Exploring an Alternative to Hubble's Law

[hellfire]

If you apply the assumption of adiabatic expansion to a photon gas with $P \propto u/3 = E/3V$...

$$dE + PdV = 0$$

$$dE = - \frac{EdV}{3V}$$

$$E \propto V^{-1/3}$$

With $V \propto R^3$, this is:

$$E \propto R^{-1}$$

For the internal energy per unit volume:

$$u \propto R^{-4}$$

But I am confused. This follows merely from the assumption of adiabatic expansion, without taking into consideration the expansion of space. If a photon gas expands adiabatically in a piston of some characteristic lengt L (in static space), it will also increase its wavelengh, because $E \propto L^{-1}$ and $E \propto \lambda^{-1}$. However, in an expanding space the same relation applies. Why?

 

[Garth]

If the photon number remains constant there is no problem. Adiabatic expansion yields $E \propto R^{-1}$, which means the energy of each photon $E_\nu \propto R^{-1}$ that reveals itself as cosmological red shift.

The expansion of space dilution of each photon's energy is consistent with the time dilation cosmological red shift.

$1 + z = R_0/R(t)$

Garth

 

[hellfire]

OK, the evolution of the internal energy of a photon gas that is expanding adiabatically in static space is the same as the evolution of the internal energy of the photon gas that is comoving (constant comoving volume) in an expanding space.

But consider a photon gas in a cylinder with a piston. The piston moves due to the gas pressure that expands adiabatically. Additionally, consider that during the adiabatic expansion the space within the cylinder expands. Would this scenario imply a $E \propto R^{-2}$ dependence (due to gas expansion and space expansion)? (Such a scenario is not relevant for cosmology because the photon gas would increase its comoving volume which is against the cosmological principle, at least for the CMB).

 

[Garth]

I do not understand what you mean by: "Additionally, consider that during the adiabatic expansion the space within the cylinder expands."

Is everything expanding? The whole cylinder and the piston? Rulers as well? How are you measuring this expansion?

In cosmology, the work done by the pressure in the expanding co-moving volume of the universe is equal to the change in total energy. If that pressure is mediated by a photon gas then that photon gas loses energy, even though the total photon number remains constant. Thus each photon loses energy and is red shifted as a result.

Garth

 

[hellfire]

The piston would be moving, as in a usual thermodynamical experiment in laboratory. Additionally there would be a nonnegligible expansion of space within the cylinder.

In cosmological terms you could imagine that the CMB would increase its comoving volume (it would be flowing radially "outwards" of the observable universe) instead of mantaining it constant.

 

[gptejms]

But I am confused. This follows merely from the assumption of adiabatic expansion, without taking into consideration the expansion of space. If a photon gas expands adiabatically in a piston of some characteristic lengt L (in static space), it will also increase its wavelengh, because $E \propto L^{-1}$ and $E \propto \lambda^{-1}$. However, in an expanding space the same relation applies. Why?

I haven't carefully gone thru your derivation,but P seems to be constant.Why?What makes the calculation adiabatic?

If the universe were to expand like an ordinary gas,its temperature wouldn't change at all---because unlike an ordinary gas which does work against external pressure and loses internal energy,there is nothing for the universe's 'background photon gas' to work against.Its internal energy would remain constant in this scenario.So the only thing that causes temperature loss for the background radiation is the cosmological expansion.

 

[hellfire]

I haven't carefully gone thru your derivation,but P seems to be constant.

The pressure is P = u/3 and therefore has the same dependence with R as u.

 

[gptejms]

Ok---you are using the first law(with dQ=0 for adiabaticity).But the gas is working against the external constant pressure P (distinguish this from internal pressure of the gas which is not constant and as you say given as u/3).

Anyway,as I have said there is no similarity here with the expansion of the universe's photon gas--the only effect that causes its cooling is the cosmological red shift due to expanding space.