Temperature and time dilation

Question

Say you had some liquid radioactive isotope with a half-life equal to X. If it was cold, the molecules would move slowly, and thus there would be virtually no time dilation involved, resulting in the standard half-life of X.

That same liquid radioactive isotope is now heated up to a very hot temperature. The molecules are moving faster, and thus should experience some time dilation, right? If so, shouldn't the half-life become X + τ? Granted that, given practical realities, τ would be really small, but in theory, if we got the heat up high enough to where the velocity of the molecules was a non-trivial percentage of c, wouldn't the half-life increase significantly?

If so, is time dilation really a micro effect? In the spaceship example, where a twin travels to Antares or something, and then comes back, younger than his twin, would it be more accurate (and yes, I realize that the differences would be negligible) to say that each piece of him is a slightly different age?

Answer 1

This kind of time diltion is a real and observable effect, although it it most easily seen in solids rather than gasses. As the temperature of a crystal increases, the atomic vibrations inside it increase in amplitude. Because their average speeds are higher, they experience greater time dilation at higher temperatures, which may be observed with ultra-high precision Mössbauer spectroscopy.

This was first noticed in the course of the Pound-Rebka experiment, which measured the gravitational redshift using a Mössbauer source and detector placed far apart at the top and bottom of a long shaft. They initially did not get the precision they had anticipated, until it was pointed out (by a young Brian Josephson, I think) that the temperature difference between the source and the detector was skewing the observed frequencies of the experimental γ-rays (a frequency shift being equivalent to an inverse shift in the rate of passage of time). The temperatures at the top and bottom of the experimental setup needed to be known to within a fraction of a degree to get a measurement accurate enough to see the gravitational redshift (which had a size of only a few parts per $10^{15}$).

I have personally observed this phenomenon in the lab. Blowing hot air from a handheld hair dryer on the $^{57}$Co source at one end of a Mössbauer apparatus was sufficient to see the absorption peak, from the detector at the other end, move.

Answer 2

For an ideal gas, the classical Maxwell-Boltzmann distribution is replaced by the Maxwell-Jüttner-Synge distribution. How relativistic a temperature is can be parametrized by the ratio of thermal energy to rest mass energy:

$$ \theta = \frac{kT}{mc^2} $$

The MJS function is (https://en.wikipedia.org/wiki/Maxwell–Jüttner_distribution):

$$ f(p) = \frac 1 {4\pi m^3c^3\theta K_2(\theta^{-1})} e^{-\frac {\gamma(p)} {\theta}} $$

Note that, with

$$ p = \gamma m\beta c $$

and the usual:

$$ \gamma = \frac 1 {\sqrt{1-\beta^2}}$$

then also:

$$ \gamma = \sqrt{1+\big(\frac p {mc}\big)^2} $$

which is a far less common expression.

Maybe go to Wolfram integrate to get the average lifetime boost:

$$ \bar{\gamma} = \langle \gamma(p)f(p) \rangle $$

luckily the Bessel function of the second kind comes in as a constant:

$$ K_2\big(\frac{mc^2}{kT}\big)$$

so your integral is of the form:

$$ \int \sqrt{a^2+x^2} e^{-\sqrt{a^2+x^2}} dx$$

anyway, there seem to be answers here:

Does Heat Cause Time Dilation?