Is temerature frame dependant ?

Question

What is the temperature of a body ?

" The temperature of a system is defined as simply the average energy of microscopic motions of a single particle in the system per degree of freedom. For an ideal monatomic gas, the microscopic motions are the translational motions of the constituent gas particles. "

So, if the temperature is a measure dependant on average kinetic energy, will it be frame dependant like kinetic energy ?

To illustrate, consider a closed massless container filler with an ideal mono-atomic inert gas. Let is also be rest with any frame of referance A. According to the kinetic theory of gasses, the molecules inside the gas will experience random motion in all directions. This will contribute to heat and in turn temperature. Since there is no external force acting on this adiabatic and closed system, its center of mass be at rest.

Now, consider frame of referance B : Sit on any one molecule. From this frame, the velocity of the molecule becomes zero and hence so does its kinetic energy. If there are other molecules with the same energy, they will now also have no kinetic energy. In this new frame of referance there is another problem, the center of mass has some velocity. As a result the net kinetic energy of the container changes in this new frame of referance. But will also the temperature ???

It is against classical mechanics for such a thing to happen. Considering very small temperatures, and speeds NOT approaching c, how can this problem be solved. The temperature of a substance MUST be the SAME in any frame of referance.

But, if it depends of kinetic energy = 1/2mv^2 and that depends on velocity with varies with frame of referance, it is deducable that so is temperature.

HOW can this be explained?

Answer 1

Temperature is defined in terms of internal energy of a system. That is, in the center of mass frame. So there is no frame dependence.

Answer 2

I disagree with other responders. I think temperature is NOT frame dependant. In the examples cited (e.g. heat of re-entry on a space capsule), my interpretation is that the rise in temperature is caused by the fact that the MACROscopic kinetic energy of the air mass is converted into thermal energy due to the collisions with the spacecraft. And as a result, the colliding mass of air loses some of its macroscopic KE.

Another example: Suppose a tree (enclosed in a bubble of air) were found hurtling through outer space. If temperature were frame-dependant, then one would expect that in some reference frame (i.e. one that is traveling very fast w/respect to the tree), both the tree and the surrounding air would be at an extremely high "temperature". If that's the case, why doesn't the tree burst into flame?

Another point: An alternative definition of temperature is in terms of the spectrum of radiation that permeates the volume you're measuring. (In volumes that contain matter, the random thermal motion always corelates with the EM spectrum.) But clearly (at non-relatvistic speeds) the characteristics of the EM spectrum don't change that much in different reference frames. And even if you consider the change due to doppler shift as being characteristic of temperature, you'd have to conclude that a mass of air moving _away_ from you should be "colder" than a similar mass of air moving _toward_ you (even though both masses have the same relative speed with respect to your frame!)

I think the only sensible thing is to define the temperature of a given mass in terms of the random motion of its particles, RELATIVE TO THE CENTER OF MASS of the collection. That way, the definition is frame-invariant.

Answer 3

I will adderess only the part about non-relativistic
container with ideal gas.

Accurate kinetic theory of ideal monoatomic gas
does not state that average kinetic energy is
proportional to emperature.

Instead, it states that
- - = 3/2 kT,

where
KE is kinetic energy of molecule,
p is momentum of molecule,
v is velocity of container,
L is angular momentum of molecule,
Ω is angular velocity of container, and
<> denotes average over time.

If you consider motionless container,
when v = 0, and Ω=0, then you get familiar formula
= 3/2 kT.

The former formula is covariant wrt Galilean
transformations of frame of reference, and
temperature remains invariant under such
transformations. The reason for two extra
terms - - is that just like energy KE,
momentum p and angular momentum L of
all molecules are added together, and total
amount of each property is conserved.
All this goes far beyond high-school physics.

Answer 4

Are you referring to relativistic effects, where two frames are moving relative to one another at speed approaching c? I don't have the patience or interest to think this thru rigorously, but:

I suspect that the increased mass might exactly offset the slowing of time; so my guess is the temperature is not frame dependent.

Answer 5

Temperature is frame dependent:

1) Supersonic jet expansion into vacuum. Transverse temps are near absolute zero. Temp in the direction of flow is not espcially cold.

2) Atmospheric re-entry from orbit. The vehicle comes in around 5 miles/second air speed or 18,000 mph. That is an apparent rms temperature to the spacecraft of 74,000 C given still air vs. ground. Heat shields are necessary.

http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/kintem.html
Middle; calculator

Answer 6

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Answer 7

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