What is the Apparent Temperature of Relativistically Moving Bodies ?

1. 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

2. Max Karl Ernst Ludwig Planck Cooler by a Lorentz factor 1858 Apr. 23. Kiel 1947 Oct. 04. Göttingen

3. Albert Einstein Cooler by a Lorentz factor 1879 Mar. 14. Ulm 1955 Apr. 18. Princeton

4. Danilo Blanusa Math professor in Zagreb Glasnik mat. fiz. i astr. v. 2. p. 249, (1947) Sur les paradoxes de la notion d’énergie Hotter by a Lorentz factor 1903 Osijek 1987 Zagreb

5. Heinrich Ott 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 a Lorentz factor 1892 - 1962

6. Peter Theodore Landsberg 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 1930 -

7. So far it sounds like a Zwillingsparadox for the temperature BUT

8. Christian Andreas Doppler Doppler-crater on the Moon Doppler red-shift / blue-shift 1803 Nov 29 Salzburg 1853 Mar 27 Venezia

9. 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

10. 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)

11. 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 ?

12. 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?

13. 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

14. Relativistic energy-momentum density and currents

15. Relativistic energy-momentum conservation

16. Homogeneity of a body in volume V no acceleration of flow locally no local gradients of energy density and pressure

17. Integrals over set H() of volume V volume integrals of internal energy change, work and heat combined energy-flow four-vector; energy-current = momentum-density (c=1 units)

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

19. Entropy and its change Clausius: integrating factor to heat is 1/T The integrating factor now is: Aa

20. Temperature and Gibbs relation New intensive parameter: four-vector g (Jüttner: g is the four-velocity of the body)

21. Canonical Entropy Maximum Carried and conducted (transfer) energy and momentum, and volumes add up to constant

22. The meaning of g Jüttner v < 1: velocity of body, w < 1: velocity of heat conduction ga = ua + wa splitting is general, S=S(Ea,V) suffices!

23. Spacelike and timelike vectors v: velocity of body, w: velocity of heat conduction w  1 means causal heat conduction

24. One dimensional world 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

25. One dimensional equilibrium Take their ratio; take the difference of their squares!

26. One dimensional equilibrium The scalar temperatures are equal; T-s depend on the heat transfer!

29. The transformation of temperatures Four velocities: v1, v2, w1, w2 Max. one of them can be Lorentz-transformed to zero T ratio follows a general Doppler formula with relative velocity v!

33. Cases of apparent temperature Landau frame: w=0, but which w ?

34. http://demonstrations.wolfram.com/TransformationsOfRelativisticTemperaturePlanckEinsteinOttLanhttp://demonstrations.wolfram.com/TransformationsOfRelativisticTemperaturePlanckEinsteinOttLan

35. t u2a T2 = 2 T1 u1a w2a w1a x Doppler red-shift

36. t u2a T2 = 1.25 T1 u1a w2a = 0 w1a x No energy conduction in body 2

37. t u2a T2 = 0.8 T1 u1a w1a = 0 x w2a No energy conduction in body 1

38. t u2a T2 = T1 u1a w1a w2a x Energy conductions in bodies 1 and 2 compensate each other

39. t u2a T2 = 0.5 T1 u1a w1a x w2a Doppler blue-shift

40. 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)

41. 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 Summary and Outlook Biro, Molnar, Van: PRC 78, 014909, 2008