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

Indeed, examine the simplest case represented in the Fig.1. As seen, there are two reference frames. One of them (with primes) is moving uniformly and rectilinearly with the velocity v. There is a cylindrical object at rest in the moving reference frame. There is a cylindrical cavity in the object. The walls of the cavity are a black-body. They are emitting and absorbing photons. There is a very small hole on a face-wall of the object (see Fig.1). The flux of photons is flowing out the cavity. Since the hole is very small, the equilibrium of the photon gas in the cavity does not disturb practically. The photon radiator is at rest in the moving reference frame. An observer is at rest in the laboratory reference one. The observer is detecting photons (the energy of the electromagnetic wave). Here Maxwell’s 3-D tensor of energy-momentum αβ σ has only one component - 11 σ . It is equal to the density of energy in the wave [9]. Knowing the density of energy in the flux of photons, we can estimate the density of energy of the photon gas in the cavity in practice. If the angle θ between v and the observer is 2/3 π (the object is moving away from the observer), then the radiation frequency of the oscillator ω will be for this case equal to β β ω ω + − = 1 1 2 0 . (8a) If the object is moving to the observer i.e., the velocity of the system is equal to -v, then β β ω ω − − = 1 1 2 0 . (8b) Denote the frequencies ω in (8a) and (8b) as 1 ω and 2 ω then , 1 2 2 0 2 1 β ω ω ω ω − = + = (8c) If the angle θ were )2/ ( π − , then the formula for the frequency transformation would have another form, namely: 2 0 1 β ω ω − = . (9) for the observer in the laboratory reference frame. Thus without taking into consideration (8) and (9), we cannot evidently use the well-known Bose- Einstein distribution for obtaining the Stefan-Boltzmann law when the object under study is moving with relativistic speed. The aforesaid allows us to formulate a main goal of our work – obtaining a radiation law for the black-body moving with a relativistic velocity when the angle θ between the moving velocity v and the observer is 2/3 π (see Fig.1). A solution of the problem will be performed by the methods given in [9]. Here we must be added the following. Attempts have been made to obtain the law connecting the radiation intensity with the temperature when relativistic effects are involved [10, 11]. For example, in [11] an ultrarelativistic plasma is examined containing electrons and positrons. Their annihilation generates electro- magnetic radiation. Its intensity is defined, in particular, with the help of a one-dimensional Bose-Einstein distribution. It is proportional to the plasma temperature to the fourth power, with the velocity of the object as a whole being equal to zero. It is plasma particles that are in motion. II. M ethods and R esults a) Definition of the number of field oscillators with a given frequency when the angle θ is 3π/2 (Fig.1) Assume that we have an opaque object with an inner cylindrical cavity. Its surface is a black body heated up to some temperature T. There is a thermodynamical equilibrium in the cavity between its inner surface and electromagnetic radiation. There is a very small hole in the object cover, through which electromagnetic waves radiate out of the cavity (see Fig.1). The object is moving uniformly and rectilinear with the velocity v together with the reference frame. The radiation from the cavity is detected with a device being at rest in a laboratory reference frame. First of all, we will show that the Stefan-Boltzmann law (3) is incorrect over the whole range of object motion velocities, i.e., from zero up to v → c. Indeed, according to X. Ott [3], the radiation energy in the cavity is equal to: ( ) 2 1 0 1 β ω − = ∑ n n h E , n =1,2,…. l , (10) then the electromagnetic energy density ( ) ( ) ( ) 2 0 1 0 2 0 0 1 1 β ω β ε − = − = ∑ V h V E n n , n =1,2,… l , (11) where n is an oscillator serial number, ( ) n ω is the frequency of its oscillations. For the flux of photons moving away from the cavity (see Fig1), the equations (10) and (11) are also correct, Indeed, there is only one component 11 σ of Maxwell’s stress tensor (see above). The component is equal 1 Year 2023 3 © 2023 Global Journals Global Journal of Science Frontier Research Volume XXIII Issue ersion I VI ( A ) The Heat Transfer by Radiation under Relativistic Conditions