h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Ultimately the c^2 comes from the concept of four-dimensional space time (doesn't it?). In relativity, measurements of time, and measurements of space, can couple together and 'interact' with one another; but, like you can't add apples and oranges, you can't add metres (or intervals of length) to seconds (or intervals of time); to be able to connect the two you have to convert concepts of space and time into compatible units. The derivations of the Lorentz transformation equations (which describe time dilation / length contraction) do this by considering (at least indirectly) intervals of space, and---rather than intervals of time itself---the distance that a beam of light would have covered in that interval of time. This is a fair absolute method because, according to Einstein's first postulate, the speed of light is measured to be the same for all observers; so if two observers agree (or can measure) the same time, they'll always measure the same light-distance.

The E=mc^2 can be derived using four-vectors (four dimensional analogs to position, velocity, momentum, etc, which contain and extra 'time'-like component) in which the c^2 is necessarily included; for example, four-position (the four-dimensional space-time equivalent of position) is (ct, x, y, z), where t is the time, and (x, y, z) is 'conventional' position (three-position).

Something like that... Wikipedia has a fair amount on four vectors, and the derivation of E=mc^2 using that method.

Sorry, people have probably already said all of this.

Quickest (and most dodgy and probably incorrect) derivation of E=mc^2:
Imagine a box in space, one atom on one internal surface is in an excited state. It decays and emits a photon which travels to the other end of the box. The photon carries momentum, and conservation of momentum will require the box to move *back* in the opposite direction to the photon. When the photon hits the other end of the box, it'll be absorbed, and the box will come to rest after 'colliding' with the photon (and more conservation of momentum). At the end of the process, the centre of mass of the box has moved in space (because the box has moved). But Newton's laws of motion prevent the centre of mass of any system from moving ('absolutely') in space unless there's an external force acting (and unless it's already moving, obviously!). There *is* no external force acting, so you must consider that the photon has carried a unit of mass with it, in order to shift the centre of mass of the box-*system* towards one end, and thus allow the centre of mass of the system to remain fixed in space. ==> The photon carries mass.

The energy of the photon is E = pc, where p is it's momentum (which is a classical concept originating from Maxwell's laws of electromagnetism). Momentum is p = m v, where m is the mass and v is the velocity. For a photon, v = c, therefore E = (m c) c = mc^2.

Remember, for E = mc^2, m is the relativistic mass, not the rest mass (in general). If the object has no kinetic energy, then m becomes identical to the rest mass.....................

Either way is fine.

My method is to define the mass as a function of velocity, then the momentum, force, work and then finally kinetic energy. They all work, even the quick and dirty method you just mentioned.

Actually: the c^2 bit comes from making the units add up, but that still doesn't explain *why* mass and energy are equivalent. I think the right hand side of E=mc^2 (where m is relativistic mass) is really a modification and re-expression of concepts of momentum (most derivations of the equation I've seen come from the view point of momentum, most thought-experiment-derivations are considering details of momentum). There's always a connection between energy and momentum, because energy is an expression of the ability of an object to do *work* (which is a force acting through some interval of space), while momentum is the expressino of the ability of an object to impart an *impulse* (which is a force acting through some interval of time). Perhaps energy = mass (dropping the units) is an expression of the space-time link between these two concepts?

""My method is to define the mass as a function of velocity, then the momentum, force, work and then finally kinetic energy. They all work, even the quick and dirty method you just mentioned.""

Well my quick method is completely useless because it formally only applied to photons, probably, and doesn't even begin to explain what relativistic mass actually *means*, anyway...

I think you put your finger on it in that last sentence.

energy = mass -> Mass is really just one form of energy. E=mc^2 merely gives the coefficient of the conversion.

>> There's always a connection between energy and momentum <<

E^2 - (p.c)^2 = (m.c^2)^2

We used to call this the law of 'Conservation of Momenergy' as a SR version of the conservation of matter/conservation of energy. The two terms on the left vary with reference frame but the difference is invariant. (Obviously this reduces to E=m.c^2 for v=0).

Following on from your lines of thought, could the 'reason' for the mass energy equivalence lie in understanding why inertial mass and gravitational mass are the same? (In which case the answer lies outside the framework of special relativity?)