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Comment by eloff | Hacker News Reader

Has anyone ever wondered why the C^2 in E=MC^2 I'm sure it wasn't chosen at random just because a large constant was needed, it must fall out of some other physics equations.

But still, what does that mean? What does the transmutation of energy into mass or visa versa have to do with the universal speed limit?

I am always struck by the inexplicable power of mathematics to describe the physical laws of the universe, in often simple equations like this. It doesn't have to be that way as far as we know.

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Mass is a state of energy. c^2 turns out to be the conversion factor when you're turning things from our units of mass to our units of energy.

From the cosmic perspective, 'mass' is energy in a state that, instead of traveling at the speed of light, wants to sit on one place (for reasons that were discovered later), and needs a push to get it moving. When you measure mass, you're really measuring how much you need to push a given amount of this inertially-fixed energy to impart a given velocity. Our ordinary units of energy, mass, and velocity came from observing these facts of physics before we figured out what mass and energy "are", and E=mc^2 ties the knot between the different units we came up with.

The really fascinating thing is that Einstein was able to propose E=mc^2 before any of that was known. It was just a funny term that showed up in the equations for relativistic energy and momentum, and somehow Einstein's intuition was that it might mean something more.

I read a lot about physics trying to understand this stuff on pop-science level at least, but your comment really gave me a new perspective about it. Thanks.

Can you recommend an idiot-friendly source to read about this mass-energy relationship in more detail?

A recent attempt by a physicist that makes science education videos to explain how mass exists (through different fields):

https://www.youtube.com/watch?v=Ztc6QPNUqls

https://terrytao.wordpress.com/2012/10/02/einsteins-derivati...

It is one of the assumptions of Einstein, namely that there is a fixed max velocity c. To get Newton's 1/2mv^2 in the Taylor expansion requires a factor c^2.

Something else would lead to measurable deviations in slowly moving objects.

E = mc^2 is the energy contained within a particle of mass m, IF it is at rest (has zero velocity with respect to your frame). If it is moving, it's energy is given by E = ymc^2, where y (gamma) is 1/sqrt(1 - v^2/c^2). This may sound like the particle has somehow "gained" or "lost" energy just by switching frames, but so has everything else. The differences in energy have been kept the same, which is all that really matters.

When deriving this formula for the energy it becomes clear that _any_ multiple of ymc^2 is conserved. We choose E = ymc^2 and not, say, E = 5ymc^2 because in the Newtonian limit (as speeds approach zero), the ymc^2 formula reduces to the well-known E = 1/2 mv^2, which is the kinetic energy of an object obtained from classical mechanics.

I'd really recommend reading a mathematical treatment of special relativity, because you'll be exactly clear on where this formula comes from.

I think the answer we want now is a non-mathematical one. It gets tiresome how most of the "answers" are just too afraid( or maybe naive?) to just say "we don't know" why the speed of light is the constant of proportionality in the mass-energy conversion equation.

You can answer the "how" very well which is what the mathematical derivation is, and which is what most people are answering in this thread but you see that none of those answers start with "Because..." but rather just keep on spitting out more math. And what we want is an explanation for why c^2 is there and what it means.

We don't know what it means, we haven't answered that question and we don't know how to answer it. It's similar to how pi keeps coming up in a lot of equations and we struggle to understand why.

> I think the answer we want now is a non-mathematical one.

I'm just an interested lay person, but my understanding is the past century of physics is a giant argument for trusting the math, and accepting it as explanation.

If someone asks "Why does xyz act this way?" then the best answer for it, once you get to non-intuitive quantum scales is: "because the math says so." Dirac's prediction of the existence of antimatter, the Higgs, and pretty much all quantum mechanical behavior are great examples of this.

Nima Arkani-Hamed, a kickass theoretical physicist, has a great lecture series on the philosophy of physics:

• http://www.cornell.edu/video/nima-arkani-hamed-philosophy-of...

In an aside, he mentions that he used to explain the Higgs field with analogies such like calling it a kind of "molasses" slowing down particles and giving them mass. He's stopped doing that, because the physical analogy is really misleading. The true reason the Higgs as we conceive of it exists is that it addresses mathematical issues in our treatment of bosons that have mass.

The math said there was an issue. 50 years ago the math said the Higgs fixed it. It took 50 years and construction of the LHC, but now we've verified the existence of the Higgs.

The c² is not central, you can work in a set of units such that the equation is E²=m²+p². If you decide to add a unit for mass, then you naturally need a conversion factor with units of speed.

This is the first part of the Explanation. It is indeed E²=m²+p², but the p is the _relativistic_ impulse. For practical proposes you usually want the classical impulse p_0 in this equitation. And is a way easier to see how c shows up in the conversion between these two.

It's the proportionality constant between rest mass and energy. Mass times speed squared has units of energy. Same thing for the pc term: momentum times speed is energy.

To explain how we figured out the constant is c is a bit more complicated. The wiki article Introduction to Special Relativity is a good way to start if you want to understand everything well.

Not exactly an answer to your question, but this will be useful for the interpretation: https://www.youtube.com/watch?v=Xo232kyTsO0

more completely, it's E^2 = (m c^2)^2 + (p c)^2 where m is the rest mass of the particle and p is the current momentum. It's a nice triangle relationship.

As for the interpretation thereof, you're going to need someone else.

No one has answered your question. There is no answer really to your question. At least, not to the level that you are asking. Why is c^2 in that equation? What does it mean? We don't know. It's a deeper level of understanding that we haven't achieved.

Another similar question would be why does pi come up in everything? There's a lot of physics where pi comes up and we don't know why.

The original question has perfectly satisfying explanations accessible to anyone who knows algebra and is willing to spend a few weeks studying special relativity. cynicalkane above provides a great intuitive summary.

Likewise, generally speaking, when pi comes up in physics equations it is a geometric factor, either having something to do with the bona fide geometry of the system (e.g. the capacitance of a sphere) or with normalizing some relevant probability distributions. Name one example where we struggle to understand where pi comes from in physics formula!

I strongly disagree. We don't know the answer to everything, but we have solved this question - in terms of other, more fundamental things that we haven't fully explained yet. It would be disingenuous to pretend that because we have solved the deeper layer, we can't give an answer at the shallower one.

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