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Today I was reading about mass-energy equivalence and their respective conservations laws. I have come up with two definitions about mass:

  1. Amount of matter an object consists of.
  2. A measure of an object's inertia.

I was always satisfied with the first definition. That is, mass is about how many particles an object have. To find the total mass just $$\sum_{i=1}^nm_i$$ where $m_i$ is the mass of the i-th particle.

But reading about mass-energy equivalence I found out that when a chemical reaction occurs and energy is lost then an equivalent of mass is also lost. But this mass can not be lost as an electron/proton/neutron. A particle doesn't annihilate from a molecule. So if this is not the case then the mass of a particle we actually measure it comes from the "energy" it contains? Is it like currency equivalence? I mean if I pay a bill in euros then I lose an equivalent amount of dollars and the person I pay he gains the same amount of euros (and equivalent amount of dollars). So they don't convert to each other they just go/leave in a system in an equivalent way.

Einstein relation is: $$E=mc^2$$ The total energy of an object is $$E_T=mc^2+KE=mc^2+(γ-1)mc^2=γmc^2$$ So both kinetic and rest energy have an equivalent amount of mass. What doesn't appear in the equation is the potential energy (we referred to energy of an object and not a system). How can we make the above relation for systems in order to contain potential energy? If potential energy (a form of energy) doesn't have an equivalent amount of mass then it doesn't make sense to say that mass and energy are equivalent. Also photons have energy but no mass. So we can't say that everything that has mass has also an equivalent amount of energy and vice versa. Why we say that mass and energy are equivalent then?

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What Einstein proved with his famous formula, is that your first definition is just wrong. Mass isn't a measure of the amount of matter an object consists of. Instead, it is a measure of how much energy that is contained in the object or system. I like your analogy with currencies. Just like you can convert from dollars to euros using some conversion factor, you can convert from mass to energy using a factor of c^2.

As for your question about potential energy: the formula you showed only gives you the amount of energy in one particle. If you instead want to calculate the energy of a whole collection of particles, then you need to include the potential energies too. You see, potential energy is a quantity associated to the whole system, not to individual particles. Thus, the mass of one particle won't depend on the potential energies in the system. Interestingly, though, the mass of the whole system is its total energy divided by c^2, and since you need to include the potential energies in the total energy, you might find that the mass of the system is less than the sum of the masses of all its particles (it's less because potential energy can often be negative). A hydrogen atom, for example, weighs less than the sum of the masses of the electron and the proton.

I really suggest watching the YouTube video on E=mc2 by PBS Space Time. It really helped me better understand this interesting formula.

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  • $\begingroup$ Thanks for the answer. I will check the video. But I don't understand how we can calculate the total mass of a system (e.g. 2 particle system). The total energy is: $$E=E_1 + E_2 + V \rightarrow E-V= γ_{1}m_{1}c^2 + γ_{2}m_{2}c^2$$ Why I must equate $E=Mc^2$? $\endgroup$ – Antonios Sarikas Jul 16 at 13:51
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    $\begingroup$ There you go, that's the right formula. The total energy of the system is the sums of the energies of the individual particles, plus the potential energies. Remember, the E in the E=mc2 means all the energies in the system, whether its kinetic or potential. And in the case of a system of two particles, these are the only energies in the system. $\endgroup$ – Felis Super Jul 16 at 13:55
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    $\begingroup$ If you want to know why the formula works, then you should search for a derivation on the internet. For example, it's easy to find a pdf of Einstein's original paper from 1905 where he discusses this for the first time (name of the paper: "Does the inertia of a body depend upon its energy content?"). The argument is really beautiful and simple. $\endgroup$ – Felis Super Jul 16 at 13:58
  • $\begingroup$ Do you know any good introductory book in special relativity? $\endgroup$ – Antonios Sarikas Jul 16 at 14:12
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Actually, in a thought experiment with a box with perfectly reflective interior walls, the box will have an observable additional mass if there is electromagnetic radiation inside the box.

Conceptually, you can understand this as the photons getting a redshift (from general relativity) when they travel upwards and a blueshift when they travel downwards. That causes the momentum transfer to the walls to be a bit less on the upper wall than on the lower wall, which causes a net force down.

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You said that if potential energy does not have an equivalent mass, then the entire relation can't be true. Actually, the $E = mc^2$ corresponds to an sort of 'internal energy', something which you may think of as needed to hold the entire object together. When reactions like nuclear fusion or fission occur, some of the internal energy is released. Similarly, while every object that has mass has this 'internal energy', it does not mean that every object having energy would have mass. Like, photons. Photons have energy but they don't have mass.

So what exactly is mass? Depends on who you are asking. Ask a rocket scientist, it is the Newtonian definition, 'the amount of matter'. Ask, a relativist, and his reply is a 'form of condensed energy'. Ask a quantum physicist, the reply you get is 'a property of fundamental particles provided by the Higgs field'. And this can happen. We can define something in multiple ways, depending on the subject of interest. Mass is just a property objects have, and that property can have various meanings when looked from various perspectives.

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