Has $E=mc^2$ been experimentally verified for macroscopic objects with potential energy?

Question

In relation to this question: What is potential energy truly?, I'm wondering if $E=mc^2$ has been experimentally verified to hold true for macroscopic objects with increased potential energy? I'm particularly interested in whether the following examples have been tested:

• Does a macroscopic object at a higher position in a gravitational field have more mass due to the gravitational potential energy?
• Does a spring weigh more when it is compressed compared with uncompressed?
• Does a charged object weigh more when it is in an electric field?

If anyone could post links that provide more details on actual experiments that have shown these, that would be great.

EDIT: I've edited the question to try to be more specific about what I am asking. Apologies to those who have posted answers already if it makes your answer seem less relevant.

Answer 1

The fact that you can't compute the weight of an isotope by adding up the mass of the neutrons, protons, and electrons is because the kinetic and potential energy of the interactions affects the total energy, and hence mass. But you object because that include nuclear interactions.

OK. Look at molecules. The reason a molecule's weight isn't the sum of the weights of the isotopes of the atoms involved is because the the kinetic and potential energy of the interactions affects the total energy, and hence mass. So that counts.

Has this been measured? Yes. Firstly, we can measure the weight of molecules, and we can measure the weight of a macroscopic amount of molecules. Secondly, the energy difference is available as produced heat or required heat. When we measure the heat released or absorbed, we are measuring the energy associated with the change in mass. Just like in a nuclear reactor, except smaller amounts of energy per gram are needed/released since electromagnetic potential energies and kinetic energies are so much smaller.

Answer 2

For the Spring

I know the answer for a compressed spring. The compressed spring has higher mass than the uncompressed spring. The stored energy in a compressed spring shows up as somewhat higher electric fields, overall, between the atoms of the spring.

To minimize confusion, consider a small bar of metal made of a single crystal of aluminum atoms. In the crystal, the nuclei and electron clouds of the atoms configure themselves to a lowest energy configuration. The details of this are complex, but for this example we just need to accept that the atoms wind up at a spacing from each other where if they were closer together they would be repelling each other because (primarily) of the positive charges on the nuclei pushing against each other, and if they were further apart they would be pulling towards each other, primarily because the electron cloud of each atom is distorted to attract the positive nucleus of the neighboring atom.

SO now consider compression. We push on the ends of the bar, forcing the atoms of metal to be somewhat closer to each other then in their relaxed state. The atoms push back, and they do it by the extra electrostatic repulsion of the positively charged nuclei. But when you push those nuclei closer to each other you are increasing the net electric field between them.

Electric fields store energy. The energy density of an electric field is proportional to field-strength squared. So the compressed metal, with somewhat higher electric fields because its positively charged ions are closer together than in relaxed state, has its compression energy stored in the higher electric fields!

BUt of course by relativity, that energy density of an electric field means the electric field has mass density $m=E/c^2$.

So the answer for the compressed spring is that in compression, interatomic electric fields are slightly higher than when the spring is relaxed, and the energy of that slightly higher field is the energy that can be recovered by decompressing that spring, and that higher electric field has a mass from its energy density due to relativity. So a compressed spring weighs more than an uncompressed spring because of the mass of its slightly higher interatomic electric fields.

Unfortunately, I can find no reference to someone measuring the mass difference of a compressed spring and an uncompressed spring. The mass of electric and magnetic fields is indirectly verified by the gravitational red-shift or blue-shift of radio waves. This is verified, oscillators on satellites are measured at higher frequency on the earth's surface, and the reason is that as the radio energy goes down to earth it is increased in energy by the earth's gravitational pull on the mass of its electric and magnetic fields. The energy of a quantum of radiation, a photon, is $hf$ where $h$ is Planck's constant and $f$ is the frequency of the radio. The increase in the frequency of the radio wave as it gets to the earth's surface is exactly consistent with the $mgh$ energy of a "falling" photon from satellite height $h$, photon mass of $hf/c^2$, in gravitational pull $g$.

For the Gravitational Potential

Just as the compressed spring has higher electrostatic field in compression than relaxed, moving a small mass away from the earth increases the net gravitational field found by summing the gravitational field of the earth with the gravitational field of the object being raised. Fields add linearly: linear superposition is the solution of the net gravitational field of two objects. But the energy density in the field proportional to field strength squared. So the field configuration with the objects slightly further apart has more energy in it, found by integrating the energy density over space.

Without actually proving it or having been told it authoritatively by someone who did the math, I will assert by analogy with the spring case that this gravitational field energy density must balance, must account for, the higher potential energy of the mass lifted higher above the earth. This would then suggest that the weight of the object when it is at higher height is the same as when it is on the ground, because the extra energy, the potential energy, is not physically located in the object lifted, but in the gravitational field around and in between the object and the earth.

As far as I know, the energy density of a gravitational field is a lot less obvious than the energy density of a electric and magnetic fields. Gravitational radiation is much less a part of our daily lives than the nearly omnipresent electromagnetic radiation, and so experimental evidence that gravitational field has mass when it is in gravitational radiation may not be available.

Conclusion

The energy stored in a spring is associated with higher electric fields inside the lattice of the spring. Those electric fields have a known energy density which by $E=mc^2$ has a known mass density. The mass of electric and magnetic fields is known to be consistent with gravitational "acceleration" of radio waves known as blue shift.

The energy stored in height above a planetary mass it would seem should, by analogy, be stored in the gravitational field. But since Gravity is weak compared to electromagnetic forces, and gravitational radiation is not something we can do much with experimentally, the actual experimental evidence that gravitational potential energy is stored in the gravitational field between and around the objects is harder to come by.

Answer 3

The special theory of relativity has been validated innumerable times in particle physics experiments and nuclear physics experiments. Macroscopic verification of the special theory of relativity also exist.

I will quote from this article, "Einstein's relativity and everyday life"

But what about Einstein's theories of special and general relativity? One could hardly imagine a branch of fundamental physics less likely to have practical consequences. But strangely enough, relativity plays a key role in a multi-billion dollar growth industry centered around the Global Positioning System (GPS).

The system is based on an array of 24 satellites orbiting the earth, each carrying a precise atomic clock

....

The satellite clocks are moving at 14,000 km/hr in orbits that circle the Earth twice per day, much faster than clocks on the surface of the Earth, and Einstein's theory of special relativity says that rapidly moving clocks tick more slowly, by about seven microseconds (millionths of a second) per day.

The Lorenz transformations are what defines special relativity, and have to be used.

Also, the orbiting clocks are 20,000 km above the Earth, and experience gravity that is four times weaker than that on the ground. Einstein's general relativity theory says that gravity curves space and time, resulting in a tendency for the orbiting clocks to tick slightly faster, by about 45 microseconds per day. The net result is that time on a GPS satellite clock advances faster than a clock on the ground by about 38 microseconds per day.

But at 38 microseconds per day, the relativistic offset in the rates of the satellite clocks is so large that, if left uncompensated, it would cause navigational errors that accumulate faster than 10 km per day! GPS accounts for relativity by electronically adjusting the rates of the satellite clocks, and by building mathematical corrections into the computer chips which solve for the user's location. Without the proper application of relativity, GPS would fail in its navigational functions within about 2 minutes.

The fact that the theory works to this accuracy validates it, and this includes the E=mc^2 which is part of special and general relativity.

Answer 4

One can offer a simple logic to clarify the relation between rest mass, relativistc mass, energy and momentum and connecton to E=m c^2. The essence of it has been told by many, but not all in one place unfortuanely.see http://geocalc.clas.asu.edu/pdf/ZBW_I_QM.pdf. https://arxiv.org/pdf/1602.07534.pdf and I have many others.

The origin of everything is the EM field. This field has mechanical attributes, in the form of momentum, as you can rotate a small turbine placed in an evacuated jar using radiation. And it has energy, as it can go on and evaporate the blades by cranking up the intensity. In any case, energy is derived from momentum. Radiation has an electric attributes as we know, in the form of an electric field, and a magnetic field that is a direct consequence of it.

An energy beam does not have a rest mass, but a trapped beam between two mirrors- as in lasers, has a rest-mass. Hence one can conclude that rest mass is nothing more than a trapped EM field. The trapped momentum part gives rise to mass- as momentum is conserved, leading to gravitation and the inverse square forces (as shown in Bertrand theorem). This is also supported by pair creation and annihilation events.

The trapping of a field can be between two mirrors, but can also be in going round infinitely in circles- as in an electron. It is self trapping in this case. The resulting circulating field produces spin, magnetic dipole moment, and it produces no hard core, again as in the case for an electron. But you can have double trapping too- as in the case of the proton, wherein masses(trapped energy) have a very large kinetic energy, and the lot are further trapped in one structure forming the proton or the neutron. All composite particles follow the same logic.

The momentum for radiation is E=pc, and for a particle E=m ingeral(v.dv)=.5 m v^2. Put v=c, and get E=m c^2, with the factor 2 disappearing, as we need two opposite radiation beams to produce a trapped radiation, and a resulting rest, orzero momentum of the product. The total energy of a composite/sum particle would be that of radiation and mass giving by the more general Einstein formula; E^2=(pc)^2 + (mc^2)^2. The momentum direction inside a circulating wave is tangential and thus normal to the direction of motion of the particle. This is the reason for adding the energy vectorially and the square on the terms.. because the momenta from which the energies are obtained is orhogonal too.

Therefore, we can say that all measurement of pair production and annihilation processes and all those for particle creation and disintigration, are proofs that the formula E=m c^2 is exact and accurate.