If energy $E$ can neither be created nor destroyed, then presumably there is a constant total E in the universe. Consider E=mc^2. If total E in the universe is constant, then as mass m changes c must also change for the equation to balance. The mass m of the universe is reduced by nuclear fusion in stars with no loss of energy, for example. Also, if the above is true then the speed of light would have been high in the very early universe before massive particles condensed. Let me know why I am wrong, I have little experience with GR.
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You have somewhat misunderstood $E=mc^2$. Energy and mass are different ways of measuring the same attribute of physical systems - energy has mass and mass has energy. Let's call this combined attribute mass/energy. We can represent mass/energy in units of mass - say kilogrammes - or we can represent it in units of energy - say Joules. The conversion factor between mass/energy measured in kilogrammes and mass/energy measured in Joules is $c^2$.
Let's sidestep the whole "is the total energy of the universe constant and what does it even mean" question, and think instead about some system that is isolated from its environment. The total mass/energy of that system is constant. In other words the total energy of that system (including the energy represented by what we might conventionally think of as mass) is constant and its total mass (including the mass represented by what we might conventionally think of as energy) is also constant.
In particular, if nuclear reactions take place within that system, they might reduce the conventional mass of the system, and increase its conventional energy by an equivalent amount $E=mc^2$. But the total mass/energy of the system remains constant, and if we represent that total mass/energy in units of mass, that value will be constant (as it will if we represent the total mass/energy in units of energy).
gandalf61
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There are two misconceptions here. Noether's theorem tells us energy is not always conserved, and this is true for the universe as a whole. Secondly, $E=mc^2$ is a popular abbreviation. The relativistic energy momentum relationship is actually: $$ E = \sqrt{ (pc)^2 + (mc^2)^2} ,$$ which only reduces to the more well known $E=mc^2$ when there is no motion.
$p$ is the momentum. Photons have momentum proportional to their frequency. When a star undergoes fusion, the mass loss is converted to the energy of the emitted light. This means the overall energy of the star does not change due to the fusion process. It also means the gravitational energy of the star does not change due to the fusion process.
In your question, you are assuming $$E=mc^2 \implies c = \sqrt{\frac{E}{m}}$$ and then concluding: if $E$ is constant and $m$ is reducing, then $c$ must be changing. It is not as simple as that.
