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## Monday, July 24, 2017

### Relativizing Hubble's Constant

Albert Einstein had good reasons to believe nothing could go faster than light in a vacuum, including the vacuum itself (aka: dark energy). Consider his famous equation along with equation 2:

If the total energy (E) equals mass (m) times c^2, then c is the maximum potential velocity. If a greater velocity is possible, then Einstein was wrong. Equation 2 above shows the folly of trying to increase the velocity beyond light speed. To have more speed requires more energy, but more energy has more mass (or mass equivalence), since the equation represents mass energy. This is why c^2 is a constant.

Consider equation 3 below. (To see how it was derived, click here.) It shows how spacetime expansion velocity (Hr) relates to the gravitational constant G. The term on the right side has the cosmological constant in the denominator, but you'll notice there is also c^2. At equation 4 we set the expansion velocity at the cosmological horizon (Hr[u]) equal to c. After a little algebra, we derive equation 7. The idea is to test whether the expansion velocity can exceed c.

If we try to increase the left side of equation 7, the right side must also increase, but how? If we increase mass density (p), then spacetime curvature must also increase; i.e., the cosmological constant must increase. If we try reducing the cosmological constant (i.e. spacetime curvature), then mass density will decrease. In other words, we can't make the universe expand faster than c.

But surely if the universe continues to expand, the radius (r) should grow and the velocity (Hr) should also grow beyond c. But if we use all the energy the universe has to offer, the maximum potential is mc^2, not m(infinite velocity)^2. Perhaps the following crude diagram can assist us:

The outer circle represents the cosmological horizon; the inner circle is an arbitrary distance an observer may be looking into space (Do). If HDo is the expansion velocity observed, then the remaining universe must be expanding at a rate of HDr. The sum of these velocities is c or Hru. With this information we can write the following mathematical proof:

The result is a system of equations at 21 above. Any observer will see the universe expanding at an accelerated rate up to the cosmological horizon. Before that point, Hubble's constant remains reasonably constant (perhaps due to the fact that the mass density of spacetime, which could alter the value of H, is fairly consistent throughout the universe). At the horizon and beyond, Hubble's constant shrinks to accommodate the maximum potential velocity c.

Given this information we can hypothesize that any point in spacetime moves at a maximum potential velocity of c relative to any point that is separated by a distance of Do + Dr. Hence we have c^2 as a maximum potential. If we couple it with a mass, we have mc^2. We can also show the following relationships between gravity and dark energy:

Equations 22 through 24 show that when mass (m) is added to an expanding universe, proper time (t') shrinks. This would slow the rate of expansion if it were not for gravity making up the difference. For more details click here.

Update: Here is another mathematical proof showing that Hubble's constant shrinks beyond the cosmological horizon. Take special note of equation 31. It shows how the constant c^2 is maintained.