# 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. Max Karl Ernst Ludwig Planck. Cooler by a Lorentz factor. 1858 Apr. 23. Kiel 1947 Oct. 04. Göttingen. - PowerPoint PPT PresentationTRANSCRIPT

What is the Apparent Temperature of Relativistically Moving Bodies ?T.S.Bir and P.Vn (KFKI RMKI Budapest)EMMI, Wroclaw, Poland, EU, 10. July 2009. arXiv: 0905.1650

Max Karl Ernst Ludwig Planck1858 Apr. 23. Kiel1947 Oct. 04. GttingenCooler by aLorentz factor

Albert Einstein1879 Mar. 14. Ulm1955 Apr. 18. PrincetonCooler by aLorentz factor

Danilo Blanusa1903 Osijek1987 ZagrebMath professor in Zagreb

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

Sur les paradoxes de la notion dnergie

Hotter by aLorentz factor

Heinrich Ott1892 - 1962Student of Sommerfeld

LMU Mnchen PhD 1924, habil 1929

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

Lorentz - Transformation der Wrme und

der Temperatur

Hotter by aLorentz factor

Peter Theodore Landsberg1930 - 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 temperatureBUT

Christian Andreas Doppler1803 Nov 29 Salzburg1853 Mar 27 VeneziaDoppler-crater on the MoonDoppler red-shift / blue-shift

The Temperature of Moving BodiesPlanck-Einstein: coolerBlanusa - Ott: hotterLandsberg: equalDoppler - van Kampen: v_rel = 0T.S.Bir and P.Vn (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 observerOnly one of them can be Lorentz-transformed away; another one equilibratesDepending on the factual velocity of heat current all historic answers can be correct for the temperature ratioThe 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 volumeslocal expansion factor (Hubble) What is the covariant form eos?functional form of S(E,V,N,) How does T transform?

Relativistic thermodynamics based on hydrodynamicsNoether currents Conserved integralsLocal expansion rate Work on volumesE-mom conservation locally First law of thermodynamics globallyDissipation, heat, 1/T as integrating factor (Clausius)Homogeneous bodies in terms of relativistic hydro

Relativistic energy-momentum density and currents

Relativistic energy-momentum conservation

Homogeneity of a body in volume Vno acceleration of flow locallyno local gradients of energy density and pressure

Integrals over set H() of volume Vvolume integrals of internal energy change, work and heatcombined energy-flow four-vector; energy-current = momentum-density (c=1 units)

Dissipation: energy-momentum leak through the surfacerelativistic four-vector: heat flowfour-vector: carried + convected (transfer) energy-momentum l.h.s.: Reynolds transport theorem; r.h.s: Gauss-Ostrogradskij theorem

Entropy and its changeClausius: integrating factor to heat is 1/TThe integrating factor now is: Aa

Temperature and Gibbs relationNew intensive parameter: four-vector g(Jttner: g is the four-velocity of the body)

Canonical Entropy MaximumCarried and conducted (transfer) energy andmomentum, and volumes add up to constant

Our special ansatzTwo temperatures: scalar and energy-associated T

ConsequencesRe-splitting is necessary!

Orthogonal splittingw is a spacelike four-vector

The meaning of gv < 1: velocity of body, w < 1: velocity of heat conductionga = ua + wa splitting is general, S=S(Ea,V) suffices!

The meaning of gv < 1: velocity of body, w < 1: velocity of heat conductionga = ua + wa splitting is general, S=S(Ea,V) suffices! Jttner

Spacelike and timelike vectorsv: velocity of body, w: velocity of heat conduction w 1 means causal heat conduction

1+1 dimensional worldg has only two componentswe deal with four different velocities in the equilibrium conditionsonly one of them can be Lorentz-transformed awaythe body movements and the energy-momentum currents are all subluminal

One dimensional worldv is the velocity of body,subluminal,w is the velocity of heat,subluminal;Lorentz factor for observeris related to vLorentz factor for temperatureis related to w

One dimensional equilibriumTake their ratio; take the difference of their squares!

One dimensional equilibriumThe scalar temperatures are equal; T-s depend on the heat transfer!

The transformation of temperaturesT 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 temperatureLandau frame: w=0, but which w ?

w2 = 0T1 = T2 / w1 = 0T1 = T2 w2 = 1, v > 0T1 = T2 red shiftw2 = 1, v < 0T1 = T2 blue shiftw1 + w2 = 0T1 = T2

http://demonstrations.wolfram.com/TransformationsOfRelativisticTemperaturePlanckEinsteinOttLan

txu1au2aw1aw2aDoppler red-shift T2 = 2 T1

txu1au2aw1aw2a = 0No energy conduction in body 2 T2 = 1.25 T1

txu1au2aw1a = 0w2aNo energy conduction in body 1 T2 = 0.8 T1

txu1au2aw1aw2aEnergy conductions in bodies 1 and 2 compensate each other T2 = T1

txu1au2aw1aw2aDoppler blue-shift T2 = 0.5 T1

Cases of apparent temperaturesLandsbergBlanusa

Our statements:In the relativistic thermal equilibrium problem between two bodies four velocities are involved for a general observerOnly one of them can be Lorentz-transformed away; another one equilibratesDepending on the factual velocity of heat current all historic answers can be correct for the temperature ratioThe Planck-Einstein answer is correct for most common bodies (no heat current)

Summary and OutlookS = S(E,V,N)E exchg. in movecooler, hotter, equalDoppler shiftrelative velocity v equilibrates to zero

S = S(Ea,V,N)ga / T equilibratesga = ua + waS = S( ||E||, V, N)T and w do not equilibrate and w v equilibrateT: transformation dopplers w by v rel.New Israel-Stewart expansion, better stability in dissipative hydro, cools correctBiro, Molnar, Van: PRC 78, 014909, 2008