Equations of State for Photon Gas and Relativistic Electron Gas

This Insight develops equations of state that are useful in calculations about cosmology and about the insides of stars. The first calculation is for a photon gas and the second is for a ‘relativistic’ gas of particles with mass.
Photon Gas
Exercise 22 on p108 of Bernard Schutz’s ‘The first course in General Relativity (Second Edition) is to prove that, for anisotropic, monochromatic, photon gas, p=ρ/3, where p is pressure and ρ is mass-energy density.
Say all photons have frequency ##\nu## and the number of photons per cubic metre is ##n##. Then ##\rho=n h \nu##.
Now consider one face of a cube of side length 1m. The pressure on that face is the component, parallel to the normal to the face, of the impulse delivered to that face in one second, by photons striking it from outside the cube. We can ignore components of impulse that are parallel to the face because the isotropy will make such components from different particles cancel each other out.
We measure that impulse by...

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Hello,

Thank you for sharing this insight about equations of state and their applications in cosmology and the study of stars. Your explanation of the photon gas and the relativistic gas of particles with mass is clear and concise. I appreciate the reference to Bernard Schutz's 'The first course in General Relativity' for further reading and understanding.

I would like to add that equations of state are essential in understanding the behavior and properties of matter in extreme environments, such as the insides of stars and the early universe. They help us make predictions and calculations about the behavior of matter under extreme conditions, which would not be possible otherwise.

In addition to the photon gas and relativistic gas, there are other types of equations of state that are used in cosmology and astrophysics, such as the polytropic equation of state and the ideal gas law. Each of these equations is useful in different scenarios and helps us gain a better understanding of the universe.

Thank you again for sharing this insight and for highlighting the importance of equations of state in our understanding of the cosmos. I look forward to reading more of your contributions on this topic in the future.