B-E,F-D Statistics and the Lorentz Transformations

[Anamitra]

Let us consider the B-E and F-D statics:

$${<}{n}_{i}}{>}{=}{\frac{1}{{exp}{(}{{\epsilon}_{i}{-}{\mu}{)}{/}{kT}}{\mp}{1}}$$

Now we observe the formula from a boosted frame.The left side is a scalar and should not change in response to the Lorentz transformations.What about the right hand side?The quantity
$${(}{\epsilon}_{i}{-}{\mu}{)}{/}{kT}$$ does not look like a Lorentz invariant.

 

[Anamitra]

$${(}{\epsilon}_{i}{-}{\mu}{)}{/}{kT}$$ is a dimensionless quantity.But is it Lorentz invariant?

[Quantities like v1/v2 are dimensionless. But when we observe the ratio from a boosted frame its value changes]

 

[Meir Achuz]

"The left side is a scalar and should not change in response to the Lorentz transformations."
Neither side is a Lorentz scalar.

 

[Dale]

.The left side is a scalar and should not change in response to the Lorentz transformations.

In GR a scalar is not simply a number, it is a number that transforms in a specific way (ie as a rank 0 tensor)

 

[Anamitra]

Plausible Physical Mechanism
A box of particles having distinct energy levels[ e1,e2,e3,e4 etc] is considered. On giving it a boost we have:
1)Change of energy levels.For example e1 becomes e3 and e4 becomes e7.Previously e1 had n1 particles and e4 had n4. Now the old e1[its energy has changed] has n3 particles instead of n1 particles and the old e4[whose energy has changed to e7] has n7 particles instead of n4.The laws[BE and FD statistics ] should retain their form.
2)New energy levels may get created due to the boost and get populated according to the BE or FD statistics.

Transition of particles between energy levels serve to favor the explanation.The new equation may be the old equation for some level which had the same value of energy..

A boost is quite synonymous with what we call mass motion and that should not change temperature.One may keep a box of oxygen on a table and start running backwards--that should not change the temperature!

 

[Dale]

A boost is quite synonymous with what we call mass motion and that should not change temperature.One may keep a box of oxygen on a table and start running backwards--that should not change the temperature!

Why not?

 

[Anamitra]

There is a problem with posting #5.
If we leave a box filled with particles on a table and start moving,the energy levels will get changed values. But there is no physical cause that can move particles between different levels to restore the old forms of the equations--we don't have any physical agent causing the transfer of particles.
A change in temperature may support the issue. But in such a case we are assuming that$${(}{\epsilon}_{i}{-}{\mu}{)}{/}{kT}$$ is not changing.

The problem seems to persist if both sides of the equations in #1 are not scalars

 

[Dale]

Honestly, I don't know anything about relativistic thermodynamics, but at first glance I would not expect temperature to be a relativistic invariant. After all, it is associated with things like black body radiation which would be subject to relativistic Doppler, and energy, pressure, and volume which are all frame variant. Temperature might be a scalar, but if so it is not obvious to me.