# Meaning of E=mc^2

I recently came across an article from Stanford Encyclopedia of Philosophy titling "The Equivalence of Mass and Energy"
(link: http://plato.stanford.edu/entries/equivME/#2.2 )
which has really confused me about the nature of mass an energy.

The article contains many interpretations of E=mc$^{2}$
And one of them titled Same-property interpretations of E = mc$^{2}$ says something like this:
"Mass and Energy are different measurements of the same thing. By choosing units properly we can make c=1 and thus there is actually no difference between them. Mass and energy are the same property of physical systems. Consequently, there is no sense in which one of the properties is ever physically converted into the other."
(The article also gives book References:
1. Torretti, R. (1996), Relativity and Geometry, New York: Dover.
2. Eddington, A. (1929), Space, Time, and Gravitation, London: Cambridge University Press)

Now, does E = mc$^{2}$ actually means that mass and energy are same?

Also I find many other interpretations and many book references supporting them like:
Bondi and Spurgin argued that Einstein's equation does not entail that mass and energy are the same property. But they can not be converted into each other.
(Ref: Bondi, H. and Spurgin, C.B. (1987), “Energy has mass,” Phys. Bull., 38: 62–63)

Another one:
Mass and Energy are different properties but they can be converted. (Ref: Rindler, W. (1977), Essential Relativity, New York, N.Y.: Springer-Verlag.)

As a student of introductory level, I am absolutely confused about the nature of mass and energy (are they same or different?) and what E=mc$^{2}$ actually suggests?

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## Answers and Replies

Nugatory
Mentor
Much of the confusion here stems from the way that in natural language, "the same thing" is somewhat ambiguous. Is ice the same thing as water? Not if you want to swim in it, or stand on it, or drink it, or throw it... But a chemist will tell you that they're both the exact same thing, namely good old H2O.

So the descriptions above are all about equally right, and indeed we sometimes switch from one to the other according to the problem we're working on.

If I tell you that in a nuclear explosion 100 kg of uranium turns into 99.998 kilograms of fission products when the reaction is done, you can use ##E=mc^2## to calculate that .002 kg of mass turned into 5x1014 joules of explosive energy - and you are probably thinking of mass and energy as different things that can be turned into one another, like ice and water.

However, if you're working with highly energetic particles in a collider, it's generally more convenient to think like the chemist above. He sees a mixture of ice, water, and steam, and says "Well, that's just some number of water molecules doing their thing" and he knows that no matter how the mixture changes, there's always the exact same amount of H2O involved. If you choose to think of energy and mass this way, then they're just different ways of measuring the same thing, and the ##c^2## in ##E=mc^2## has about the same significance as the ##1000## in the equation ##milliseconds = 1000\times{seconds}##.

[Edit: No one caught the typo above? .002 kg is 2x1014 joules, not 5x1014]

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1 person
Is ice the same thing as water? Not if you want to swim in it

Really great example. Thanks!

Dale
Mentor
2020 Award
So, in the previous thread I spoke about relativistic mass, and I wish that I had not done so. Relativistic mass is related to energy by ##m=E/c^2##, but when physicists use the unqualified term "mass" they are usually refering to the quantity ##m^2 c^2 = E^2/c^2 - p^2## which is often called "invariant mass".

This definition of mass is, in fact, distinct from energy and therefore a valuable independent concept. This is also the meaning of mass that is intended when photons are described as massless. In this concept, all reference frames agree on the amount of invariant mass that an object has, although they will generally disagree on the amount of energy and momentum. Furthermore, an object with positive mass has non-zero energy in all reference frames, while an object with 0 mass may still have non-zero energy. So using invariant mass you can say that all mass has energy, but not all energy has mass.

Also, note that the invariant mass of a system (in flat spacetime) is both frame invariant and conserved. The energy and momentum are also conserved, but not frame invariant.

1 person
HallsofIvy
Science Advisor
Homework Helper
Is ice the same thing as water? Not if you want to swim in it, or stand on it, or drink it, or throw it...
You just have to wait a while!

I think I got that "Once and for all" reply I was looking for.

Thanks DaleSpam and Nugatory!