How can relativity make Navier-Stokes unstable? Lorenzo Gavassino - PowerPoint PPT Presentation

27 October 2020 How can relativity make Navier-Stokes unstable? Lorenzo Gavassino Nicolaus Copernicus Astronomical Center of the Polish Academy of Sciences Based on In collaboration with 10.1103/PhysRevD.102.043018 Dr. Marco Antonelli Prof. Brynmor Haskell

Diffusion equation 𝜖𝑢 = 𝐸 𝜖 2 𝑈 𝜖𝑈 Standard Newtonian equation of the evolution of a temperature profile over time (1D case) 𝜖 𝑦 2

Diffusion equation 𝜖𝑢 = 𝐸 𝜖 2 𝑈 𝜖𝑈 Standard Newtonian equation of the evolution of a temperature profile over time (1D case) 𝜖 𝑦 2 Initial profile (𝑢 = 0)

Diffusion equation 𝜖𝑢 = 𝐸 𝜖 2 𝑈 𝜖𝑈 Standard Newtonian equation of the evolution of a temperature profile over time (1D case) 𝜖 𝑦 2 Initial profile (𝑢 = 0) 𝑢 = 5

Diffusion equation 𝜖𝑢 = 𝐸 𝜖 2 𝑈 𝜖𝑈 Standard Newtonian equation of the evolution of a temperature profile over time (1D case) 𝜖 𝑦 2 Initial profile (𝑢 = 0) 𝑢 = 20

Diffusion equation 𝜖𝑢 = 𝐸 𝜖 2 𝑈 𝜖𝑈 Standard Newtonian equation of the evolution of a temperature profile over time (1D case) 𝜖 𝑦 2 Initial profile (𝑢 = 0) 𝑢 = 50

Diffusion equation 𝜖𝑢 = 𝐸 𝜖 2 𝑈 𝜖𝑈 Standard Newtonian equation of the evolution of a temperature profile over time (1D case) 𝜖 𝑦 2 Initial profile (𝑢 = 0) 𝑢 = 300

Diffusion equation 𝜖𝑢 = 𝐸 𝜖 2 𝑈 𝜖𝑈 Standard Newtonian equation of the evolution of a temperature profile over time (1D case) 𝜖 𝑦 2 Initial profile (𝑢 = 0) 𝑢 = 1000

Does it work in relativity? 𝑓𝑦𝑞 − 𝑦 2 1 It describes how an initial 𝐻 𝑦, 𝑢 = The Green function is condition 𝜀 𝑦 evolves in time 4𝐸𝑢 4𝜌𝐸𝑢 The tail of the Gaussian is a signal which propagates outside the light-cone Localized Faster than light communication. source Causality broken!

What if we “Boost” it? ቊ𝑢 ′ = 𝛿 𝑢 − 𝑤𝑦 If we apply the Lorentz transformation: 𝑦 ′ = 𝛿 𝑦 − 𝑤𝑢 𝜖𝑢 = 𝐸 𝜖 2 𝑈 𝜖𝑈 becomes 𝜖𝑦 2 𝜖𝑦 ′2 − 2𝑤 𝜖 2 𝑈 𝜖 2 𝑈 𝜖𝑦 ′ 𝜖𝑢 ′ + 𝑤 2 𝜖 2 𝑈 𝜖𝑢 ′ − 𝑤 𝜖𝑈 𝜖𝑈 𝜖𝑦 ′ = 𝐸𝛿 𝜖𝑢 ′2 Warning! 𝑈 −1 ≔ −𝛾 𝜉 𝛾 𝜉 so I do not need to “transform” 𝑈 A second-order term in time The state-space in the boosted frame is larger! There are more degrees of freedom: 𝑈, 𝜖𝑈 𝑈 In the frame of the medium In the boosted frame 𝜖𝑢 ′

Instability Let us study the homogeneous solutions in the boosted frame 𝜖𝑦 ′2 − 2𝑤 𝜖 2 𝑈 𝜖 2 𝑈 𝜖𝑦 ′ 𝜖𝑢 ′ + 𝑤 2 𝜖 2 𝑈 𝜖𝑢 ′ − 𝑤 𝜖𝑈 𝜖𝑈 𝜖𝑦 ′ = 𝐸𝛿 𝜖𝑢 ′2

ሶ ሶ Instability Let us study the homogeneous solutions in the boosted frame T 𝜖𝑦 ′2 − 2𝑤 𝜖 2 𝑈 𝜖 2 𝑈 𝜖𝑦 ′ 𝜖𝑢 ′ + 𝑤 2 𝜖 2 𝑈 𝜖𝑢 ′ − 𝑤 𝜖𝑈 𝜖𝑈 𝜖𝑦 ′ = 𝐸𝛿 𝜖𝑢 ′2 𝜖𝑢 ′ = 𝐸𝛿𝑤 2 𝜖 2 𝑈 𝜖𝑈 2 parameters to set in the 𝜖𝑢 ′2 initial conditions instead of 1 Thermal runaway! 1 𝑈 0 𝑓 Γ + 𝑢 ′ − 1 Γ + = 𝐸𝛿𝑤 2 > 0 𝑈 = 𝑈 0 + Γ + t creates a class of solutions which explode for 𝑢 ′ → +∞ 𝑈 0 ≠ 0 Our freedom of setting This instability has no Newtonian analogue.

If I go back to the rest-frame of the medium 𝑈 𝑈(𝑦, 𝑢) ~ 𝑓 Γ + 𝛿(𝑢−𝑤𝑦) 𝑈(𝑦, 0) ~ 𝑓 −Γ + 𝛿𝑤𝑦 Initial thermal profile: 𝑤 −1 𝑦 − 1 Space-time dependence 𝑤 𝑢 of the kind: Exponential profile which shifts rigidly faster than light! Completely non-realistic situation: 1) Strong acausality 𝑦 2) Infinite temperature for 𝑦 → −∞ This instability is unphysical, but, working in 3) Incompatible with the assumptions which lead to the diffusion the boosted frame, we would need to fine- equation in the first place tune the initial conditions to avoid it. Kostaedt & Liu (2000): 10.1103/PhysRevD.62.023003

The Eckart approach to dissipation 𝑈 𝜈𝜉 = 𝜍 + 𝑄 𝑣 𝜈 𝑣 𝜉 + 𝑄𝑕 𝜈𝜉 + 𝑟 𝜈 𝑣 𝜉 + 𝑟 𝜉 𝑣 𝜈 + Π𝑕 𝜈𝜉 + Π 𝜈𝜉 Heat flux Bulk-viscous stress Shear-viscous stress In Newtonian physics: Fourier Law: 𝒓 = −𝑙∇𝑈 All the dissipative pieces in the stress-energy tensor are assumed proportional to spatial gradients 𝑘 𝑣 𝑘 Π = −𝜂𝜖 Navier-Stokes: Π 𝑘𝑙 = −𝜃 𝜖 𝑘 𝑣 𝑙 + 𝜖 𝑙 𝑣 𝑘 − 2 3 𝜖 𝑚 𝑣 𝑚 𝜀 𝑘𝑙

The Eckart approach to dissipation 𝑈 𝜈𝜉 = 𝜍 + 𝑄 𝑣 𝜈 𝑣 𝜉 + 𝑄𝑕 𝜈𝜉 + 𝑟 𝜈 𝑣 𝜉 + 𝑟 𝜉 𝑣 𝜈 + Π𝑕 𝜈𝜉 + Π 𝜈𝜉 Heat flux Bulk-viscous stress Shear-viscous stress In Newtonian physics: Fourier Law: 𝒓 = −𝑙∇𝑈 − kT a All the dissipative pieces in the stress-energy tensor are assumed proportional to spatial gradients 𝑘 𝑣 𝑘 Π = −𝜂𝜖 Navier-Stokes: Π 𝑘𝑙 = −𝜃 𝜖 𝑘 𝑣 𝑙 + 𝜖 𝑙 𝑣 𝑘 − 2 3 𝜖 𝑚 𝑣 𝑚 𝜀 𝑘𝑙 Eckart in essence: almost all the Newtonian relations still hold… in the reference frame of the fluid element.

Again… a second derivative! If the fluid element is moving the derivatives in space are boosted: 𝜖 𝜖𝑦 ′ − 𝑤 𝜖 𝜖 𝜖𝑦 = 𝛿 𝜖𝑢 ′ This produces derivatives in time with no Newtonian analogue. The dissipative pieces, thus, acquire time-derivative terms, e.g. Boost Π = −𝜂𝜖 𝜉 𝑣 𝜉 = −𝜂 𝜖 𝑘 𝑣 𝑘 + 𝜖 𝑢 𝑣 𝑢 𝑘 𝑣 𝑘 Π = −𝜂𝜖 On the other hand, the equations of motion are simply the energy-momentum and particle conservations which, in turn, involve an other derivative in time 𝜖 𝜈 𝑈 𝜈𝜉 = 𝜖 𝑘 𝑈 𝑘𝜉 + 𝜖 𝑢 𝑈 𝑢𝜉 = 0 Same as the heat equation : the Navier-Stokes equations, which were first-order in time in Newtonian physics, become second order in relativity!

Again… |𝜕| Linear stability: we look for solutions of the form 0 + 𝜀𝑔 𝑓 𝑗 𝑙𝑦−𝜕𝑢 𝑔 𝑦, 𝑢 = 𝑔 Equilibrium solution Small perturbation Hydro We obtain a collection of dispersion relations 𝜕 𝑘 = 𝜕 𝑘 (𝑙) 𝑙 In Newtonian Navier-Stokes one only finds the Hydro-modes: SO 𝑙 2 + 𝑃 𝑙 3 𝜕 𝑇𝑃 = ±𝑑 𝑡 𝑙 − 𝑗 Γ 𝛥 𝑇𝑃 > 0 Sound waves 𝑙→0 𝜕 𝑙 = 0 lim Gapless: 𝜕 𝑇𝐼 = −𝑗 Γ S𝐼 𝑙 2 + 𝑃 𝑙 3 𝛥 𝑇𝐼 > 0 Shear waves Stable: 𝐽𝑛 𝜕 ≤ 0 They are gapless and stable.

… an explosion! |𝜕| Linear stability: we look for solutions of the form Gapped 0 + 𝜀𝑔 𝑓 𝑗 𝑙𝑦−𝜕𝑢 𝑔 𝑦, 𝑢 = 𝑔 Equilibrium solution Small perturbation Hydro We obtain a collection of dispersion relations 𝜕 𝑘 = 𝜕 𝑘 (𝑙) 𝑙 In Eckart theory one still finds the Hydro-modes SO 𝑙 2 + 𝑃 𝑙 3 𝜕 𝑇𝑃 = ±𝑑 𝑡 𝑙 − 𝑗 Γ 𝛥 𝑇𝑃 > 0 Sound waves 𝑙→0 𝜕 𝑙 ≠ 0 lim Gapped: 𝜕 𝑇𝐼 = −𝑗 Γ S𝐼 𝑙 2 + 𝑃 𝑙 3 𝛥 𝑇𝐼 > 0 Shear waves Unstable: 𝐽𝑛 𝜕 > 0 But also some gapped mode (some mode which survives in the homogeneous limit) G + 𝑃 𝑙 2 𝜕 𝐻 = 𝑗 Γ 𝛥 𝐻 > 0 which turns out to be unstable! The gapped mode exists because of the higher order in time of the equations.

What went wrong? • Every thermodynamic system admits a maximum entropy state. • Since the entropy grows, the system will eventually converge to this state for every initial condition. • This state is the thermodynamic equilibrium (which is necessarily stable under perturbations: Lyapunov criterion). A system in thermodynamic equilibrium can never exhibit instabilities

The origin of the problems Equilibrium = Maximum Newtonian Navier-Stokes: entropy state S The total entropy is always The only degrees of freedom maximised in homogeneous are the thermodynamic fields states (if basic thermodynamic conditions are respected). 𝑈, 𝜈, 𝑣 𝑘 Therefore the Hydro-modes The number of constants of necessarily reduce the motion equals the number of entropy. thermodynamic fields 𝐹, 𝑂, 𝑞 𝑘 In conclusion: the equilibrium is As a consequence, for fixed stable in Newtonian constants of motion, there is Navier-Stokes only one homogeneous state. Hydro-mode Gapped modes cannot exist.

The origin of the problems Relativity opens new path in Eckart theory: Newtonian state-space which the entropy can grow with no bound The degrees of freedom now are the thermodynamic fields It happens that in the and their derivatives in time Eckart theory the entropy grows along the gapped 𝑈, 𝜈, 𝑣 𝑘 , 𝜖 𝑢 𝑈, 𝜖 𝑢 𝜈, 𝜖 𝑢 𝑣 𝑘 modes. Their number exceeds the number of constants of motion. The instability is Therefore there is room for a thermodynamical! The large variety of new obedience of the homogeneous configurations system to the second which are dynamically accessible. Gapped modes are a law is the very origin necessity. of the runaway!