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

2 )03(02 )3(2 2 )03(02 )3(2 2 2 03 2 02 2 03 2 02 2 3 2 2 2 3 2 2 11 1 ; 1 , 1 ~ β β β σ − = − = − + + + = + + + H H E E H H E E H H E E , where )03(02 )3(2 , E E are the intensity components of the electric field in the directions 2 and 3 for the observers in the laboratory reference frame and for the observer moving with the system under study correspondingly; )03(02 )3(2 , H H are the intensity components of the magnetic field in the directions 2 and 3 for the above observers. No matter how the temperature of the system transforms, i.e., according to Planck or to Ott or to Callen and Horwitz, we shall always arrive at the point of absurdity. Indeed, let the temperature transform, e.g., according to Planck, i.e., to (1). In this case the right side of (3) will have the following form ( ) 22 4 0 1 β − aT . Then, as seen from (11), the right side of (3) appears to tend to zero as v → c , while the left side of this formula to increase infinitely. This indicates a close connection between the radiation law of a moving black body and the temperature transformation under relativistic conditions. Now find the number of oscillators g ( 2 1 , ω ω ) 2 1 ω ω dd with frequencies in intervals 1 1 1 , ω ω ω d + and 2 2 2 , ω ω ω d + and a given polarization in the cavity using the well-known procedure [9]. The following fact should be pointed out at once. The number of these oscillators is a function of two variables. The reason for that was explained above but here the following should be noted. If a spherical coordinate system is used for the case v<<c , then in our case it is convenient to use a cylindrical one taking account of formulae (8) and (9). The classical approach to finding the quantity ( ) ω ω d g is based on using the number space n followed by transition to a spherical space of the wave vector k= │ k │ = L n π 2 , where L is the normalized cube edge, and finally to the spherical space of frequencies ω . In the case studied we use a cylindrical space representable as two spaces – flat circular and linear perpendicular to one another. Then to define the necessary quantity we shall use two coordinate systems: polar and one-dimensional Euclidean, i.e., a straight line. The amount of numbers within the spherical layer dn of the spherical space is dnn 2 4 π [9] (the spherical coordinate system). The amount of numbers 1 n in the circular layer is equal to 1 1 2 dnn π (the polar coordinate system). As to 2 n in a linear interval of one- dimension space, it will be equal to 2 dn . As a result, we have for the whole system: g ( 2 1 , ω ω ) 2 1 1 2 1 2 dn dnn d d π ω ω = . (12) Turning from a number space to a wave vector space and finally to a frequency one, we shall have: g ( 2 1 , ω ω ) = 2 1 ω ω d d ( ) ( ) 2 2 1 3 2 2 1 1 2 2 1 3 2 1 1 2 1 1 2 2 2 2 LL c d d LL dk dkk dn dnт ∆∆ = ∆∆ = π ω ω ω π π π = = ( ) V c d d ∆ 3 2 2 1 1 2 π ω ω ω . (13) In case of electromagnetic waves should be taken into account two polarizations, and then we shall have: ( ) 2 1 , ω ω g = 2 1 ω ω d d V c d d ∆ 3 2 2 1 1 2 π ω ω ω . (14) Here it is important to emphasize that formula (14) is correct for the observer at rest in a real space monitoring, from the referring frame, the object moving then uniformly and rectilinearly with the relativistic velocity v. Since the radiation is thermal the average volume of the oscillators with a given polarization will almost be independent of time. In this case, it is unnecessary to define oscillator numbers in Minkowski space. b) Relativistic temperature as either a vector or a tensor Now we should make a new attempt to solve some problems connected with the relativistic temperature. First of all, we should clarify if this thermodynamic parameter is a scalar or appears to be a vector or a tensor. In this connection we should first recall the formulae for velocity addition in SR. As known, the components of the total velocity in the directions X 2 or X 3 will tend to zero for the observer in the laboratory reference frame as v → c (see Fig.1). In turn, the component parallel to axes the X 1 will not do that. This suggests immediately that the temperature becomes a mathematical object different from a scalar. What is the object? Until very recently the temperature in the above case is considered to be either a scalar or a quantity © 2023 Global Journals 1 Year 2023 32 Global Journal of Science Frontier Research Volume XXIII Issue ersion I VI ( A ) The Heat Transfer by Radiation under Relativistic Conditions