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 Presentation

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  • 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

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