# Would neutron average life expectancy be shorten by gaining (relativistic) mass?

A free neutron can survive up to 887.7s +/-3 (count from the bottle) or 878.5s +/-1 (pump kinetic energy). Could the difference of 9s due to its total energy at that moment when observation is made?

• I think I'll add this to my list of "Questions that seem plausible if you think in terms of relativistic mass even though they are obvious unreasonable from a proper point of view." That is, this is another reason why talking about 'relativistic mass' leads only toward error. – dmckee --- ex-moderator kitten Oct 13 '17 at 1:27
• Normally the life expectancy of a moving particle is higher by exactly the same gamma factor as the relativistic mass is higher than the rest mass. The only reason against the term relativistic mass that I know of is that some geniuses might confuse it with the rest mass and plug it into Newton's equations, but that is just because they are stupid so that is not a valid reason to abandon the term completely. The neutron problem is yet unsolved, that won't go away by simply avoiding the term relativistic mass (which still gives the correct answear for the muon phenomenon). – Gendergaga Oct 13 '17 at 1:58
• VERY naughty not to give a link for the general reader. – anna v Oct 13 '17 at 4:13
• I have found this scientific american article that gives the numbers scientificamerican.com/article/… , there are no "pumps" mentioned though. – anna v Oct 13 '17 at 4:23
• @annav: that's the article I read. – user6760 Oct 13 '17 at 4:27

## 3 Answers

I believe you're referring to the embarrassing problem that measurements of the neutron lifetime made with neutron beams have in recent decades given systematically longer lifetimes than measurements made with ultracold neutrons trapped in bottles:

[source]

Your idea is reasonable: since the beam neutrons are moving faster, perhaps time dilation plays a role. But the math doesn't work out. The modern beam measurements are made with thermal or cold neutron beams. A thermal neutron is drawn from an an ensemble with temperature $T\approx 300\rm\,K$, and the equipartition theorem tells us that such neutrons will typically have $kT \approx \frac12 mv^2$ --- that corresponds to a speed of about $v\approx 2000\,\mathrm{m/s} \approx 10^{-5}c$. But that gives a relativistic factor of

$$\gamma = \frac1{\sqrt{1-{v^2}/{c^2}}} \approx \left( 1-10^{-10} \right)^{-1/2} \approx 1+\frac12 10^{-10}$$

The ratio of the lifetimes from the two types of measurements is $$\frac{887\rm\,s}{880\rm\,s} \approx 1.008$$ which is different from one in the fourth significant figure, rather than the tenth significant figure. The relativistic correction isn't large enough to explain the discrepancy.

(This argument is still sound if you're not sloppy with factors of two.)

You mention in a comment that you learned about this discrepancy from this Scientific American news story. If you'd like more technical information you might read this feature article in the same magazine, or this academic review article which I think probably precipitated both of the Scientific American stories.

• Since the decay of neutrons is mediated by the weak interaction, and the weak interaction is almost maximally violating of parity symmetry (thus, chiral), could there be a difference in the chirality balance of the beam vs cold trap? Admittedly, chirality isn't relativistically invariant for massive particles like this, so I'm grasping at straws. The other candidate factor, in my mind, would involve how accurately they are able to start the timer in both experiments. Is the start trigger the same in both cases? – Sean E. Lake Oct 13 '17 at 2:35
• Chirality/helicity matter for weak interactions between particles, which is the basis for experiments that look for parity violation in capture or scatter processes with polarized particles. But the neutron decay happens in the particle's rest frame, where the chirality mixture is always the same. You're not the only one grasping at straws --- the community is also stumped. There's a whole pile of new neutron lifetime measurements to try and understand the difference between the techniques. – rob Oct 13 '17 at 2:49

In the system where it has the relativistic mass (rest mass times gamma factor), which is the system in which it is not at rest but moving, the live expentancy is not shorter but longer (because of the relative time dilation it ages slower, so it lives longer by the same factor the relativistic mass is higher than the rest mass, which is again the gamma factor).

So if you observe the moving particle to have a lower life expectancy than the one at rest that must be for some other reasons than time dilation.

To expand on @СимонТыран's answer, the fact that muons reach Earth's surface at all was one of the early forms of evidence for special relativity. The mean life of the muon is about 2 microseconds. In that time, a muon could travel only about 600 meters, at the speed of light. Since muons are produced thousands of meters up, only a tiny fraction should be able to reach the ground. What we observe is not consistent with that, and can only happen if the mean life is a rest frame phenomenon and the high energy muons gets extended by time dilation. For more details, see Wikipedia's article and the Hyperphysics article.