Cavity black-body radiation aboard a relativistic rocket

[Methavix]

Homework Statement

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 flux phi, energy density epsilon, energy E, pressure p, heat Q, and entropy S.

Homework 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

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

 

[dextercioby]

First, get a book on relativistic thermodynamics. Second, i hope you don't insinuate that sigma' is any different from sigma.

Daniel.

 

[Methavix]

i don't say that sigma' is different from sigma, my calculations said this. in fact at the end of my post i wrote: "where is my error" ? :)

 

[MeChaState]

Hi,
Where do you have this relationship?

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

It is saying that the change of heat in the BB is related to the change of volume?
Thanks
David

 

[MeChaState]

I think this is not the right way of using the formula:
E = (E’+p’*psi’*beta^2)/sqrt(1-beta^2)
Where ever you got the second term from, for sure in the rest frame E' there is no division by sqrt(1-beta^2).
the Energy-Momentum states: E^2(moving frame) = E'^2(rest frame) + c^2.p^2.