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Proving the Schwartz Inequality and Heisenberg's Uncertainty Principle

In this post we once again derive the Heisenberg uncertainty principle, but this time we make use of the Schwartz inequality and the posit...

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.

Friday, July 7, 2017

How to Derive Dark Energy, Etc. From Heisenberg's Uncertainty Principle

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:

Wednesday, July 5, 2017

The Expanding Universe Conundrum

How fast is the universe expanding? To figure this out, we could take the radius of the known universe and divide it by the universe's age. That should give us the average velocity the universe has expanded since the beginning of time:

Wow! It appears the universe has been expanding at the speed of light since day one! If that's true, if the galaxies, for example, have always been moving apart from each other at light speed, how did their photons reach us? How can we see them? If the space between galaxies were expanding at light speed, the light from the neighboring galaxies would never have made up the increasing distance. As soon as a photon, say, from NGC-2419 covered a light year, another light year of distance would have been added. The result? There would have been no discovery of good old NGC-2419 and more distant galaxies.

But what if the science is wrong? What if we are at rest and space does not expand--and those distant galaxies are moving away from us through space? Their light would eventually reach us no matter how fast they're moving away. Unfortunately this is not the case. If it were, the most distant galaxies would become dimmer and dimmer as they move further and further away. To see stuff beyond our known universe, all we would need is a more powerful telescope. If the science is correct (and it is), there should be a point in the distant cosmos, a cosmological horizon, beyond which we can see nothing (the light can't reach us due to expanding space).

So then how did the light from distant galaxies reach us? Imagine the universe is a sphere. Let's pretend we can draw a small circle anywhere we want on that sphere. That small circle encircles some galaxies (including the Milky Way). The sphere, represented by the 2D larger circle below, expands at light speed (c). The smaller circle within also expands over the same time period (t):

Now, let's double-check the velocity of the circles:

The larger circle expanded at velocity c as expected, but the smaller circle expanded less over the same time period; its velocity is only v or Hr. Therefore, the observable universe has indeed expanded at the speed of light (or more), but galaxies nearer to us have not. Their photons were able to reach us and we can see them.

Sunday, July 2, 2017

Deriving Hubble's Constant, Etc.

Here is one way to derive Hubble's constant. First, let's define the variables:

We start with an arbitrary total distance (D) which is the product of a small coefficient (a) and distance (x). As x grows, so does D, and vice versa. At equation 2 below, we find the time derivative of D (which equals V).

At equation 3 above, we multiply the left side by x/x. At equations 4 through 7 we do some algebraic slight of hand to derive Hubble's constant (H). Equation 8 shows the approximate numerical value of Hubble's constant. Its units are s^-1--the reciprocal of time.

Next, let's see what its relationship is to proper or relativistic time. To accomplish this we use 1/H in lieu of time (t).

Equation 14 shows how Hubble's constant stays constant. If proper time t' changes, so does variable u. The two are proportionate. Equation 15 below is the velocity the universe expands at distance D. When D increases, so does t'. You would think this amounts to a constant velocity for any distance D, But variable u also increases and offsets t'. As a result, velocity increases as D increases.

Equations 16 through 20 express distance D in terms of time (ctD). When examining the relativistic consequences of Hubble's constant, it is important to recognize that the time that makes up distance (ct) is a function of D only. Whereas t' is a function of D, mass and energy. High mass, for example, will contract time t to time t', and will contract length from ct to ct'. But distance D equals ct or act'; i.e., it can be measured out even if the units shrink due to relativity.