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Methavix
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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
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