What is the Apparent Temperature of Relativistically Moving

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. ar. Xiv: 0905. 1650

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

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

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

Heinrich Ott Student of Sommerfeld LMU München Ph. D 1924, habil 1929 Zeitschrift für Physik v. 175. p. 70, (1963) Lorentz - Transformation der Wärme und der Temperatur 1892 - 1962 Hotter by a Lorentz factor

Peter Theodore Landsberg Prof. emeritus Univ. Southampton MSc 1946 Ph. D 1949 DSc 1966 Nature v. 212, p. 571, (1966) Nature v. 214, p. 903, (1966) Does a Moving Body appear Cool? Equal temperatures 1930 -

So far it sounds like a Zwillingsparadox for the temperature BUT

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

The Temperature of Moving Bodies T. S. Biró and P. Ván (KFKI RMKI Budapest) • Planck-Einstein: cooler • Blanusa - Ott: hotter • Landsberg: e q u a l • Doppler - van Kampen: EMMI, Wroclaw, Poland, EU, 10. July 2009. v_rel = 0 ar. Xiv: 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 thermal equilibration looks like between relatively moving bodies at relativistic speeds. • Is this a Lorentzscalar 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

Relativistic energy-momentum conservation

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

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)

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

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

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

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

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!

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

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

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

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

The transformation of temperatures Four velocities: v 1, v 2, w 1, w 2 Max. one of them can be Lorentz-transformed to zero T ratio follows a general Doppler formula with relative velocity v!

Cases of apparent temperature w 2 = 0 T 1 = T 2 / γ w 1 = 0 T 1 = T 2 γ w 2 = 1, v > 0 T 1 = T 2 ● red shift w 2 = 1, v < 0 T 1 = T 2 ● blue shift w 1 + w 2 = 0 T 1 = T 2 Landau frame: w=0, but which w ?

http: //demonstrations. wolfram. com/ Transformations. Of. Relativistic. Temp erature. Planck. Einstein. Ott. Lan

t u 2 a T 2 = 2 T 1 u 1 a w 2 a w 1 a x Doppler red-shift

t T 2 = 1. 25 T 1 u 2 a u 1 a w 2 a = 0 No energy conduction in body 2 x

t T 2 = 0. 8 T 1 u 2 a u 1 a w 1 a = 0 x w 2 a No energy conduction in body 1

t T 2 = T 1 u 2 a u 1 a w 2 a x Energy conductions in bodies 1 and 2 compensate each other

t T 2 = 0. 5 T 1 u 2 a u 1 a w 1 a x w 2 a Doppler blue-shift

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 = u a + w a • 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