# Rest Energy in special relativity

• I

"In special relativity, however, the energy of a body at rest is determined to be mc2. Thus, each body of rest mass m possesses mc2 of “rest energy,” which potentially is available for conversion to other forms of energy. The mass-energy relation, moreover, implies that, if energy is released from the body as a result of such a conversion, then the rest mass of the body will decrease."

The bit in bold is what I am having trouble understanding.

If we change the rest energy do we not change the rest mass? I am thinking of an electron. If that rest energy changes (converted to another form of energy) doesn't the rest mass change and therefore it is no longer an electron? (Which surely is not possible).

Could someone explain what it means by potentially available for conversion to other forms of energy? (Or do I just need to read a better source?!)

Thanks.

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Orodruin
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If we change the rest energy do we not change the rest mass? I am thinking of an electron. If that rest energy changes (converted to another form of energy) doesn't the rest mass change and therefore it is no longer an electron? (Which surely is not possible).
The electron would cease to exist. The type of conversion alluded to would be, for example, electron-positron annihilation in which the rest mass of the electron and positron is converted to energy carried by photons.

vanhees71
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Particles can decay into other particles with a lower total rest mass. The extra rest mass is converted into kinetic energy of the resultant particles and/or photons.

vanhees71
Ibix
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For example, a Uranium nucleus that undergoes fission splits into a bunch of neutrons and a few "daughter nuclides". If you trap them all and weigh them the total mass is slightly less than the Uranium nucleus you started with. The missing mass has been converted to energy (which can be used to generate electricity or flatten a city).

But there are cases where a nucleus just emits a photon. This is the result of the nucleons rearranging into a lower energy structure; it isn't changing into a different type of atom. Again, the emitted photon carried away energy and the re-structured nuclide will be slightly lighter than what you started with.

An electron has no internal structure. So, as Orodruin notes, the only way it can release energy is to be destroyed by an anti-electron.

vanhees71
Thanks.

So the rest energy of the elementary particles can not be converted without releasing all of the rest energy and being destroyed in the process.

Whereas the phrase they use is referring to the rest energy of a system made up of a number of elementary particles which are just being re-arranged, emitting energy in the process.

Ibix
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You may be over-stating it a bit there. Some elementary particles can decay into different elementary particles, not just into photons. For example, beta decay is a quark changing into a different kind of quark and an electron and an anti-neutrino. However, I'm not certain if there are any examples of this happening to an elementary particle in isolation.

There are plenty of people here who will be certain. Watch this space...

Orodruin
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However, I'm not certain if there are any examples of this happening to an elementary particle in isolation.
##\mu^- \to e^- + \nu_\mu + \bar\nu_e##

Edit: There are many other examples. Also, elementary particles can decay into states containing composite particles as well, such as hadronic decays of ##\tau##s.

Ibix
jtbell
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<<delete half-finished LaTeX>>

Oro took the symbols off of my fingers as I was beginning to type them!

Ibix
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##\mu^- \to e^- + \nu_\mu + \bar\nu_e##

Edit: There are many other examples. Also, elementary particles can decay into states containing composite particles as well, such as hadronic decays of ##\tau##s.
Thank you! (It's been a long time since I studied particle physics...)

So in the rest frame of the muon there is only rest energy, but the decay products have both mass and kinetic energy that must have come from the rest energy of the original particle.

Whereas the phrase they use is referring to the rest energy of a system made up of a number of elementary particles which are just being re-arranged, emitting energy in the process.
Yes, this is more or less the crux of the matter. Think systems.

Rest energy isn't really a "form" of energy so much as it is a "category" of energy, if that makes sense. It's the amount of energy a system has as measured in the system's rest frame (the frame in which the system has no momentum). All the kinetic and potential energies "inside" the system contribute to the system's rest energy, as do the rest energies of the constituent molecules and atoms. But we can "zoom in" and further categorize the rest energies of the molecules and atoms as the potential/kinetic/rest energies associated with their constituent subatomic particles. We can keep doing this all the way down to the elementary particles, whose rest energies are irreducible and arise from the Higgs mechanism.

So yes, when it comes to elementary particles like electrons, rest energy is an all-or-nothing kind of thing. But rest energy is a much broader concept that applies to all sorts of systems. In general, open systems gain and lose rest energy all the time without ceasing to exist.

Mass and rest energy are the same quantity/concept, just expressed in different units.

Ibix
So yes, when it comes to elementary particles like electrons, rest energy is an all-or-nothing kind of thing.
https://en.wikipedia.org/wiki/Positronium

And how about hydrogen, an electron and a proton?

Orodruin
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What about them? They are not elementary particles.

What was my point again? Let's see ... I was disagreeing with the claim that an electron can not transform partially, but it can transforms completely.

Electron-positron system can partially transform to photons, by becoming positronium.
Electron-positron system can completely transform to photons, by becoming ... photons.

A single electron can not partially transform to anything.
A single electron can not completely transform to anything.

vanhees71
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That's a strange language. What indeed can happen is that an electron and a positron form a bound state, analogous to a hydrogen atom consisting of a proton and an electron, by emitting a photon. The rest mass of the positronium is smaller than ##2m_e## by ##E_{\text{binding}}/c^2## according to Einstein's famous formula:
$$M_{\text{positronium}}=2 m_e + \frac{E_{\text{binding}}}{c^2}.$$
Note that ##E_{\text{binding}}<0##.

SiennaTheGr8
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