Understanding the Mass-Energy Equivalence Concept behind E=mc2

[Jarwulf]

For E=mc²

I'm having trouble understanding intuitively how every kilogram of m conveniently is associated with a neat c² joules since as far as I know neither kg or joules were formulated with c in mind. I've seen that the mathematical derivation works out but I can't quite put it together on a qualitative level.

 

[Matterwave]

If you're worried why the units "work out" mysteriously, it's just that the Joule is defined in terms of kilograms, meters, and seconds.

 

[atyy]

It isn't. Mass, momentum and energy are redefined so that the formula EE=ppcccc+mmcc is true. However, the redefinition is not cavalier. It approximates the Newtonian definitions for low velocities. It makes energy conservation true in special relativity. And the relativistic quantities correspond to quantities that can be measured.

It's not any more mysterious than the Lorentz transformations which mix space and time, again with "c" coming into make the units right.

 

[Matterwave]

I think ou have 2 too many c's by the p's and 2 too few c's by the m's. Should be EE=ppcc+mmcccc in your notation.

 

[atyy]

Ooops, yes.

 

[Jarwulf]

If you're worried why the units "work out" mysteriously, it's just that the Joule is defined in terms of kilograms, meters, and seconds.

I guess I'm missing something here but I find it odd that an arbitrary unit of mass has the energy to have 1 Newton of force applied to it over the distance of exactly a lightyear squared. Its the same way if you use grams or whatever. I don't know how to explain my bemusement properly.

It isn't. Mass, momentum and energy are redefined so that the formula EE=ppcccc+mmcc is true. However, the redefinition is not cavalier. It approximates the Newtonian definitions for low velocities. It makes energy conservation true in special relativity. And the relativistic quantities correspond to quantities that can be measured.

It's not any more mysterious than the Lorentz transformations which mix space and time, again with "c" coming into make the units right.

So the units are redefined? I usually see kg meters and seconds in the equation used although I heard any consistent set of units would work though.

 

[JesseM]

For E=mc²

I'm having trouble understanding intuitively how every kilogram of m conveniently is associated with a neat c² joules since as far as I know neither kg or joules were formulated with c in mind. I've seen that the mathematical derivation works out but I can't quite put it together on a qualitative level.

In units of kilograms and joules the equivalence isn't very "neat" at all, for example one kilogram of mass would have a rest energy of 8.9875517873681764 * 10^16 joules. A more neat system would be one where one unit of mass had a rest energy equal to one unit of energy, as in Planck units. The fact that the equivalence involves such arbitrary-looking numbers when expressed in kilograms and joules is a good sign that these units weren't designed by people who knew about the equivalence in advance!

I guess I'm missing something here but I find it odd that an arbitrary unit of mass has the energy to have 1 Newton of force applied to it over the distance of exactly a lightyear squared.

But that's not actually true, c^2 is the speed of light squared, which is totally different from "a light year squared" or any other arbitrary unit of distance squared. The equivalence says nothing about the energy needed to pushing a given amount of mass a certain distance (which would in any case depend on how quickly you wanted the mass to traverse the distance).