kodu.ut.ee/~manuel/talks/finsler/2015_03_19_berlin2.pdf · Fluid dynamics on Finsler spacetimes Manuel Hohmann Laboratory of](https://reader035.fdocuments.in/reader035/viewer/2022063015/5fd1e8e45bb62a653420866c/html5/thumbnails/23.jpg)
Examples of fluids
Geodesic dust fluid:φ(x , y) ∼ δ(y−u(x)) .
“Jenkka”
Collisionless fluid:Lrφ = 0 .
“Polkka”
Interacting fluid:Lrφ 6= 0 .
“Humppa”
Manuel Hohmann (University of Tartu) Finsler fluid dynamics DPG Berlin 19. März 2015 7 / 11
![Page 24: Laboratory of Theoretical Physics Physics Institute ...kodu.ut.ee/~manuel/talks/finsler/2015_03_19_berlin2.pdf · Fluid dynamics on Finsler spacetimes Manuel Hohmann Laboratory of](https://reader035.fdocuments.in/reader035/viewer/2022063015/5fd1e8e45bb62a653420866c/html5/thumbnails/24.jpg)
Example: dust fluid in Finsler geometry
Classical dust with density ρ(x) and velocity ua(x).One-partical distribution function on O:
φ(x , θ) =1mρ(x)
δ(θ − v(x))√det hF (x , θ)
.
No collisions⇒ Liouville equation Lrφ = 0.⇒ Equations of motion for ρ and ua:
∇ua = 0 and ∇δa(ρua) = 0 .
Dynamical covariant derivative ∇.Horizontal part of Cartan linear connection ∇δa .
Metric background geometry F (x , y) =√|gab(x)yayb|:
ub∇bua = 0 and ∇a(ρua) = 0 .
⇒ Well-known Euler equations of fluid dynamics.
Manuel Hohmann (University of Tartu) Finsler fluid dynamics DPG Berlin 19. März 2015 8 / 11
![Page 25: Laboratory of Theoretical Physics Physics Institute ...kodu.ut.ee/~manuel/talks/finsler/2015_03_19_berlin2.pdf · Fluid dynamics on Finsler spacetimes Manuel Hohmann Laboratory of](https://reader035.fdocuments.in/reader035/viewer/2022063015/5fd1e8e45bb62a653420866c/html5/thumbnails/25.jpg)
Example: dust fluid in Finsler geometry
Classical dust with density ρ(x) and velocity ua(x).One-partical distribution function on O:
φ(x , θ) =1mρ(x)
δ(θ − v(x))√det hF (x , θ)
.
No collisions⇒ Liouville equation Lrφ = 0.⇒ Equations of motion for ρ and ua:
∇ua = 0 and ∇δa(ρua) = 0 .
Dynamical covariant derivative ∇.Horizontal part of Cartan linear connection ∇δa .
Metric background geometry F (x , y) =√|gab(x)yayb|:
ub∇bua = 0 and ∇a(ρua) = 0 .
⇒ Well-known Euler equations of fluid dynamics.
Manuel Hohmann (University of Tartu) Finsler fluid dynamics DPG Berlin 19. März 2015 8 / 11
![Page 26: Laboratory of Theoretical Physics Physics Institute ...kodu.ut.ee/~manuel/talks/finsler/2015_03_19_berlin2.pdf · Fluid dynamics on Finsler spacetimes Manuel Hohmann Laboratory of](https://reader035.fdocuments.in/reader035/viewer/2022063015/5fd1e8e45bb62a653420866c/html5/thumbnails/26.jpg)
Example: dust fluid in Finsler geometry
Classical dust with density ρ(x) and velocity ua(x).One-partical distribution function on O:
φ(x , θ) =1mρ(x)
δ(θ − v(x))√det hF (x , θ)
.
No collisions⇒ Liouville equation Lrφ = 0.⇒ Equations of motion for ρ and ua:
∇ua = 0 and ∇δa(ρua) = 0 .
Dynamical covariant derivative ∇.Horizontal part of Cartan linear connection ∇δa .
Metric background geometry F (x , y) =√|gab(x)yayb|:
ub∇bua = 0 and ∇a(ρua) = 0 .
⇒ Well-known Euler equations of fluid dynamics.
Manuel Hohmann (University of Tartu) Finsler fluid dynamics DPG Berlin 19. März 2015 8 / 11
![Page 27: Laboratory of Theoretical Physics Physics Institute ...kodu.ut.ee/~manuel/talks/finsler/2015_03_19_berlin2.pdf · Fluid dynamics on Finsler spacetimes Manuel Hohmann Laboratory of](https://reader035.fdocuments.in/reader035/viewer/2022063015/5fd1e8e45bb62a653420866c/html5/thumbnails/27.jpg)
Fluids with cosmological symmetry
Most general cosmological Finsler function F (t , y , w).Cosmological time t .Velocity component y in t-direction.Velocity component w perpendicular to t-direction.
Homogeneity: F determined by F as
F (t , y , w) = y F (t , w/y) .
Observer space O with y F (t , w/y) = 1.Most general fluid φ(t , w/y) with cosmological symmetry.Liouville equation Lrφ = 0:
Fwwφt = Ftwφw .
Robertson-Walker metric: F =√
1− a2(t)w2/y2:
φt = − wy
(w2
y2 a2 − 2)
aaφw .
Manuel Hohmann (University of Tartu) Finsler fluid dynamics DPG Berlin 19. März 2015 9 / 11
![Page 28: Laboratory of Theoretical Physics Physics Institute ...kodu.ut.ee/~manuel/talks/finsler/2015_03_19_berlin2.pdf · Fluid dynamics on Finsler spacetimes Manuel Hohmann Laboratory of](https://reader035.fdocuments.in/reader035/viewer/2022063015/5fd1e8e45bb62a653420866c/html5/thumbnails/28.jpg)
Fluids with cosmological symmetry
Most general cosmological Finsler function F (t , y , w).Cosmological time t .Velocity component y in t-direction.Velocity component w perpendicular to t-direction.
Homogeneity: F determined by F as
F (t , y , w) = y F (t , w/y) .
Observer space O with y F (t , w/y) = 1.
Most general fluid φ(t , w/y) with cosmological symmetry.Liouville equation Lrφ = 0:
Fwwφt = Ftwφw .
Robertson-Walker metric: F =√
1− a2(t)w2/y2:
φt = − wy
(w2
y2 a2 − 2)
aaφw .
Manuel Hohmann (University of Tartu) Finsler fluid dynamics DPG Berlin 19. März 2015 9 / 11
![Page 29: Laboratory of Theoretical Physics Physics Institute ...kodu.ut.ee/~manuel/talks/finsler/2015_03_19_berlin2.pdf · Fluid dynamics on Finsler spacetimes Manuel Hohmann Laboratory of](https://reader035.fdocuments.in/reader035/viewer/2022063015/5fd1e8e45bb62a653420866c/html5/thumbnails/29.jpg)
Fluids with cosmological symmetry
Most general cosmological Finsler function F (t , y , w).Cosmological time t .Velocity component y in t-direction.Velocity component w perpendicular to t-direction.
Homogeneity: F determined by F as
F (t , y , w) = y F (t , w/y) .
Observer space O with y F (t , w/y) = 1.Most general fluid φ(t , w/y) with cosmological symmetry.Liouville equation Lrφ = 0:
Fwwφt = Ftwφw .
Robertson-Walker metric: F =√
1− a2(t)w2/y2:
φt = − wy
(w2
y2 a2 − 2)
aaφw .
Manuel Hohmann (University of Tartu) Finsler fluid dynamics DPG Berlin 19. März 2015 9 / 11
![Page 30: Laboratory of Theoretical Physics Physics Institute ...kodu.ut.ee/~manuel/talks/finsler/2015_03_19_berlin2.pdf · Fluid dynamics on Finsler spacetimes Manuel Hohmann Laboratory of](https://reader035.fdocuments.in/reader035/viewer/2022063015/5fd1e8e45bb62a653420866c/html5/thumbnails/30.jpg)
Fluids with cosmological symmetry
Most general cosmological Finsler function F (t , y , w).Cosmological time t .Velocity component y in t-direction.Velocity component w perpendicular to t-direction.
Homogeneity: F determined by F as
F (t , y , w) = y F (t , w/y) .
Observer space O with y F (t , w/y) = 1.Most general fluid φ(t , w/y) with cosmological symmetry.Liouville equation Lrφ = 0:
Fwwφt = Ftwφw .
Robertson-Walker metric: F =√
1− a2(t)w2/y2:
φt = − wy
(w2
y2 a2 − 2)
aaφw .
Manuel Hohmann (University of Tartu) Finsler fluid dynamics DPG Berlin 19. März 2015 9 / 11
![Page 31: Laboratory of Theoretical Physics Physics Institute ...kodu.ut.ee/~manuel/talks/finsler/2015_03_19_berlin2.pdf · Fluid dynamics on Finsler spacetimes Manuel Hohmann Laboratory of](https://reader035.fdocuments.in/reader035/viewer/2022063015/5fd1e8e45bb62a653420866c/html5/thumbnails/31.jpg)
Conclusion
Basic idea:Model fluids by particle trajectories.Lift trajectories from spacetime to observer space.Describe geometry of observer space using Finsler geometry.Measure particle density by distribution function.Derive fluid dynamics from geodesic motion.
Presented examples:Classical dust fluid on Finsler spacetime.Most general cosmological fluid on Finsler spacetime.
Future research goals:Coupling of fluids to non-metric gravity theories.Cosmological solutions of gravity with non-metric geometry.Extension of parameterized post-Newtonian formalism.
Manuel Hohmann (University of Tartu) Finsler fluid dynamics DPG Berlin 19. März 2015 10 / 11
![Page 32: Laboratory of Theoretical Physics Physics Institute ...kodu.ut.ee/~manuel/talks/finsler/2015_03_19_berlin2.pdf · Fluid dynamics on Finsler spacetimes Manuel Hohmann Laboratory of](https://reader035.fdocuments.in/reader035/viewer/2022063015/5fd1e8e45bb62a653420866c/html5/thumbnails/32.jpg)
Conclusion
Basic idea:Model fluids by particle trajectories.Lift trajectories from spacetime to observer space.Describe geometry of observer space using Finsler geometry.Measure particle density by distribution function.Derive fluid dynamics from geodesic motion.
Presented examples:Classical dust fluid on Finsler spacetime.Most general cosmological fluid on Finsler spacetime.
Future research goals:Coupling of fluids to non-metric gravity theories.Cosmological solutions of gravity with non-metric geometry.Extension of parameterized post-Newtonian formalism.
Manuel Hohmann (University of Tartu) Finsler fluid dynamics DPG Berlin 19. März 2015 10 / 11
![Page 33: Laboratory of Theoretical Physics Physics Institute ...kodu.ut.ee/~manuel/talks/finsler/2015_03_19_berlin2.pdf · Fluid dynamics on Finsler spacetimes Manuel Hohmann Laboratory of](https://reader035.fdocuments.in/reader035/viewer/2022063015/5fd1e8e45bb62a653420866c/html5/thumbnails/33.jpg)
Conclusion
Basic idea:Model fluids by particle trajectories.Lift trajectories from spacetime to observer space.Describe geometry of observer space using Finsler geometry.Measure particle density by distribution function.Derive fluid dynamics from geodesic motion.
Presented examples:Classical dust fluid on Finsler spacetime.Most general cosmological fluid on Finsler spacetime.
Future research goals:Coupling of fluids to non-metric gravity theories.Cosmological solutions of gravity with non-metric geometry.Extension of parameterized post-Newtonian formalism.
Manuel Hohmann (University of Tartu) Finsler fluid dynamics DPG Berlin 19. März 2015 10 / 11
![Page 34: Laboratory of Theoretical Physics Physics Institute ...kodu.ut.ee/~manuel/talks/finsler/2015_03_19_berlin2.pdf · Fluid dynamics on Finsler spacetimes Manuel Hohmann Laboratory of](https://reader035.fdocuments.in/reader035/viewer/2022063015/5fd1e8e45bb62a653420866c/html5/thumbnails/34.jpg)
References
Kinetic theory on the tangent bundle:J. Ehlers, in: “General Relativity and Cosmology”,pp 1–70, Academic Press, New York / London, 1971.O. Sarbach and T. Zannias,AIP Conf. Proc. 1548 (2013) 134 [arXiv:1303.2899 [gr-qc]].O. Sarbach and T. Zannias,Class. Quant. Grav. 31 (2014) 085013 [arXiv:1309.2036 [gr-qc]].
Finsler spacetimes:C. Pfeifer and M. N. R. Wohlfarth,Phys. Rev. D 84 (2011) 044039 [arXiv:1104.1079 [gr-qc]].C. Pfeifer and M. N. R. Wohlfarth,Phys. Rev. D 85 (2012) 064009 [arXiv:1112.5641 [gr-qc]].MH, in: “Mathematical structures of the Universe”,pp 13–55, Copernicus Center Press, Krakow, 2014.
Manuel Hohmann (University of Tartu) Finsler fluid dynamics DPG Berlin 19. März 2015 11 / 11