Local Rates and Temperature in Twin Paradox

In the "Twin Paradox," the space-traveling twin ages more slowly than the earth-bound twin. So it would seem that all processes occur more slowly inside the spaceship. But if that is so, wouldn't atoms and molecules also move more slowly (i.e., take a longer time to cover the same distance)? And wouldn't that imply that the temperature in the spaceship drops as its speed increases? If you're looking at an ideal gas in a box, how do you tell the difference between a gas which is cooling, and a gas whose "time is slowing down" like in the spaceship?
- Kevin Karn (age 52)
Osaka, Japan

That's a great, subtle question.

The first part is easy to answer. Say that all rates appear reduced in the spaceship, from our point of view, because of its motion relative to us. We have no trouble figuring out what the local temperature is in the spaceship, bcause we know exactly how that apparent slowing down depends on the relative velocity. So it's easy to calculate what all the velocities are in the local frame. So that's a classical way to figure out that temperature.

Here's another way to think about the temperature. It tells you, among other things, what fraction of atoms are in their various excited states. (A key search term here is "Boltzmann factor".) There's a discrete list of these quantum states, so you can ask things like "what fraction of the H atoms are in 2P states?" and tell from that what the temperature is. Now that fraction doesn't change no matter how you look at the collection of atoms. So there is a direct way of reading what the local temperature is in terms of these observable numbers, without having to go through Lorentz transforms to calculate velocities in the spaceship frame.

Mike W.

(published on 03/18/2014)