What is the Apparent

Temperature of Relativistically

Moving Bodies ?

T.S.Biró and P.Ván (KFKI RMKI Budapest)

EMMI, Wroclaw, Poland, EU, 10. July 2009. arXiv: 0905.1650

Max Karl Ernst Ludwig Planck

1858 Apr. 23. Kiel1947 Oct. 04. Göttingen

Cooler by aLorentz factor

Albert Einstein

1879 Mar. 14. Ulm1955 Apr. 18. Princeton

Cooler by aLorentz factor

Danilo Blanusa

1903 Osijek1987 Zagreb

Math professor in Zagreb

Glasnik mat. fiz. i astr. v. 2. p. 249, (1947)

Sur les paradoxes de la notion d’énergie

Hotter by aLorentz factor

Heinrich Ott

1892 - 1962

Student of Sommerfeld

LMU München PhD 1924, habil 1929

Zeitschrift für Physik v. 175. p. 70, (1963)

Lorentz - Transformation der Wärme und

der Temperatur

Hotter by aLorentz factor

Peter Theodore Landsberg

1930 -

Prof. emeritus Univ. Southampton

MSc 1946 PhD 1949 DSc 1966

Nature v. 212, p. 571, (1966)

Nature v. 214, p. 903, (1966)

Does a Moving Body appear Cool?

Equal temperatures

So far it sounds like a

Zwillingsparadox for the

temperature

BUT

Christian Andreas Doppler

1803 Nov 29 Salzburg1853 Mar 27 Venezia

Doppler-crater on the Moon

Doppler red-shift / blue-shift

The Temperature of Moving Bodies

• Planck-Einstein: cooler

• Blanusa - Ott: hotter

• Landsberg: equal

• Doppler - van Kampen: v_rel = 0

T.S.Biró and P.Ván (KFKI RMKI Budapest)

EMMI, Wroclaw, Poland, EU, 10. July 2009. arXiv: 0905.1650

Our statements:

• In the relativistic thermal equilibrium problem between two bodies four velocities are involved for a general observer

• Only one of them can be Lorentz-transformed away; another one equilibrates

• Depending on the factual velocity of heat current all historic answers can be correct for the temperature ratio

• The Planck-Einstein answer is correct for most common bodies (no heat current)

This is not simply about the relativistic Doppler-shift!

• The question is: how do the thermal equilibration looks like between relatively moving bodies at relativistic speeds.

• Is this a Lorentz-scalar problem ?

Some Questions

• What moves (flows)?– baryon, electric, etc. charge ( Eckart : v = 0)– energy-momentum ( Landau : w = 0)

• What is a body?– extended volumes– local expansion factor (Hubble)

• What is the covariant form eos?– functional form of S(E,V,N,…)

• How does T transform?

Relativistic thermodynamics

based on hydrodynamics

• Noether currents Conserved integrals

• Local expansion rate Work on volumes

• E-mom conservation locally First law of thermodynamics globally

• Dissipation, heat, 1/T as integrating factor (Clausius)

• Homogeneous bodies in terms of relativistic hydro

Relativistic energy-momentum

density and currents

ababbaab

b

abab

a

a

a

abbababaab

guupP

0uP,0Pu,0qu

PuqquueuT

Relativistic energy-momentum

conservation

ddaaa

b

bdd

abab

b

a

ababd

dub

b

aadd

b

b

aaaaddab

b

u,u

0uqp

uquuup

uqeuqeuTb

Homogeneity of a body in

volume V

0u,0p,0e

0uu0ub

aaa

a

a

a

a

a

no acceleration of flow locally

no local gradients of energy density and pressure

Integrals over set H() of volume V

aaa

)(H

abba

b)(H

b

b

aaa

qeu

dVqudVu)pu(

volume integrals of internal energy change, work and heat

combined energy-flow four-vector;

energy-current = momentum-density (c=1 units)

Dissipation: energy-momentum

leak through the surface

aaa

H

aaH H

aa

a

Hb

abbaaaa

GuEE,dVqG

eVedVE,dV)x(uV

1u

QdAquVupGuE

relativistic four-vector: heat flow

four-vector: carried + convected (transfer) energy-momentum

l.h.s.: Reynolds’ transport theorem;

r.h.s: Gauss-Ostrogradskij theorem

Entropy and its change

pdVdGdEdS)V,E(SS :e.o.s

ddSAdVupdEQ

bb

aa

bb

ab

b

aa

AA

uAa

AA

A

AA

uA

a

aaaaa

Clausius: integrating factor to heat is

1/T

The integrating factor now is: Aa

Temperature and Gibbs relation

pdVdEgTdS

pdVdGgdETdS

AA

A

T

g,

AA

uA

T

1

a

a

a

a

b

b

aa

b

b

a

a

New intensive parameter: four-vector g

(Jüttner: g is the four-velocity of the body)

Canonical Entropy Maximum

2

2

1

1

2

a

2

1

a

1

2

a

21

a

1

a

2

a

121

T

p

T

p,

T

g

T

g

0)V,E(dS)V,E(dS

0dEdE,0dVdV

Carried and conducted (transfer) energy and

momentum, and volumes add up to constant

The meaning of g

v < 1: velocity of body, w < 1: velocity of heat conduction

ga = ua + wa splitting is general, S=S(Ea,V) suffices!

Jüttner

Spacelike and timelike vectors

1ww

0Tww1gg

0ww0wu,1uu

a

a

22a

a

a

a

a

a

a

a

a

a

v: velocity of body, w: velocity of heat conduction

w 1 means causal heat conduction

One dimensional world

2

2

a

a

w1

Tv1

1

)w,vw(w

)v,(u

v is the velocity of body,

subluminal,

w is the velocity of heat,

subluminal;

Lorentz factor for observer

is related to v

Lorentz factor for temperature

is related to w

One dimensional equilibrium

2

222

1

111

2

222

1

111

T

)wv(

T

)wv(

T

)wv1(

T

)wv1(

Take their ratio; take the difference of their squares!

One dimensional equilibrium

2

2

2

1

2

1

22

22

11

11

T

w1

T

w1

wv1

wv

wv1

wv

The scalar temperatures are equal; T-s depend on the heat transfer!

The transformation of temperatures

2

2

1

2

1221

2211

v1

vw1

T

T

v)v(v)w(w

vwvw

T ratio follows a general Doppler formula with relative velocity v!

Four velocities: v1, v2, w1, w2

Max. one of them can beLorentz-transformed to zero

Cases of apparent temperature

w2 = 0 T1 = T2 / γ

w1 = 0 T1 = T2 γ

w2 = 1, v > 0 T1 = T2 ● red shift

w2 = 1, v < 0 T1 = T2 ● blue shift

w1 + w2 = 0 T1 = T2

Landau frame: w=0, but which w ?

http://demonstrations.wolfram.com/

TransformationsOfRelativisticTemp

eraturePlanckEinsteinOttLan

t

x

u1a

u2a

w1a

w2a

Doppler red-shift

T2 = 2 T1

t

x

u1a

u2a

w1a w2

a = 0

No energy conduction in body 2

T2 = 1.25 T1

t

x

u1a

u2a

w1a = 0

w2a

No energy conduction in body 1

T2 = 0.8 T1

t

x

u1a

u2a

w1a

w2a

Energy conductions in bodies 1 and 2 compensate each other

T2 = T1

t

x

u1a

u2a

w1a

w2a

Doppler blue-shift

T2 = 0.5 T1

Our statements:

• In the relativistic thermal equilibrium problem between two bodies four velocities are involved for a general observer

• Only one of them can be Lorentz-transformed away; another one equilibrates

• Depending on the factual velocity of heat current all historic answers can be correct for the temperature ratio

• The Planck-Einstein answer is correct for most common bodies (no heat current)

Summary and Outlook• S = S(E,V,N)• E exchg. in move• cooler, hotter, equal• Doppler shift• relative velocity v

equilibrates to zero

• S = S(Ea,V,N)• ga / T equilibrates• ga = ua + wa

• S = S( ||E||, V, N)• T and w do not

equilibrate and w v equilibrate• T: transformation

dopplers w by v rel.• New Israel-Stewart

expansion, better stability in dissipative hydro, cools correct

Biro, Molnar, Van: PRC 78, 014909, 2008