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In E=mc2, Why C?

  1. Jan 6, 2011 #1
    Basic Question:

    Of the infinite number of other values which could have been the multiplier in E=mc2, it surely cannot be a coincidence that the value of the speed of light squared was the number. So why c?

  2. jcsd
  3. Jan 6, 2011 #2


    Staff: Mentor

    Hi Mileman10, welcome to PF!

    What other combination of dimensionful universal constants could you use to get dimensions of L²/T²?
  4. Jan 6, 2011 #3
    Thanks for the reply, but to me, your answer just begs the question.

    And, (big disclosure here), I'm just an interested novice, trying to understand things conceptually, if that's possible.

    In my college years, I studied Greek at Oxford, and was privileged to play chess once with Paul Dirac. (He won.) He explained basic stellar evolution in an hour, and was so clear and simple even I could get it.

    That's what I was hoping for here. So, please, if possible, keep it basic.


  5. Jan 6, 2011 #4

    Doc Al

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    Staff: Mentor

    Why do you say that? I think it was a reasonable answer.

    Do you realize that the speed of light plays a special role in relativity? Assuming you do, is it so surprising that c appears in many of the equations of relativity, including this one?
  6. Jan 6, 2011 #5


    Staff: Mentor

    OK, let me try again.

    We are looking for a universal factor which converts a mass into an energy. So just by looking at the units we know that the factor that we are looking for must have units of L²/T². So we look at the universal constants (e.g. c, G, h, etc.) and see how we can compose a factor with the right units of L²/T². There are only a small number of such constants so we quickly find that the only way to do so is to use c².

    So the c² should be completely clear just from dimensional considerations. What is not obvious is why it is 1c² instead of 0.5c² or something else. But for some reason nobody ever asks about the subtle question and everyone focuses on the obvious question.
  7. Jan 6, 2011 #6
    I appreciate your explanations, but on a very basic level, c is just a very large number, and c squared even more so.

    Forgetting desired units for the moment, this large number is most often used in simple discussions of the power of atomic fusion to explain how a pea-sized amount of Uranium could level Hiroshima, for example. The potential energy in even tiny amounts of matter is enormous, by a factor of c2.

    Please explain, why it is that particular factor, and not a google, for example.

  8. Jan 6, 2011 #7


    Staff: Mentor

    No, c is not just a very large number. In fact, when doing relativity most of the time we use units where c=1. The actual numerical value of c is not very relevant and is completely arbitrary since we can set it to any number we wish simply by choosing our units appropriately.

    The important thing about c is not its numerical value, but its units, and the fact that it is frame invariant, finite, and isotropic.
    Last edited: Jan 6, 2011
  9. Jan 6, 2011 #8
    OK, I think I'm beginning to follow, a little.

    So is it still accurate to say that "Energy equals mass times the speed of light squared", or is this technically incorrect?
  10. Jan 6, 2011 #9


    Staff: Mentor

    Yes, that is accurate.
  11. Jan 6, 2011 #10
    Thank you, sir, for your patient help.

  12. Jan 7, 2011 #11


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    Again, one has to be very careful about rest mass vs inertial mass. E = mc² holds if m is the inertial mass, but the symbol m is rarely used for it in relativity. For object in motion, a better equation is E² = p²c² + (mc²)². Here, m is the rest mass of the object, and p is momentum of the object. Note that this equation gives correct energy even for massless particles, such as light.
  13. Feb 9, 2012 #12

    Piggybacking off Ted's question, why not just say that the equation is E=ML? [ Where L = kinetic energy in light MV^2 with no M]

    The equation E= MC^2 is solving for total energy in a quantity of matter and since light is weightless, Kinetic energy in light is simply its velocity, no? What is the difference? Are we really just parsing terms?

    The implication is that matter isn’t simply energy like Einstein said. Matter is a specific type of energy, its just light in another form. It is bound up and tied in loops or knots or whatever. I can see why Einstein might not want to make that claim. It sounds outrageous and maybe a little unscientific. Is that the reason we stick to C^2 or am I off base?

    How much energy is in one Kilogram of matter?
    E=MC^2 => E =C^2 => E= the amount of energy in light

    Summary: The total amount of energy in a particular bit of matter is just a question of how much light is wrapped up in the matter.. The answer is the quantity of matter multiplied by energy of light (C^2)

    Is there something I am missing here? If light were the basic building block of matter, it would come as no surprise that C^2 keeps popping up all over the place.
    Last edited: Feb 9, 2012
  14. Feb 9, 2012 #13


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    What's L in that equation of yours?

    You are getting a lot of the basics of physics wrong. Try reading some faqs or something. The equation E=mc^2 does not read like "energy is equal to matter times light squared". A statement such as that makes no sense.
  15. Feb 9, 2012 #14


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    Staff: Mentor

    We don't make a unit equal to C^2 because the value C is a constant and appears in more than one place. It makes it much easier to use it squared or whatnot instead of making a new term in a couple of equations.

    Also, E=MC^2 is not useful for light as it has no mass. E² = p²c² + (mc²)² is actually the full equation. We simplify it to E=MC^2 to convert the invariant mass of an object to energy. In the full equation you would still have to square whatever you set C^2 equal to, so we can just use C and square it and it will be easier.
  16. Feb 9, 2012 #15
    I think that Mileman10 comes up with a very good question that deserves a better answer. Perhaps someone has a link to the original paper where Einstein is supposed to have come up with this relation?

    Regarding the energy containment in a kilogram of matter it is also dependent on the gravitational field. A one kilogram steel ball at rest placed very close to the Schwarzschild radius of a black hole should contain a lot less energy than the same steel ball placed on earth right?
  17. Feb 9, 2012 #16
    Thanks for the response. I didn't say "energy is equal to matter times light squared". I said that it seems like E=MC^2 is similar to Energy = (the amount of matter) ( the amount of energy in light) and that the amount of energy in light is equal to the kinetic energy of light. I guess it would be fair to say that the kinetic energy of light is 0 since Ke = mv^2 would be a large number multiplied by 0.
  18. Feb 9, 2012 #17


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    c^2 is by no means some measure of "amount of energy in light". The energy of a single photon is given by E=hf where f is the frequency and h is the planck's constant. It is also E=pc. c^2 doesn't even have units of energy.

    All of light's energy is kinetic. It has no rest energy.
  19. Feb 9, 2012 #18
    Why does it use c? Is it because at that speed matter would have to be equivalent to energy in order to travel that speed?
  20. Feb 9, 2012 #19


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    Matter (with mass) can never move at c. The speed of light, c, is simply a given constant in the theory of relativity. We assume a "maximum speed of information transmission", and call that c. This constant pops up obviously because we have used it as our main axiom. What value you give c depends purely on the choice of units. We just have to posit that such a maximum exists.

    Why does such a maximum exist? Because the experiments show that it does. As far as special relativity is concerned, this is a postulate that requires experimental evidence, but is not proved by the theory itself. No theory can prove it's own axioms. That would be tautological.
  21. Feb 9, 2012 #20
    Understood.. E² = p²c² + (mc²)² adjusts for relative inertia of a mass. For the initial intent E=MC^2 was fine since the momentum of the matter in question wasn't significant.
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