What is the essence of light that allows it to be a conversion factor between energy and mass? I’m not talking about dimensional analysis. I’m asking a deeper question about the very nature of light. It has to bee more than the numbers work out. Why do the numbers work out.
I hope this is more clear now. If not please help me. Thanks so much.
Why did Einstein use light as a conversion factor between energy and mass? I’m not talking about the math working out, or why it needs to be c^2. But rather, what is the nature/essence of light that allows it to be a conversion factor? A very basic question, but I’ve never seen a good answer.
The nature of light that interested Einstein was the fact that it always travels at a fixed speed $c$ in vacuum regardless of the velocity of the observer.
The above fact about light cannot be understood in any reasonable way in the context of Galilean relativity, wherein transformations from one frame to another involve Galilean boosts (and thus would change the speed of light, contrary to the physical facts).
The above fact about light is easily understood in the context of special relativity, wherein transformations from one frame to another involve Lorentz boosts (which leave the speed of light constant, consistent with the facts of nature).
Therefore, we understand that Lorentz transformation are the proper transformation between frames moving at different velocities. These transformation involve the speed of light and define an invariant squared spacetime distance: $$ ds^2 = -c^2 dt^2 + dx^2 + dy^2 + dz^2\;, $$
The integral along the world line $ds$ for a given massive particle is invariant and so is used to construct the action of a free particle like this: $$ S = -mc\int ds = -mc\int dt \sqrt{c^2 - v^2} \equiv \int dt L\;, $$ where $L$ is the Lagrangian. Note that the speed $c$ appears in the above equation, but this is the equation of action for a point mass of mass $m$. The speed $c$ is showing up because of the nature of space-time itself, through which the point mass moves.
From the Lagrangian we can find the Hamiltonian (energy) of a free particle, which turns out to be: $$ E = \sqrt{p^2 c^2 + m^2 c^4}\;, $$ where, as usual $p = \frac{\partial L}{\partial v}$.
When the particle is at rest ($p=0$) the above equation for the energy reduces to: $$ E_{at\; rest} = mc^2\;, $$ where $m$ is the rest mass and $c$ is the speed of light.
