What is the Apparent Temperature of Relativistically Moving Bodies ?
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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
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Max Karl Ernst Ludwig Planck
1858 Apr. 23. Kiel1947 Oct. 04. Göttingen
Cooler by aLorentz factor
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Albert Einstein
1879 Mar. 14. Ulm1955 Apr. 18. Princeton
Cooler by aLorentz factor
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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
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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
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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
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So far it sounds like a
Zwillingsparadox for the
temperature
BUT
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Christian Andreas Doppler
1803 Nov 29 Salzburg1853 Mar 27 Venezia
Doppler-crater on the Moon
Doppler red-shift / blue-shift
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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
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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)
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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 ?
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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?
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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
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Relativistic energy-momentum
density and currents
ababbaab
b
abab
a
a
a
abbababaab
guupP
0uP,0Pu,0qu
PuqquueuT
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Relativistic energy-momentum
conservation
ddaaa
b
bdd
abab
b
a
ababd
dub
b
aadd
b
b
aaaaddab
b
u,u
0uqp
uquuup
uqeuqeuTb
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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
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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)
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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
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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
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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)
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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
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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
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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
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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
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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!
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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!
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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
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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 ?
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http://demonstrations.wolfram.com/
TransformationsOfRelativisticTemp
eraturePlanckEinsteinOttLan
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t
x
u1a
u2a
w1a
w2a
Doppler red-shift
T2 = 2 T1
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t
x
u1a
u2a
w1a w2
a = 0
No energy conduction in body 2
T2 = 1.25 T1
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t
x
u1a
u2a
w1a = 0
w2a
No energy conduction in body 1
T2 = 0.8 T1
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t
x
u1a
u2a
w1a
w2a
Energy conductions in bodies 1 and 2 compensate each other
T2 = T1
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t
x
u1a
u2a
w1a
w2a
Doppler blue-shift
T2 = 0.5 T1
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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)
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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