(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

If we have a cavity filled with black-body radiation, an this is placed onboard a relativistic rocket (uniform linear motion).

I want to yield the transformation laws (from the proper frame S' joint to the rocket and moving with constant velocity V in respect to a coordinate frame S) for energy fluxphi, energy densityepsilon, energyE, pressurep, heatQ, and entropyS.

2. Relevant equations

I know these equations expressed in the proper frame:

phi' = sigma'*T'^4

epsilon' = (4/c)*phi' = (4/c)*sigma'*T'^4

E' = epsilon'*psi' = (4/c)*sigma'*T'^4*psi'

p' = epsilon'/3 = (4/c)*(sigma'*T'^4)/3

dQ' = (16*sigma'*T'^4*dpsi')/(3*c)

S' = (16*sigma'*T'^3*dpsi')/(3*c)

where

sigma' = Stefan-Boltzmann constant in the proper frame

T' = temperature in the proper frame

psi' = volume of the cavity in the proper frame

3. The attempt at a solution

Being (it is a known formula):

E = (E’+p’*psi’*beta^2)/sqrt(1-beta^2)

by subsituting, i yield:

E = [(1+(beta^2)/3)/sqrt(1-beta^2)]*E’

And, if I want to transform pressure? I know in general that:

p = p’ ----> (4/c)*(sigma*T^4)/3 = (4/c)*(sigma'*T'^4)/3 ----> sigma*T^4 = sigma'*T'^4

then:

sigma = sigma’*gamma^4

but this implies also:

epsilon = epsilon’

and for the Boltzmann constant k:

k = k’*gamma (by the definition of sigma, if we suppose invariant the Planck constant, is it right?)

The heat transforms as follows:

dQ = (16*sigma'*T'^4*dpsi')/(3*c*gamma)

Now, there is a problem. In fact, if a calculate E directly from epsilon (considering right its transformation):

E = epsilon*psi = epsilon’*(psi’/gamma) = E’/gamma

But this is in disagree with the transformation known of the energy.

Where is my error?

Thanks

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# Cavity black-body radiation aboard a relativistic rocket

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