Earth Stands Still
Now the equation looks like:
Klaatu then writes in the solutions of his more advanced species. Today, as was done 2 years ago, we can guess at what he wrote:
Today's physicists, trained for years in our primitive science, may raise petty objections when someone alters their equations. On the question of units, the Friedmann equations are valid in units of mass density or energy density.
The stress-energy tensor T_uv can have units of mass density ρ or energy density ρ(c^2). Removing for a moment the c^2 from Friedmann we would have:
8πGρ/3 = ⅓κρ (c^2)
4πGρ/3 = ⅙κρ (c^2)
To normalise the left-hand and right-hand sides, some physicists chose κ=8πG/(c^2). This led to decades of misconception that Relativity requires a fixed c. Some "geometrized" unit systems give κ=8πG/(c^4), which is too convoluted to describe.
If T_uv has units of energy density, we must use the same for Friedmann:
8πGρ(c^2)/3 = ⅓κρ(c^2)
4πGρ(c^2)/3 = ⅙κρ(c^2)
This is very trivial. The (c^2) on both sides simply cancels out. Einstein called constant κ "related to the gravitational constant" without mentioning c.
Now the Einstein equation becomes Ruv-½guvR=8πGTuv. The Bianchi identities become:
The world is much simpler without that pesky (c^2) factor.
Finally, the Einstein-Hilbert action becomes:
Thus we can do everything General Relativity can without a fixed c. Some problems, like the deflection of bodies by the Sun, work even better with a varying c.
In reality, a human professor would throw Klaatu out for messing with his blackboard. A Universe that can be described in a few equations just might be beyond human understanding. Humans tend to complicate their Universe with epicycles, luminiferous ether, or cosmic constants. Are humans ready for new physics? Perhaps we should ask Klaatu.