Global Journal of Science Frontier Research, A: Physics and Space Science, Volume 23 Issue 1

The Heat Transfer by Radiation under Relativistic Conditions Emil V. Veitsman Abstract- An expression was obtained for the energy density of the moving black-body radiation, i.e., the Stefan-Boltzmann law valid in the interval of object velocities from zero to the velocity of light in vacuo when the angle of observation θ equals zero. The object temperature is shown to comprise two parts. The first one is a scalar invariant under the Lorentz transformations. The second one is a vector depending on the velocity of system motion. The scalar component of the temperature is a contraction of two tensor components of rank 3. Under normal conditions this mathematical object is a scalar. Taking account of a tensor character of the temperature a new formulation is given for the second thermodynamics law. The results obtained are of the great practical importance, in particular, while designing devices to measure the radiation temperature of moving cosmic objects, e.g., quasars. Keywords: heat transfer; radiation; relativistic conditions; temperature; thermodynamics; entropy. I. I ntroduction he problem of the moving black-body radiation arose in 1907 – almost immediately after the creation of Special relativity (SR). It is in this year that Kurd von Mosengeil’s big article was published in der Annalen der Physik [1]. This work supervised by Max Planck underlies his relativistic thermodynamics [2]. The great scientist considered the theory of the black-body radiation to be well-studied and the most suitable for formulating foundations of thermodynamics correct over the entire whole interval of object velocities v, i.e., ranging from zero to the velocity of light in vacuo. In article [1] a system is studied comprising a radiator of electromagnetic waves, receiver and reflector (mirror). The radiators are receivers at the same time. The three elements are moving uniformly and rectilinearly in space with a relativistic velocity forming an acute angle with one another. As a result, the temperature transformation law was obtained under relativistic conditions: 2 0 1 β − = T T , (1) where 0 T is the temperature if v<<c (here and below index “0” means that the given quantity concerns normal conditions); . / cv = β For more than 50 years formula (1) had not been called in question until X.Ott’s article was published [3], in which the relativistic temperature was shown to transform following another law: 2 0 1/ β − = T T . (2) The expression (2) was obtained by X.Ott for a variety of physical processes including electromagnetic radiation. However unlike Mosengeil, X.Ott elected another approach for studying the process of electromagnetic wave radiation under relativistic conditions. He examined wave emission of individual atoms, whereas Mosengeil studied black-body radiation, as we have noticed above. In particular, in [1] Stefan-Boltzmann’s law was obtained: 4 0 0 0 0 aT V E = = ε , (3) based on the famous Planck formula derived first semiempirically: ( )         − = 1 8 , 3 3 kT h e c d h dT ω ω ω π ω ω ρ , (4) where 0 E is the radiation energy of the black-body; V 0 is the volume; а is Stephan-Boltzmann’s constant (J/ с c ∙grad 4 ); ( ) T , ω ρ is the radiative energy density (J/ с c); k is Boltzmann’s constant; ω is the frequency of oscillator radiation. As known, Stefan-Boltzmann’s constant equals: 33 2 4 15 c k a  π = . (5) X.Ott’s article has induced a long-term polemic on the temperature transformation under relativistic conditions. Some researchers adhered to Planck- Einstein’s viewpoint; the others adhered to X.Ott’s. Some scientists considered the temperature to be a relativistic invariant [4]. There appear absolutely exotic opinions. For example, the authors of Ref. [5] arrived at a conclusion of the temperature under relativistic conditions being changed both according to Planck, and to Ott, and to Callen and Horwitz as the able T 1 Year 2023 29 © 2023 Global Journals Global Journal of Science Frontier Research Volume XXIII Issue ersion I VI ( A ) Author: Independent Researcher, 28 department, 5 Klimashkin Str., 123557. Moscow. e-mail: evveitsman@gmail.com