General Relativistic Thermo-Dynamics

Bad Honnef September 2014

STAG RESEARCHCENTERSTAG RESEARCH

CENTERSTAG RESEARCHCENTER

Nils Andersson

“Thermodynamics is a branch of physics concerned with heat and temperature and their relation to energy and work. It defines macroscopic variables, such as internal energy, entropy and pressure that partly describe a body of matter or radiation.” (the wisdom of Wikipedia)

Classic description builds on a set of phenomenological “laws” that govern systems that can be (effectively) described by a small number of “state variables”.

The entropy is a key ingredient, essentially measuring how random a system is on the microscopic level.

A “mysterious” quantity with a number of different interpretations (Boltzmann – disorder, Shannon – information, etcetera).

IT'S THE LAW:Energy can neither be created nor destroyed. It can only change form.

If two systems are in thermal equilibrium with a third system, then they are in equilibrium with each other.

The entropy of an isolated system not in equilibrium will tend to increase over time, approaching a maximum value at equilibrium.As temperature approaches absolute zero, the entropy of a system approaches an absolute minumum.

As a “warm-up exercise”, consider a single species system described by an equation of state E=E(N,S,V), with N the number of particles, S the entropy and V the volume.

The first law

defines the chemical potential, temperature and pressure; dE = µdN +TdS ! pdV

µ = !E!N S ,V

, T = !E!S N ,V

, p = " !E!V N ,S

If we instead work with densities, n=N/V, s=S/V and ρ=E/V, (and assume additivity for E, N and V) the above implies the Gibbs relation;

p + ! = nµ + sT ! d! = µdn +Tds with µ= "!"n s

and T= "!"s n

Rewriting this, we interpret the chemical potential as the “adiabatic injection” energy for a particle;

0=Tds = d! ! µdn

The marriage of thermodynamics and relativity has a complicated history.

The discussion is “pointless”.

If heat is exchanged then there will also be momentum flow, so the system cannot be in thermal equilibrium if there is relative motion.

Must not confuse T as a thermodynamic parameter with the method of measurement (eg. the length of a moving alcohol thermometer).

hot or cold?

Q. Is a moving body hot/cold?

A. Yes/no/maybe

A more relevant question may be:

How do we build on the “elegance of thermodynamics in its covariant guise” (Israel) ?

Important lesson: Heat does not necessarily flow from hot to cold if there is momentum exchange!

averaging First, we need to decide which relativistic effects we are trying to account for.

-  at high temperatures, the individual particle velocities may be large,

-  the average velocity of the fluid elements may be large,

-  the curved spacetime may play a role.

This lecture will focus on the larger scale, after averaging over a “fluid element”, and how heat can be accounted for in a general relativistic setting.

The problem is “hard enough” already before we worry about the microphysics.

why?

G!" =8#Gc2T!"

Gravity is due to the geometry of spacetime. The motion of all bodies is described by Einstein’s field equations;

general relativity

Need to solve these equations for the spacetime metric gαβ given some “realistic” matter stress-energy tensor Tαβ .

Need to account for all contributions to the energy/stresses/momenta etcetera.

Know that

!!G !" = !!T !

" = 0

Can think of this as the equations of motion for matter, but if we want to study out-of-equilibrium systems we need more information.

For a perfect fluid, we have where projects out components orthogonal to the flow (local observer uα). The energy is given by and p is the (isotropic) pressure. To close the system, we need an equation of state. Simplistically, this is some relation p=p(ρ), but it really should incorporate the appropriate micro-physics. Warning: Don’t solve the problem “backwards”!

the “text-book” approach

!!" = g!" + u!u! , !!" u! = 0

T !" = p#!

" + ( p + $)u"u! = $u"u! + p !!"

! = u"u#T #

"

Working out the perfect fluid equations of motion we get;

Contracting this with uβ we have

To see that this is what we expected, we need a bit of thermodynamics…

Assuming that we have a cold single particle system, we have ρ=ρ(n) where n is the number density. The associated chemical potential is

!!T!" = !! # + p( )u!u! + pg!""# $% =

= u!!" # + p( )u!"# $% + " + p( )u!!!u! + g!"!! p = 0

µ = d!dn

and the Gibbs relation µn = p + ! !

( ) 0 0u n n u nu nα α α αα α α αµ µ µ∇ + ∇ = ∇ = ⇒ ∇ =

In other words, the particle flux is conserved.

u!u!( )

=!1!"#

"! " + p( )u!#$ %& + u!g!"

=u!!

"! p = !u!"!" ! p + !( )"!u! = 0

equations of motion

Now consider the projection orthogonal to the four velocity;

!!" "#T#" = !!" u

"

=0!"#

"! " + p( )u!#$ %&+ !"# $ + p( )u!"!u" + !#" g

!"

=!#!

!"# $#"! p = 0

Introducing the four acceleration

and noting that

a! = !u! = u"!"u!

!!" u#!#u

" = g!" + u! u"( )u#!#u" = u#!#u! = a!

! + p( )a! = ! "!" #" p

we can write the equation in a form that reminds us of Newton’s second law;

As expected, pressure gradients drive changes in the four-acceleration.

Main lesson: Perfect fluids do not “move” on geodesics.

So far, there are no real surprises…

Let us now go beyond the textbooks and ask what happens when we add heat to the system. This problem was first considered by Eckart in 1940. To arrive at his model, we; 1) add the heat flux qα to the stress-energy tensor, allowing for the possibility of a distinct flow relative to the matter; and work out the divergence. 2) Introduce the entropy flux as where s is the entropy per unit volume in equilibrium and T is the temperature. 3) Use the standard (equilibrium) Gibbs relation, assuming an equation of state ρ=ρ(n,s).

adding heat

T!" = !u!u! + p !!" +2q(!u! )

s! = su! + 1Tq!

p + ! = nµ +Ts ! !!" = µ!!n +T!! s

4) Let the particles be conserved

but allow the entropy to increase,

in accordance with the 2nd law of thermodynamics (note: here assumed to be a local restriction).

5) After some work… we obtain (this is the “simplest” option);

where κ is the thermal conductivity.

This results reminds us of Fourier’s law (the “acceleration term” vanishes in the Newtonian limit).

But the result is problematic:

-  heat conduction would not be causal

-  there are unphysical instabilities (growth time 10-34 s for water at room temperature)

!!n! = 0

!! s! = !s " 0

q! = !"T "!" 1T#!T + u!#! u!

$%&

'()

Tolman Pause to consider a static equilibrium star for which the spacetime metric is

ds2 = !e2!dt2 + e2!dr2 + r2 d! 2 + sin2!d! 2( )

From the momentum equations and the requirement that there is no heat flux, we have

a result due to Tolman from the 1930s (although his argument was different).

1TdTdr

= ! 1p + !

dpdr

= d!dr

! T ! = Te! = constant

Adding in some microphysics, for the n,p,e mixture of the outer core of a neutron star, we also see that

np = ne charge neutrality

µn = µ p + µe beta equilibrium

!"#

$# % µne

! = constant

Note: In a similar fashion the expansion of the Universe causes the blackbody radiation to cool.

Inspired by extended thermodynamics and kinetic theory, Israel and Stewart (1970s) introduced 2nd order “fluxes” to ensure causality. Letting,

where β1 is a “free” parameter, and working through the same steps as before we arrive at

Israel & Stewart

s! = su! + 1Tq! !

"12Tq2u!

q! = !"T "!" 1T#!T + u!#! u! + !1u

!#! q!= !q!

"#$ %$+ T2q!#"

!1u!

T$

%&'

()

*

+

,,,

-

.

///

The Israel-Stewart model has been used in a number of contexts, eg. high energy collisions, but the phenomenology is far from well understood.

We now have a complex nonlocal theory. We have a second sound, memory effects, frequency dependent coefficients etcetera…

This problem is hyperbolic, with β1 playing the role of the thermal relaxation time (Cattaneo equation). The formulation is also stable.

Expect actual signal velocities to be bounded by the “average molecular speed”, and hence be sub-luminal. How do we build this into the model?

u!!!" + ( p + !)!!u! = 0 !!n

! = 0

!!" "" p = !( p + #)u"!"u! f! = 2n

!! !!"µ" #$

= 0

To get a deeper understanding, let’s go back to the beginning. Start with the usual perfect fluid, use ρ=ρ(n) and introduce chemical potential µ;

where we have used and . n! = nu! µ! = µu!

variational model

These equations can be obtained from a variational principle based on the use of a three dimensional “matter space”.

The advantage of the variational approach is that the extension to “multi-fluid” systems is straightforward (as is the inclusion of more complex physics like elasticity, electromagnetism etectera).

!!nx! = 0

f!x = 2nx

!![!µ" ]x = 0

Note: no summation over x.

Assuming that each particle species is conserved, we arrive at:

nx! = nxux

!

multi-fluids

The only difference is that we need several matter spaces (and individual fluxes);

entrainment The Lagrangian Λ plays a central role in the model. In principle, it is a function of all scalars (assuming the system is isotropic) that can be formed from .

For the simplest model, Λ=Λ(n2), and we have the canonical momentum

In the multi-fluid case we should consider

nx!

nxy2 = !g!"nx

!ny! (y ! x)xn as well as

which leads to

µ!x = g!" Bxnx

! + Axyny!

y!x"

#

$%&

'(where Axy = ! "#

"nxy2

Main lesson: The current need not be parallel to the momentum.

This is known as the entrainment effect, and it can be expressed in terms of an “effective mass”.

µ! = !2 !"!n2

n! = Bn!

heat flow (again)

Consider a model with;

1) a conserved particle flux nα and associated four-velocity uα, such that

2) a distinct entropy flux, given by (note: work in the Eckart frame)

where

The 2nd law of thermodynamics then requires that (imposed locally!)

n! = nu! ! n2 = !n!n! and !!n! = 0

u!v! = 0 and s2 = !s! s! ! s* = s 1! v2( )!1 2

!! s! = !s " 0

s! = s* u! + v!( )

Let us return to the heat-flow problem, taking a two-fluid model as our starting point (and focussing on the entropy momentum equation).

Explicitly, the variational approach leads to the stress-energy tensor taking the form

where the heat flux is given by

and one can show that

This is the “usual” thermodynamic definition of the temperature.

Note: There is no general agreement on how one defines the “temperature” out of equilibrium, but…

… the variational model leads to consistency between the thermal momentum and the entropy chemical potential (which makes some “sense”).

T!" = ! " ! s*( )2 Bsv2#$%

&'(

=!*! "### $###

u!u! + 2u(!q! )+ )!" * + s*( )2 Bsv!v!= shear

! "# $#

q! = s* s*Bs + nAns( )v! ! s*!*v!

*

**

*, n s vs

ρ∂Θ =∂

Rewriting the entropy momentum as

and contracting the corresponding momentum equation with uβ, we arrive at

The simplest option that ensures that the 2nd law is satisfied leads to the “heat equation”;

where we have introduced the resistivity, , related to the thermal conductivity κ.

The parameter β is related to the entropy entrainment (inertia of heat!) and provides the required thermal relaxation time.

The final equation shares key features with the classic Israel-Stewart model, but there are differences (particularly in terms that do not generate entropy!).

µ!s = Bss! + A

nsn! =!*u! + "q!

!s = " 1#*

$%&

'()

2

q! *!#* +#*u!*!u! + u

!*! "q!( )" !u!*"q!+,

-.

!!" ""#* +#*u#"# u" + u

#"# "q"( )$ "u#"# q"%&

'( = $Rq!

0R ≥

second sound

Class. Quantum Grav. 28 (2011) 195023 N Andersson and C S Lopez-Monsalvo

Figure 1. This figure provides an illustration of the qualitative nature of the behaviour of heatconducting degenerate matter (as discussed in the main text), based on the consistent first-orderrelativistic model. The parameters have been chosen in such a way that the speed of sound is 10%of the speed of light, while the second sound (at short wavelengths, large k) propagates at 1/

!3

of this. The phase velocity of the waves is ! = Re "/k (cf left panel).The thermal relaxationtime # has been chosen such that the critical wavenumber at which the second sound emergesis k = 10. At lengthscales larger than this, the corresponding roots are diffusive (have purelyimaginary frequency), and in the very long wavelength limit (k " 0) we retain the expectedthermal diffusion. The damping time follows from 1/Im " (cf right panel). We also indicate theacausal region (grey area). The illustrated example is clearly both stable and causal.

(This figure is in colour only in the electronic version)

conclusion accords with the classic work by Hiscock and Lindblom [19, 20] (see also Olsonand Hiscock [25]). In fact, the conditions we have arrived at reproduce their key results.However, our analysis adds to previous work by discussing the emergence of the second soundand the nature of the associated solutions. This is a key point, especially if we are interestedin relativistic superfluid systems. The analysis also demonstrates the intricate nature of theseproblems. It is easy to see how a model that may fail one, or several, of the derived conditionsin some regime may nevertheless be valid for a different range of parameters. Hence, onereally should consider the applicability of the chosen theory on a case-by-case basis. This isprobably no more than should be expected from a phenomenological model.

Our results represent useful progress in this problem area, but one could obviouslydevelop the theory further. A natural step would be to consider the various constraintsthat we have derived for detailed equations of state, e.g., matter coupled to phonons. Itwould also be interesting to consider applications of the first-order construction. While themodel is restricted in the sense that it does not account for non-adiabatic effects, there is anexciting range of possible applications in astrophysics, cosmology and high-energy physics.Particularly interesting questions concern to what extent second sound effects are relevant inrelativistic systems and the difference between first-order results and the, considerably morecomplex, second-order theories.

16

As an exercise, we can linearize in deviations from equilibrium. This leads to the simple system;

µ! !u! + !!" !!"µ +!" !q! !

sn!

"q! = 0

!" !q! +!q! +! !!" !!"T +!T! !u! = 0

Transverse waves are unstable unless one accounts for thermal relaxation.

Longitudinal waves have the expected phenomenology:

-  long wavelengths: sound waves and thermal diffusion

-  second sound in short wavelength regime

black holes What about the gravitational field?

Is gravity a thermodynamic system with “different” microscopic degrees of freedom?

Analogous laws for classical black holes;

Bekenstein-Hawking;

Note: String theory “counting” of microstates.

dE = TdS + "work" ! dM = !8"dA+ "work"

dS ! 0 " dA ! 0

T ! !! S ! A !

Are the black-hole laws “phenomenological” or is there a fundamental link between gravity and thermodynamics?

Hint (membrane paradigm): A perturbed black hole relaxes back to equilibrium like a dissipative fluid (Damour).

holography Maximum entropy proportional to R2 rather than R3, suggests description of a volume of space is encoded on the boundary (t’Hooft, Susskind)

Combine with long wavelength “fluid” limit for (conformal) field theory.

Holographic correspondence (AdS/CFT): Dissipative dynamics of certain horizons is exactly equal to that of a quantum liquid in one lower dimension.

Current string-theory effort aims to develop a dictionary that relates fluid properties with dynamical black holes.

Provides a strategy for studying strongly coupled quantum systems for which perturbation theory breaks down.

Example: “universal” viscosity bound

Reinforces relevance of relativistic fluid dynamics, black-hole physics and thermodynamics.

! s = ! 4"

summary In this lecture I have taken a particular path through a very complicated landscape, very much based on my personal bias. You could easily choose a different path, stop at various points and the study the problem in more detail.

After centuries of investigation it is clear that many challenges remain.

Again, based on my particular preferences, here are some “deeper” questions;

-  Is there an “ultimate” nonlinear theory for thermodynamics? (derivative expansions, variations, description of shocks)

-  How do systems evolve when far from equilibrium? (structures, max/min entropy)

-  Who measures what? Can you formulate a “quasi-local” theory?

-  How does the 2nd law actually “work”? (role of observers, thermal time…)