# Do hot gasses actually weigh more than cold gasses in SR?

Where they begin to discuss a weighing a relativistic gas against a cold gas.

The idea is that the relativistic gas molecules are moving faster and therefore their "relativistic masses" are higher and therefore they weigh more, and that this could quite literally be measured by considering a balance with the same # of molecules and same container holding them, only one container is at a high temperature and the other at a low temperature.

## Why I think this is suspect:

At best if you bring out full on General Relativity then the internal pressure of the gas might have some non trivial gravitational effect to make it heavier, but I DO NOT believe that the hotter gas will be heavier on just purely Special Relativity + Newtonian Grounds. ORIGINAL ARGUMENT REDACTED BECAUSE I FOUND A FLAW.

## Original Text Below:

"Assuming the principle of equivalence, which entails that we can measure inertial mass by measuring gravitational mass with a balance, we can illustrate the difference between the Newtonian and relativistic understanding of the ideal gas as follows. Imagine that we have two otherwise identical massless vessels filled with exactly the same amount and type of gas. In one vessel, the gas is at a temperature very near absolute zero, so its molecules have very little kinetic energy. In the other vessel, the gas is at a temperature of 500° C. Place these two vessels of gas on the ends of a balance. According to Newtonian physics, the balance will be level, because both gas samples have exactly the same mass. According to relativity, the balance will not be level and will be tipped on the side of the hot gas, because the high kinetic energy of the molecules contributes to the rest energy of the gas, which contributes, through Einstein’s equation, to the rest-mass of the vessel of gas."

## 2 Answers

I agree with tparker that, through the appeal to the equivalence principle, the quoted text implicitly refers to general relativity, not just special relativity. In general relativity pressure contributes to gravitation.

If one has high speeds then one cannot use Newtonian gravity and expect to get accurate results, so indeed general relativity has to be used if one is investigating gravitational effects. However, one could instead do an experiment that involves purely inertial effects in flat spacetime, and then special relativity suffices. So for example, take the same balance as the one mentioned in the quoted text, and situate it in flat spacetime far from gravitating bodies. Then push on the balance point so as to cause it to accelerate. The two vessels of gas will resist this acceleration by different amounts, and thus reveal their differing inertial mass by tipping the balance beam.

Here, the quantity I have called inertial mass, for either vessel containing gas, is the total energy of the vessel and its contents, as observed in the rest frame of the vessel, divided by $$c^2$$.

Where in that text do you see a claim that the argument is based "just purely Special Relativity + Newtonian Grounds"? When they say "relativistic", they mean general relativity, not special relativity.

• the article only references special relativity explicitly, as its discussing mass energy equivalence, (although if you ctrl f the term "general relativity' does appear a total of 3 times as part of historical commentary included in the article) – frogeyedpeas Sep 3 '19 at 0:35
• @frogeyedpeas Nevertheless, their claim in the paragraph you cited only makes sense in the context of general relativity. Special relativity does not incorporate gravity; any time you talk about relativistic effects in gravity, you've moved beyond the realm of special relativity, and into the realm of general relativity or another relativistic theory of gravity (e.g. Brans-Dicke theory). – tparker Sep 3 '19 at 0:39