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

2 2 2 2 2 2 2 2 2 2 2 2 τ τ Τ=Τ+Τ+Τ= Τ= Τ+ Τ+ Τ z y x z y x n n n n , (26) invariant in all inertial reference frames. On the other hand the invariant of this sort gives in Euclidean space = Τ+ Τ+ Τ ' 3 3' 2 ' 2 2' 2 ' 1 1' 2 nn nn nn 2 2 2 3 3 2 2 2 2 1 1 2 z y x nn nn nn Τ+Τ+Τ= Τ+ Τ+ Τ= = invar , (27) taking into consideration that 1 1' ' 1 = nn (affine tensors), i.e., the spatial part of the invariant connected with the temperature 4-tensor is completely identical to the invariant connected with the temperature 3-tensor. It is very important since it allows one to solve our problem directly in Euclidean space. As to the ultrarelativistic plasma considered in [11], the aforesaid will be valid in this case as well, which will be discussed below. c) Radiation Intensity Dependence vs. Temperature for a Moving Black Body Consider a black body moving uniformly and rectilinearly at angle θ =3π/2 with respect to the observer in the laboratory reference frame. Based on the aforesaid and on classical methods ( i.e., for v << c , see, e.g., [9, 16]) we can now begin its solution taking into consideration the follow. Now we use a cylindrical coordinate system and certain elementary normalizing volume in it. This volume contains two independent oscillators. The first oscillator is oriented parallel to axis 1. The second one is oriented perpendicularly to this axis. Then we can write an expression for the average total energy ε of the linear oscillators with quantum numbers 1 2 1 = = n n as follows (cylindrical space, zero oscillations are neglected): ( ) ( )         −         − + = + = = 1 1 2 2 1 1 2 1 2 1 1 , 2 1 θ ω θ ω ω ω ε ε ε    e e nn , (28) where 1 ω and 2 ω are the frequencies of oscillators in the direction perpendicular and parallel to the velocity of the moving object; 1 n and 2 n are positive (quantum) integers for the oscillators in the first and second directions. In this case 1 2 1 = = n n , since photons are bosons, they can be in one quantum state; 2 1 , TT are the values of the temperature tensor components. Obtaining the formula (28), we have used the law of the probability multiplying since the both oscillators are independent one another. Then the average volume of the total energy ε of the electromagnetic field per unit volume in the moving cavity proves to equal ( ) ( ) ( ) ∫ ∫ ∫ ∫ ∞ ∞ ∞ ∞         −         − +         −         − = − = 0 0 0 2 2 1 1 3 2 2 0 1 2 1 3 2 2 0 0 1 1 2 2 1 1 2 2 1 2 2 1 1 2 2 1 1 θ ω θ ω θ ω θ ω ω ω ω ω π ω ω ω π β ε       e d e d c e d e d c V E . (29) As a result, we have obtained, in fact, four improper integrals, three of them converge. The last two integrals in (29) differ only by variables. They are easily calculated using variable transformations as follows: )2(1 )2(1 )2(1 θ ω  = y = )2(1 )2(1 kT ω  , (30) ( ) =         −         − = ∫ ∫ ∞ ∞ 0 2 2 0 1 1 3 2 1 1 1 2 2 2 2 1 1 θ ω θ ω ω ω ω ω π    e d e d c I ( ) ( ) ( ) = − − ∫ ∫ ∞ ∞ 0 2 2 0 1 1 33 2 2 2 2 1 1 1 2 2 2 1 y y e dy y e dyy c  π θ θ ( ) 33 2 2 2 2 1 2 2 c  π θ θ = ∙ 36 4 π = 2 2 2 1 33 2 2 2 1 2 4 208 .0 72 T aT c TT k =  π , (31) © 2023 Global Journals 1 Year 2023 34 Global Journal of Science Frontier Research Volume XXIII Issue ersion I VI ( A ) The Heat Transfer by Radiation under Relativistic Conditions