# Why does energy have mass?

Hey,
first of all I apologize if this isn't the right thread because I was confused of where to put this.
Recently I've learned about Quantum Field Theory and the Higgs Field. I kind of understand why the Higgs Field gives particles mass. From what I have learned it's because the field is interacting with certain particles, like affecting the spin of Electrons, and therefore slowing it down so it can't travel at light speed, this gives the Electron it's mass.
I have also learned, that Protons and Neutrons on the other hand get most of their mass from energy and that the quarks mass only accounts for 1% of the Protons mass. What bothers me is that I don't understand why energy itself has mass. I know that it has something to do with General Relativity, but I just don't understand why energy would have mass.

I'm sorry if there are some mistakes in the question itself as I am no expert, but I'm looking forward for your answers. :)

Orodruin
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By definition, mass in SR is just rest energy (divided by c^2). If you go to a reference frame where the momentum of your system is zero, then the energy in that frame is the system’s mass by definition. That this in the classical limit is the inertia of the system follows from the mass-energy equivalence. To derive it you need to look at what happens to the system when you act on it with a force.

It is not accurate to say that energy has mass. Energy and mass are both properties, not objects or substances.

I know that it has something to do with General Relativity
No it doesn’t.

Hasan Delifer
By definition, mass in SR is just rest energy (divided by c^2). If you go to a reference frame where the momentum of your system is zero, then the energy in that frame is the system’s mass by definition. That this in the classical limit is the inertia of the system follows from the mass-energy equivalence. To derive it you need to look at what happens to the system when you act on it with a force.

It is not accurate to say that energy has mass. Energy and mass are both properties, not objects or substances.

No it doesn’t.

How can most of the mass in a proton be energy then? Why does the energy have or provide mass in this case?

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Ibix
In relativistic terms, it turns out that energy of an object squared minus its momentum squared is a constant. And if you take the non-relativistic limit you find that the constant is the square of its mass. That's all there is to it (well - it can be derived rigorously rather than just stated like that, and you need some factors of c to make the units work out).

My understanding of the standard model is fairly limited, and you would probably be better asking in the QM forum (although there are plenty of people who frequent both so you may get an answer here).

vanhees71 and Hasan Delifer
dRic2
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I'm pretty dumb when it comes to Relativity, but I read this classic:
http://www.feynmanlectures.caltech.edu/I_15.html#Ch15-S8 (chapter 15, paragraphs 8 nd 9)
and I found it very enlightening (of course, you say... it is Feynman's! ). Not sure if it will help you, but it was of great help to me to figure out something I've always had trouble understanding.

Hasan Delifer
vanhees71
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But here Feynman commits a serious sin, introducing the relativistic mass. I'm very surprised, he did this, because usually he provides the utmost clear view on the fundamental concepts. In this case for some reason he failed. Already Einstein discouraged the use of a "relativistic mass". Today, in current research we only use the concept of invariant mass in relativistic physics with the good reason that everything gets simpler with the use of covariant quantities, i.e., using Minkowski-space vectors and tensors (four-vectors and four-tensors). Accordingly mass is the invariant mass
$$p_{\mu} p^{\mu}=m^2 c^2,$$
where for a point particle of invariant mass ##m## you have
$$p^{\mu}=(E/c,\vec{p}),$$
and then by definition
$$E=c \sqrt{m^2 c^2+\vec{p}^2}.$$
Note that in this choice of the energy the rest energy ##E_0=m c^2## is always included, and this is just convenient, because then ##p^{\mu}## behaves as the components of a four-vector under Lorentz transformations, while ##m## is a scalar quantity which doesn't change.

The relation to the non-covariant velocity follows like this. This quantity is defined as
$$\vec{v}=\frac{\mathrm{d} \vec{x}}{\mathrm{d} t}.$$
It's clear that this is NOT the spatial part of a four vector. It transforms quite complicated under Lorentz transformations. Much better is to use proper time, which is a scalar measure of time for massive particles, it's defined as
$$\mathrm{d} \tau= \frac{1}{c} \sqrt{\mathrm{d} x^{\mu} \mathrm{d} x_{\mu}}.$$
Using the coordinate time ##t## as parameter of the worldline of the particle you get the relation
$$\frac{\mathrm{d} \tau}{\mathrm{d} t}=\sqrt{1-\vec{\beta}^2}, \quad \vec{\beta}=\frac{\vec{v}}{c}.$$
Now momentum is defined as a four-vector
$$p^{\mu}=m \frac{\mathrm{d} x^{\mu}}{\mathrm{d} \tau},$$
with ##m## being again the invariant mass.

It's easy to see that this definition leads by definition to the correct energy-momentum relation, i.e.,
$$p_{\mu} p^{\mu}=m^2 c^2.$$
It's also easy to get the momentum in terms of the non-covariant velocity ##\vec{v}##:
$$p^{\mu} = m \frac{\mathrm{d} t}{\mathrm{d} \tau} \frac{\mathrm{d} x^{\mu}}{\mathrm{d} t} = m \gamma \begin{pmatrix}c \\ \vec{v} \end{pmatrix}, \quad \gamma=\frac{1}{\sqrt{1-\vec{\beta}^2}}.$$
$$p^0=\frac{E}{c}=m c \gamma, \quad \vec{p}=m \vec{v} \gamma.$$

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dRic2, m4r35n357, Ibix and 2 others
timmdeeg
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How can most of the mass in a proton be energy then? Why does the energy have or provide mass in this case?
I'm not really sure, there might be an analogy to Einstein's light box. The energy of photons in the box increase it's weight and thus it's mass. Or, another example, the kinetic energy of molecules in a box and their intramolecular vibrations contribute to it's mass. In a sense the proton can be considered a box.

Hasan Delifer
vanhees71
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I'd rather say the photons increase the inertia of the box (compared to the box without the photons or better said the electromagnetic field in it).

"Weight" is a different word for gravitational interaction, and according to Einstein's General Theory of Relativity the energy-momentum tensor of matter and radiation are the sources of the gravitational field, not only the part of it due to rest mass. That's only true in the non-relativistic approximation, where Newton's theory of gravitational interactions becomes a good approximation.

A lot of confusion is avoided in relativity if you distinguish strictly between mass (being a scalar) and energy (being the time component of the energy-momentum four-vector, if you consider a closed system and total energy and momentum).

Hasan Delifer and PeroK
timmdeeg
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I'd rather say the photons increase the inertia of the box (compared to the box without the photons or better said the electromagnetic field in it).
My wording was influenced by the article: "The box would first be weighed exactly. Then, at a precise moment, the shutter would open, allowing a photon to escape. The box would then be re-weighed."
To avoid ambiguity it might be better to talk about invariant mass.

Hasan Delifer
vanhees71