# Does $E$ really equal $mc^2$?

I'm currently in a debate with a co-worker.

If mass is sped up to the speed of light, does the mass become energy?

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If you're having an argument about the FTL mechanism in Star Trek, that has been covered already in this other question. In short, it is an Alcubierre drive, which does not actually involve moving faster than light in the local space of the warp bubble. The expansion and contraction of space around the bubble moves it at faster than light speeds. – gnovice Oct 26 '11 at 17:58
Yeah, sorry. I had to fiddle with in on my end to show up as 'migrated' and not 'closed' and I guess because the name was changed on this end it reposted it. – DampeS8N Oct 26 '11 at 19:24
This should be closed. It's a double-post, and not a particularly high-quality question anyways. – Jonathan Gleason Oct 26 '11 at 20:24

This is too naive, as far as physics goes, question and the title asks a different question than the body. The short answer is yes for "Does $E$ really equal $mc^2$", and no for "If mass is sped up to the speed of light, does the mass become energy". This is due to the fact that there are two uses of the word "mass" in physics, depending on context.

Rest mass is a characteristic measure of matter. When we talk of elementary particles mass is the rest mass of each, $m_0$ which does not change no matter how fast it is going.

$E=mc^2$ either means $E=m_0 c^2$ for an object at rest, or $E=m_{rel} c^2$ when the object is moving.

The equation $E = m_{rel} c^2$ holds for moving objects. When the velocity is small, the relativistic mass and the rest mass are almost exactly the same.

It is the relativistic mass which grows with the velocity,

$$m_{rel}=\frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}$$

and is due to the kinetic energy given to the particle, the rest mass does not change. From the formula it is evident that a massive particle can only incrementally approach the velocity of light, never equal it; even approaching it, enormous amounts of kinetic energy would be needed. Further reading in the wiki article.

Studies in nuclear and particle physics as well as in astrophysics have established that even though our lives are lived in non relativistic environments, relativity rules from the microcosm up. This view becomes organized by the use of four-vectors, which distinguishes two masses:

a) into an invariant part,$m_{0}$ that characterizes a body, in the same way that a length characterizes a ruler in three dimensions,

and b) a relativistic mass $m_{rel}$ that includes the energy that a body has due to motion.

At rest the two masses are equal, and in our everyday world where velocities are much smaller than $c$ indistinguishable experimentally.

In this language all mass is energy, from $E = m_{rel} c^2$,

one sees that $E/c^2 = m_{rel}$

for all velocities, since $c$ is constant. So the answer is that a material body is energy from 0 speed on, and this does not change for very high speeds except as increase in energy. Only the kinetic part of this energy is available in the non relativistic everyday world.

Thus the real physics answer is neither yes nor no.

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@Colin Thanks. I am full of thumbs with formats. – anna v Oct 26 '11 at 19:08
Hm, I always wondered if the mass equivalence principle means the gravitational mass is equivalent to the rest mass or to the relativistic mass, i.e. would a pulk of fast electrons "weigh" more? – Tobias Kienzler Oct 27 '11 at 9:12
@Tobias the mass equivalence applies to the mass characterizing the four vector of the system under consideration, m=sqrt(E^2-p2), the invariant mass of the system. Yes, a bunch of electrons would have as a limit the sum of their masses, but collectively would be heavier according to the formula. – anna v Oct 27 '11 at 9:54
should add that the p in p^2 is the vector sum of the electron momenta, and the square is the dot product of the sum with itself. – anna v Oct 27 '11 at 11:24
so in theory if you accelerate enough mass you have a gravity generator? I'm somehow doubting that since it hasn't happened yet, but maybe it'd just require too much energy... I should post a separate question tomorrow – Tobias Kienzler Oct 27 '11 at 16:43

No it doesn't. $E = m c^2$ holds only for massive objects at rest. When they move this relation no longer holds. The more generic energy-mass-momentum relationship does hold in these situations:

$$E^2 = m^2 c^4 + p^2 c^2$$

In this equation, $m$ is the invariant mass, $E$ the total energy, and $p$ the momentum. When the object with invariant mass $m$ speeds up, $E$ and $p$ both increase, while $m$ stays the same as what it is at rest. Hence its name 'rest mass'.

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Doesn't $E = m c^2$ still apply if $m$ is taken to be the relativistic mass (which increases with increasing speed)? – Keith Thompson Oct 28 '11 at 23:11
And what is the physical meaning of relativistic mass (other than energy scaled by the speed of light)? Lev Okun refers to relativistic mass as a pedagogical virus. He is absolutely right. See: arxiv.org/abs/hep-ph/0602037 – Johannes Oct 28 '11 at 23:40
Yeah, relativistic mass is an outdated concept. These days we just call it "energy" (and work in natural units if people start complaining about missing factors of $c$ :-P). (Excellent answer by the way) – David Z Oct 29 '11 at 4:57

No. Mass can be converted into energy(it happens in e.g. the sun and in a nuclear power plant), and energy into mass(e.g. the big bang). An other form of energy is motion energy(called kinetic energy) it increases as the object is accelerated toward the speed of light(the maximum speed).

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+1 for mentioning the mass/energy conversion. – eidylon Oct 26 '11 at 19:38
This "motion energy" registers on a scale, and gravitates objects towards it--- a box of photons weighs more, and an electron going fast in a circle in a magnetic field weighs more. – Ron Maimon Oct 27 '11 at 6:04
Pushing forward the big bang as an example of energy-mass conversion is highly dependent on interpretation (to say the least), and from didactical perspective will probably create more questions than it answers. – Johannes Oct 28 '11 at 22:42
The whole "conversion" idea is idiotic. -1 There is no conversion of mass to energy, the thing we call mass colloquially is the energy. The new quantity of rest mass in relativity is not the right idea of mass. – Ron Maimon Oct 29 '11 at 5:34
I do not agree the conversion idea is not idiotic. The fermions( half-integer spin particles) that obey the Fermi–Dirac statistics(and thereby seen as matter) is indeed converted into energy in an annihilation(the word itself also suggest the disappearance of the matter). – Hans-Peter E. Kristiansen Nov 24 '11 at 21:07

Yes, $E=mc^2$ always, for everything, because what we call "mass" colloquially is just the energy contained in something. The quantity physicists call "mass" nowadays is the mass as measured when travelling along with something, and this is the rest-mass. But the rest-mass is not what weighs on a scale, nor is it what you feel as resistance to pushing, nor is it additive to give the mass of a composite system, nor is it the source of gravity. The energy (divided by $c^2$) is all these things. When the energy is converted in units using c to become a mass, it is called the "relativistic mass", which, when you choose units so that c=1, is just a synonym for energy.

Lev Okun has argued that the concept of "relativistic mass" is outdated, and pedagogically useless. He is completely wrong on this. Once a student understands that what you call "mass" day-to-day is really the energy (divided by c^2), that frees up the useful short word "mass" to mean something else, namely "rest mass", but not until you understand the equivalence of energy and colloquial mass.

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the question is "If mass is sped up", and "does the mass become energy", so a yes is not an appropriate answer to a naive and confused question. The "mass" in the question is the rest mass, as we know it in everyday life, where relativistic effects are infinitesimal. The energy in m0 is inaccessible to everyday users, only becomes evident in nuclear physics and higher energies. Of course I agree with you that m is what enters all the physics studies where relativity effects are important and it is important for physics students to learn this. I do not think the questioner is a physicist. – anna v Oct 29 '11 at 18:51
continued: on the other hand the rest mass of a composite system is the invariant mass of the added four vectors of the system. Learning to work with four vectors is more to the point than using convoluted formulae. – anna v Oct 29 '11 at 19:14
@anna v: If you speed something up, it weighs more. A box of hot gas (molecules sped up) weighs more than a box of cold gas. If you have an electron going around in circles, it weighs more than an electron at rest. The physics of "mass" is the physics of energy, and the "rest mass" is the physically less intuitive concept. – Ron Maimon Oct 29 '11 at 21:14
Ron -- I am sure we agree on all the physics, but I strongly disagree with you on your didactical views. The way you answer this question means that you are selling nothing more than an empty tautology. Let's cut the crap and explain to lay people the way it is: energy rules supreme. Energy is the source of gravity, and energy is inertia. Mass is the invariant associated with energy. – Johannes Oct 29 '11 at 22:43
@Johannes: except this isn't true--- the thing we call "mass" is just the "energy divided by c^2", and I am sick of the confusions people have about mass-energy conversion (there was another question about whether fire converts mass to energy!). In order to cure this confusion, one must explain that the quantity we call "mass" is really "energy", and physicists hijacked the term "mass" to mean something else, namely rest-mass. The notion that rest-mass corresponds to the intuitive notion of mass is absurd. – Ron Maimon Oct 30 '11 at 1:27

Logically, the answer is "Yes", but only because of the way logical implication is defined; see here.

If mass is sped up to the speed of light, then sure, mass does become energy (or a bowl of petunias, or the square root of existentialism).

The point is that the premise is not possible; speeding mass up to the speed of light would require infinite energy input. Not just energy equivalent to the mass, via E = m c^2, but quite literally infinite.

The statement "If I'm 20 feet tall, then my name is Fred" is true, because I'm not 20 feet tall (the fact that my name isn't Fred is irrelevant).

But the answer to a better-phrased version of your question would probably be "No". No matter how fast you accelerate a mass, the original rest mass is still there.

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As much as I want to upvote for the mental image of a bowl of petunias busting out of the LHC tunnel, I think it's pretty misleading to start your answer off by saying "yes." Some fraction of the audience of this question may very well take that the wrong way. Given that the premise is impossible, all you can really say is that the answer is undefined. – David Z Oct 27 '11 at 1:06
I was referring to the meaning of if-then in logic. See here. – Keith Thompson Oct 27 '11 at 2:58
Yeah, I know about Boolean logic rules, but (most) people don't naturally think that way. – David Z Oct 27 '11 at 3:27
@DavidZaslavsky: Then perhaps this is a good chance for them to learn. (I've updated the answer for, I hope, better clarity.) – Keith Thompson Oct 27 '11 at 4:02
The only error in Keith's answer lies in the fact that he forgot that petunias are always accompanied by whales. – Georg Oct 27 '11 at 10:04