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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

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.

Motivation

Finsler spacetime geometry

Generalize metric length measure to Finsler function:

∫ t2

√|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.

∫ t2

F (x , λy) = λF (x , y) ∀λ > 0 .

∫ t2

F (x , λy) = λF (x , y) ∀λ > 0 .

∫ t2

F (x , λy) = λF (x , y) ∀λ > 0 .

Point mass dynamics on observer space

Point mass follows curve γ : R→ M on spacetime M.γ is extremal curve of Finsler length measure:

∫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.

∫F (γ(t), γ(t))dt = 0 .

Γ = (γ, γ) .

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.

∫F (γ(t), γ(t))dt = 0 .

Γ = (γ, γ) .

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.

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] =

# of particle trajectories through σ.

Counting of particle trajectories respects hypersurface orientation.

N[σ] =

Collisions & the Liouville equation

Collision in spacetime! interruption in observer space.

For any open set V ∈ O,∫∂Vφω =

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.

!For any open set V ∈ O,∫

∂Vφω =

d(φω) =

∫VLrφΣ

!For any open set V ∈ O,∫

∂Vφω =

d(φω) =

∫VLrφΣ

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”

Examples of fluids

“Jenkka”

“Polkka”

“Humppa”

Examples of fluids

“Jenkka”

“Polkka”

“Humppa”

Examples of fluids

“Jenkka”

“Polkka”

“Humppa”

Examples of fluids

“Jenkka”

“Polkka”

“Humppa”

Examples of fluids

“Jenkka”

“Polkka”

“Humppa”

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.

φ(x , θ) =1mρ(x)

∇ua = 0 and ∇δa(ρua) = 0 .

φ(x , θ) =1mρ(x)

∇ua = 0 and ∇δa(ρua) = 0 .

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

y2 a2 − 2)

aaφw .

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 .

1− a2(t)w2/y2:

φt = − wy

y2 a2 − 2)

aaφw .

F (t , y , w) = y F (t , w/y) .

Fwwφt = Ftwφw .

1− a2(t)w2/y2:

φt = − wy

y2 a2 − 2)

aaφw .

F (t , y , w) = y F (t , w/y) .

Fwwφt = Ftwφw .

1− a2(t)w2/y2:

φt = − wy

y2 a2 − 2)

aaφw .

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.

Conclusion

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.

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