Does the speed of light determine the energy released in fusion reactions?

Question

The Openstax Astronomy book talks about stars converting mass into energy via fusion reactions:

So far, we seem to have a very attractive prescription for producing the energy emitted by the Sun: “roll” some nuclei together and join them via nuclear fusion. This will cause them to lose some of their mass, which then turns into energy.

It says that the amount of energy produced is described by the famous equation $E=mc^2$, in which $c$ is the speed of light.

My question is this (and maybe it's silly): does the speed of light determine the amount of energy released, or does it "just happen" to fit in this equation? In other words, if you could magically change the speed of light, would you thereby change the brightness of stars, or would this equation no longer be true?

Answer 1

$E=mc^2$ follows from the fact that the Minkowski metric is $diag(-c,1,1,1)$. This in turn follows from the fact that $c$ is the maximum allowable speed (together with Lorentz invariance). So if you could somehow change the speed of light to, say, $c/2$, while still maintaining $c$ as the maximum allowable speed, then you'd still have $E=mc^2$, not $E=m(c/2)^2$.

(Of course as others have pointed out, a whole lot of other things would have to change too and you can tell different auxiliary stories depending on what you insist on holding fixed.)

On the other hand, if there were a universe in which light still traveled at $c$ but the maximum speed was $2c$, then in that universe the metric would be $diag(-2c,1,1,1)$ and the correct relation between energy and mass would be $E=m(2c)^2$ or $E=4mc^2$. You could describe that by saying in that universe, the energy associated with a given mass is four times what it is in our universe, or you could describe it by saying that in that universe, the mass associated with a given energy is one-fourth what it is in our universe.

If you compare, say, a billiard ball in that universe to a billiard ball in ours, you might be tempted to ask whether their billiard ball has the same mass as ours but more energy, or the same energy as ours but less mass. Either description works equally well, and I very much doubt that there's any meaningful sense in which one description is more "right" than the other.

(There's also probably no meaningful difference between that universe and another where the maximum speed is still $c$ but light travels at speed $c/2$.)

Edited to add: Note that in Minkowski space, there are light-like paths, but that doesn't mean you have to postulate that anything travels along those paths. So you can completely assume away the very existence of light, and you'll still have the relationship $E=mc^2$ just from the geometry. This suffices, I think, to answer the question you're asking.

Answer 2

In other words, if you could magically change the speed of light,

How is speed defined?

Let us take the simple value in the direction of motion $dx/dt$ where x is in length units and t is in time units.

There is no problem in changing the base of units as has been done and is being done for the needs of simplifying an analysis. There are three independent values in our measurement system, used to be time, length and mass, and this has morphed into many useful others.

So the arithmetic value of $c$ can be magically changed without affecting anything.

For example for studying the mathematics of quantum chromodynamics the speed of light is set to 1.

$c = m_p = ℏ = 1$

To get any effect, the changes should come to the underlying theoretical quantum mechanical Lorentz invariant framework that describes interactions that finally make up our visible world, the standard model, so that a different world could be modeled. This means that the three forces should differ functionally in measurable ways from what we have now, generating a different local universe, which would extend to the whole universe if we assume quantization of gravity in a unified model .