This post is a sequel to "Dark Energy In - Dark Energy Out = Gravity." Today we are going to find the relationship between Hubble's observations (i.e. Hubble's constant), dark energy and gravity--and we are going to derive it from Heisenberg's uncertainty principle. Let's kick things off with defining the variables:
Equation(and inequality) 1 below is the energy-time version of Heisenberg's uncertainty principle:
The idea here is to build an expanding universe by taking a bottom-up approach. We build the very large by starting with something very small. We derive a simple energy equation (see equation 4 below).
Note the change in energy or energy difference variable on the left side of equation 4. We can substitute some arbitrary energy (E) minus the ground state (epsilon * E):
Let's bring all the terms to the left side and derive equation 10 below:
We now have an energy squared minus another energy squared minus the ground state squared equals the final energy (Ef) squared. We get equations 11 and 12 below by using Planck's reduced constant (h-bar), the wave number (k), and the light-speed constant (c)--and making substitutions.
Checking the units, we find equation 12 to be eerily similar to Einstein's field equations. Not a bad thing, by the way. It allows us to rewrite equation 12 to get 13:
Multiply both sides of 13 by the volume (D^3) to get 14 and 15:
Multiply both sides by Hubble's constant (H):
Multiply both sides by c^2/D:
From here we can derive 21 below:
Equation 21 is a power equation that has two components: the force of gravity, and the velocity the universe is expanding at distance D. However, this is only part of the story. Equation 21 does not take into account the mass density of spacetime or vacuum. A more complete equation is 22:
Note that as distance D increases, Volume V increases. The vacuum-mass-density gravity grows(dark matter effect) while classical Newton's gravity shrinks. If we utilize the cosmological constant, we can see a more precise relation between gravity and dark energy. First, we need to go back a few steps and work the cosmological constant into the math. Let's start with equation 13 and work forward:
A note re: equation 23. We want epsilon/D^2 to represent a ground state and a ground state does not increase or decrease, so epsilon must be proportionate to D^2. Thus we can set the term equal to the cosmological constant.
Equation 29 reveals something interesting: the instant velocity of expansion (HD) appears to be unaffected by gravity. The gravity in the numerator seems to be proportionate to the gravity in the denominator. This suggests the big crunch ain't gonna happen. But wait! It gets better. Suppose the universe expands to a point where Newtonian gravity (GM/r^2) is insignificant? We can drop it and get equations 30 and 31:
Look at equation 31. The only variable that isn't a constant is distance D. Now here's the awesome part: When D increases, so does gravity and so does the rate of expansion. The expansion rate (HD) is a function of gravity ... or is it dark energy? They both appear to be two sides of the same coin. And why not? They are both components of vacuum power (P).
Update: We can take equation 31 and derive the value of the cosmological constant: