# Boyles law & relativity

In relativity when an object approaches the speed of light, $c$,its volume decreases due to length contraction. As a thought experiment, consider a rectangular metal box travelling at a relativistic speed. The inside of the box is filled with a gas, such that if the pressure of this gas is beyond a certain limit then the box will explode. According to Boyle's law, as volume decreases the pressure increases. So in this case would an outside observer see the box explode?

Now consider a new case where the same box filled with gas is kept at rest.This time an observer is moving at a relativistic speed close to $c$. According to frame of this moving observer he will see the box moving close to speed $c$, so would he see the box exploding due to the increase in pressure?

• From the perspective of the box, nothing has changed except that you the observer is flying away at close to c, and are starting to look funny from relativistic effects. So, no, nothing happens to the box. – Jon Custer Mar 24 '15 at 13:50
• But from perspective of observer will the box explode – user213892 Mar 24 '15 at 14:08
• Why should it, given what I wrote? Nothing unusual is happening from the box's perspective - it is only the observers view of it that looks funny, and that is all there is to it - it will look funny if you move by it near the speed of light. – Jon Custer Mar 24 '15 at 14:09
• From frame of moving observer he is at rest and box is moving.so relativistic effects must apply to box from moving observers frame Right? – user213892 Mar 24 '15 at 16:47
• @jon the question is clear. because pressure is not a Lorentz scalar, observers disagree about it. do they also disagree about whether the pressure causes an explosion? if not, why not? – innisfree Mar 24 '15 at 18:24

## 1 Answer

Pressure is a single component of the rank-2 stress-energy tensor; it isn't a Lorentz scalar and observers will disagree about the pressure of a gas in a box.

You say that the box might explode (perhaps the walls of the box break) if the pressure exceeds a critical value. Perhaps, then, observers disagree about whether an explosion occurs?

I don't think so. The maximum pressure that the walls of the box can withstand isn't a Lorentz scalar, either, and transforms identically to a pressure. If and only if the pressure in the box's rest-frame satisfies, $$P < P_{\text{Critical}}$$ the condition $$P' < P'_{\text{Critical}}$$ where prime indicates a quantity in a boosted frame, holds for a boosted observer. Thus everybody agrees about whether the pressure in the box exceeds the critical value and whether the box explodes.