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I'm currently teaching my students about the mass-energy equivalence.

As far as I know, it is accepted that a higher temperature object should have a larger inertia, because it has more energy and therefore has more relativistic mass (I am aware the concept of relativistic mass has fallen out of favor as a term), but is there any direct experimental evidence that the temperature of an object influences its inertia?

Special relativity by itself is well established experimentally, so one should trust the implications it has in this case, but are there experimental investigations of thermal energy contributions to the mass-energy equivalence?

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  • $\begingroup$ The problem is things have low heat capacity. If you take an object with mass-energy $E_1=mc^2$ and try and raise the total mass energy to even $1.01E_1=E_2$, then you have just added $0.01mc^2\approx m\times 10^{15}\,\text{J/kg}$ of energy to the object. Even hydrogen, which has a specific heat of $1.43\times 10^{4}\text{ J/kg K}$, will be heated by trillions of degrees to accomplish this, and the experimental apparatus wouldn't survive. Any apparent mass increase would be very unimpressive, unless you count particle accelerators. $\endgroup$ Commented May 18 at 21:03
  • $\begingroup$ We might be within a decade of getting a gravitational verification of this, but inertial, maybe a lot longer wait. You should simply not even mention relativistic mass; you could have rephrased both your question, and your pedagogy, to avoid it. $\endgroup$ Commented May 18 at 21:35
  • $\begingroup$ Related: Has it been experimentally proven that energy causes gravity?, which addresses this question for gravitational mass. $\endgroup$ Commented May 18 at 22:19
  • $\begingroup$ The increase in mass with increasing energy in the rest frame of a system has nothing to do with the old concept of relativistic mass. $\endgroup$ Commented May 19 at 7:30

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With macroscopic objects, the tests you can do will probably be mostly ones involving astronomical observation, since I don't think it's easy to manipulate moving kilogram-mass objects in the lab and observe their behavior with high precision while changing their temperature by big amounts.

For astronomical tests, the answer is going to depend on whether or not you assume that the equivalence principle holds. The following an example of an astronomical test where we assume the equivalence principle does not hold, so that thermal energy has gravitational mass but lacks inertia.

The early universe was dominated by radiation, and that radiation was almost exactly black-body radiation, so it was almost completely thermalized. In a theory where thermalizing energy eliminates its inertia, this would presumably mean that the radiation in the early universe would have been almost entirely without inertia. That would be a radical change to the standard cosmological models. I'm not aware of any test theory such as PPN that could describe this type of behavior, but it seems clear that the the effect on cosmology would be profound. But in fact, cosmology is a high-precision science these days, so such a radical change doesn't seem like it could possibly be reconciled with observation. General relativity isn't compatible with such a picture, but in a rough Newtonian analogy, if you throw a rock upward but the rock has almost no inertia, then it should decelerate much more rapidly than expected. Therefore in a theory where thermal energy is without inertia, we would expect that the early, radiation-dominated universe would have had an anomalous and extremely rapid deceleration of its expansion, but then as radiation ceased to be dominant, the deceleration would have trended toward GR's predictions.

Now here's a test of the hypothesis that the equivalence principle holds but thermal energy isn't equivalent to mass. When two neutron stars inspiral and collide, they're going to convert a huge amount of nonthermal energy into thermal energy -- enough to be comparable to the system's total mass. We observe the gravitational wave chirps from this type of collision, and they're in excellent agreement with GR. Now one of the reasons that gravitational waves are hard to produce is that they're quadrupole radiation. The reason you can't get gravitational monopole or dipole radiation is that energy-momentum is conserved in GR. But if thermalizing energy eliminates its gravitational and inertial mass, then energy-momentum is not conserved. That implies that we should be able to have gravitational monopole and dipole radiation. Those types of radiation would be emitted at rates many orders of magnitude higher than quadrupole radiation, and so should be easy to detect, but that's not what we see.

If we proposed that thermal energy wasn't equivalent to mass, we would also need to spell out how this would play out for a lot of the normal assumptions of physics. Normally we assume that there is Lorentz invariance, and because of that it makes sense to talk about Lorentz vectors, and then it makes sense to say that the energy-momentum vector is conserved, and that this conservation holds in all frames. That whole scenario falls apart if thermalizing energy eliminates its inertia, so I think such a test theory would have to violate local Lorentz invariance. For example, if an object moving at nonrelativistic speeds can change its mass, but you want its momentum mv to stay constant, then its velocity has to change; but in that case an observer in an inertial frame initially co-moving with the object would say that the object had randomly chosen to accelerate in some direction. Since experimental tests of Lorentz invariance have been done to incredibly high precision, it seems like it would be unlikely that anyone could come up with a currently viable test theory that would allow even a tiny inequivalence between the inertia of thermal and non-thermal energy. (I guess you might be able to weasel out of that by saying that previous tests of Lorentz invariance haven't probed thermal effects or something.)

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  • $\begingroup$ Re "...if you throw a rock upward but the rock has almost no inertia, then it should decelerate much more rapidly than expected": Decelerate when? Compared to what other circumstance? Can you provide an example, incl. more details of exactly how rock throwing business is? $\endgroup$ Commented May 20 at 20:42
  • $\begingroup$ @PeterMortensen: That paragraph is describing a scenario in which the equivalence principle fails. So for example, say that the equivalence principle fails, and the ratio of inertial mass to gravitational mass differs by 10% between quartz and granite. Then you might measure that the deceleration of the quartz was 9 m/s2 while that of the granite was 10 m/s2. $\endgroup$ Commented May 21 at 12:44
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Sometimes the energy of particle colliders like the LHC are described in terms of temperature. A collision energy of $13 \mathrm{\ TeV}$ is a temperature of about $150 \mathrm{\ PK}$. At those temperatures the invariant mass of the collision products can be substantially larger than the invariant masses of the original particles. So that could be considered such confirmation.

Although this may seem a bit weird, it does make at least some sense. When you smash two objects together in an inelastic collision, they do get hotter. So, one way of making a high temperature is to have an energetic inelastic collision.

Note that my above description used only the invariant mass, not relativistic mass which I recommend avoiding.

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If thermal/kinetic energy did not contribute to mass, the Cosmic Microwave Background and the proton mass would be different from what is observed.

The macroscopic objects in the current universe with the largest thermal contribution (up to $10\,\%)$ to their mass may be proto-neutron stars, but observations of supernova neutron star formation cannot yet prove this thermal contribution to proto-neutron star masses.

Cosmic Microwave Background

Just to add a bit on @Fictional-Name's nice answer, thermal radiation dominated the mass of the universe up until about 50,000 years after the Big Bang, so if heat didn't have mass, the evolution of the early universe would be very different. In particular, the observed Cosmic Microwave Background power spectrum would disagree with modern cosmological models. For example, my understanding is that if the mass of the early universe was less than expected because thermal energy did not contribute, the early universe would have expanded faster and the position of the first peak in the CMB power spectrum would shift to a lower multipole moment (smaller $\ell)$.

Proton Mass

The ability of lattice QCD to correctly calculate the experimentally measured proton mass is strong evidence that (non-thermal) kinetic energy of the proton's constituents contributes to inertial mass. Since thermal energy is just the kinetic energy due to the random motion of a large number of particles, if kinetic energy contributes to mass, then thermal energy must also contribute.

Neutron Star Formation

Neutron-star-forming supernova initially produce a proto-neutron star with temperatures $10^{11}-10^{12}\,\mathrm{K}$ $(\sim 10-100\,\mathrm{MeV})$, so since the neutron mass is about $1000\,\mathrm{MeV}$, up to $10\%$ of such a proton-neutron star's mass is due to thermal energy. This contributes to the upper mass limit on neutron stars and otherwise affects the neutron star formation process, but supernova neutron star formation models and observations are not yet precise enough to measure such a few percent effect.

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