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

conditions if the temperature of the object transforms only following Ott s ' theory. The other variants are absent here. However this dependence and Clapeyron s ' equation treat the same division of physics – thermodynamics of gases and vapours. But if in the framework of this division of physics the temperature of the system may transform under relativistic conditions otherwise, then it is nonsense. Take, e.g., the equality T = T 0 ) ( ' T T = . As known, the temperature of gases or liquids depends on the velocities w their molecules (atoms) relatively the mass centre of the system under study. Let us be an observer in the laboratory references frame (see above, Fig,1). He is observing the microparticle velocities relatively the mass centre of a moving system containing a gas. The velocity components are equal according to SR: w 1 = 2 ' 1 ' 1 1 c vw v w + + , w 2 = 2 ' 1 2 ' 2 1 1 c vw w + − β , w 3 = 2 ' 1 2 ' 3 1 1 c vw w + − β . As we see, the velocity w has to vary at transition from one reference frame to another one. The equality T = T 0 contradicts to SR. It contradicts also the first principle of thermodynamics. Indeed, according to Callen and Horwitz [4] enthalpy H , chemical potential μ and temperature T are the Lorentz-invariants. However, then we will be at a deadlock because according to the Gibbs’ equation (it follows from the first principle of thermodynamics) , ; , , pS Np N H S H T       ∂ ∂ =       ∂ ∂ = µ (47) where S is the entropy, p is the pressure and N is the number of microparticles in the system. Going from one referee frame to other one, we obtain in (47) 0/0, i.e., indeterminate forms which we cannot evaluate. Of utmost interest is to consider if the dependence (20) remains valid for the case of an ultrarelativistic high-temperature spherical plasma (fireball) [11]. According to the author of [11], the spectrum of its equilibrium radiation ( ) * ω ε γ d (J/cc) due to the annihilation of electrons and positrons is described by the dependence ( ) * 2 2 * 2 * 33 2 4 * 1 * ω ω ω π ω ε ω γ d e f c T d − ∆− =  , (48) in the fireball, where T / * ω ω  = is the dimensionless frequency; Т >> mc 2 is the energy, i.e., apparently, kT = θ ( k is the Boltzmann constant, T is now the absolute temperature; θ not to be confused with the angle similarly designated (see above); T rel p / , ω  =∆ ; rel p , ω is the relativistic frequency of the plasma oscillations; f is a dimensionless constant. Formula (20) is valid for the case (48) with the vector part of the temperature dependent on the total velocity of electrons and positrons in the fireball but not on the velocity of its centre of mass. If their velocities are very high, then we have the well-known case described, e.g., in [10]. This is the case of a system of particles being widely apart and moving with very high velocities. It should be noted that these two cases are not fully identical, since the microparticles in [10] are not identical before and after the collision. In article [11], an electron-positron collision results in their annihilation. However these cases are very similar, thus the system energy ε may be given as ∑ − 2 2 2 1 ~ c v сm i i ε , (49) where i m is the microperticle mass, i v is its velocity. Then the vector part of the temperature in the ultrarelativistic case will transform in inverse proportion of the roots 2 2 / 1 c v i − . Here we immediately arrive at the conclusion that the dependence (48) is very doubtful, since the right side does not transform identically to its left side under the relativistic conditions. It should be also noted that the object studied in [11] is, in fact, a stable fireball. Evidently, when the density of electrons and positrons exceeds a certain limit, the stability will be broken, and an explosion will occur. IV. C onclusions Law was obtained for the black-body radiation in the whole interval of its (black-body) movement speed, i.e., from zero up to the speed of light in vacuum. This law is a special case when an angle θ between the movement velocity of the object under study and the observer is equal to zero. When the black-body speed is zero we obtain Stefan-Boltzmann law; when the black © 2023 Global Journals 1 Year 2023 38 Global Journal of Science Frontier Research Volume XXIII Issue ersion I VI ( A ) The Heat Transfer by Radiation under Relativistic Conditions