Today we will derive the gravitational constant G, also known as Newton's constant. Here are the variables we will be working with:

Below is a crude diagram of a satellite orbiting a star or planet at velocity v, at a distance of radius r. According to its clock, the proper time is t'. The star or planet has a mass of m.

Our starting point shall be the Lorentz equation, courtesy of Einstein's theory of special relativity:

By doing some algebra we can derive equation 7 below:

Equation 7's right side expresses v^2 in terms c^2 and a time ratio. The bigger the time ratio, the faster the velocity and vice versa.

We manipulate Einstein's energy equation to get equation 8:

We make a substitution, then do some more algebra until we derive G at equation 15:

We can now see why G is the constant it is: Any change in velocity (v^2) is offset by a change of the radius-mass ratio. Any change in radius-mass ratio is offset by a change in the time ratio.

According to Stephen Hawking, if we start with a volume of space, say, a public library, and add books and more books, eventually the total number of books will become so massive they will collapse into a black hole. This is a little difficult to verify experimentally, but we can derive a black hole from Einstein's field equations. Below are the variables we will need:

In the diagram below, the blue circle represents the compressed mass (library books); the yellow and blue circle represent the mass's initial volume (library shelf space); The largest circle has a Scharzschild radius. As more and more mass is added, the blue circle shrinks and the singularity radius (r) shrinks as well.

Let's begin the derivation with equation 1:

Equation 1 has second-order tensors. We want to convert these to easy-to-work-with scalars (aka: invariant zero-order tensors). We can do this by contracting the indices. At equation 3 we pull the metric tensor (g) out of the Ricci tensor (R). At equation 4, the contravariant and covariant indices (i) cancel and vanish.

Now let's do the subtraction at equation 4 to get equation 5:

At equation 6 we set g equal to 8pi, so we can divide both sides of equation 5 by 8pi to get equation 7:

At 8 we set R equal to 1/r^2 and make a substitution to get equation 9:

The energy-stress tensor (T) has units of energy density. We do what we must to convert energy density to mass density (equations 10 and 11). We do a little algebra at 12 and 13 to get the Scharzschild radius (equation 13).

Taking equation 12 and applying limits gives the black-hole equation 14:

Just as Stephen Hawking said: If we keep adding library books (mass), the library's radius (r) shrinks and collapses into a black-hole singularity.

Below are bonus equations that show the maximum potential velocity is light speed and the maximum rest-mass energy is still mc^2.

In the above video the Physics Girl discusses how the expanding universe causes galaxies to move apart, and in turn causes photon wavelengths to stretch out. As photon wavelengths grow, they lose energy. "Where does the energy go?" she asks.

Other physicists, including myself, have a different question: "Where does dark energy come from?" As the universe expands, there is apparently more dark energy and less photon energy? Perhaps energy is conserved after all. If nothing else, it can be mathematically demonstrated. First, let's define the variables:

Equation 1 below shows how photon energy (Ep) is a function of its wavelength (lambda). The bigger lambda gets, the smaller the photon energy.

Equation 2 is dark energy (Ed)--a function of energy density (pd) times volume (V). As volume gets bigger, so does dark energy.

Equation 3 below shows the universe's radius (r) depends on how much dark energy there is. Equation 4 shows photon wavelength depends on how little photon energy there is:

Consider the universe's history. It started out with little or no space (dark energy) and it was very hot (photon energy). Over time space grew and the universe cooled (more dark energy, less photon energy). One way to conserve energy is to multiply photon energy and dark energy together. This creates a constant: as one energy grows, the other shrinks, but their product is always constant. Below we do a little algebra to get the product of the two energies:

Now, one thing we note is both energies are motion energies. Neither is at rest. Given the fact both energies have momentum (p) (due to mass or mass equivalence) we can make a substitution and derive equation 7 below:

You might recognize the momentum-energy term on equation 7's left side. It appears in this famous equation:

Einstein's energy equation, in this instance, shall represent the universe's total momentum and rest-mass energy. If we make one more substitution we get this:

Equation 9 above says the universe's conserved energy is the square root of dark energy times boson energy plus rest-mass energy squared. It includes all matter, radiation and vacuum energy.

Now you have probably been told a gazillion times that the Standard Model does not include gravity, that Einstein's field equations are incompatible with the big messy equation you see above. Let's have a look at the field equations:

The energy-stress tensor (Tij) provides a clue to how we can unify gravity with the Standard Model. Its units are energy density or energy over a volume (V). The Lagrangian has units of energy. Hmmmm ... if we contract the tensor indices and do a little algebra, we get equation 5 below:

Equation 5 shows that the scalar or zero-order tensor T is equivalent to the Lagrangian (L) divided by a volume (V). That leads us to equation 6 below:

We can now see the relationship between gravity and the other forces. If we do a little more algebra, we get an interesting result:

At equation 9 we add the newly-formed gravity Lagrangian to the Standard Model Lagrangian. Doing this yields the ground-state vacuum energy--and this quantity is consistent with the Wilkinson Microwave Anisotropy Probe measurement. So if we add gravity to the Standard Model as illustrated above, we not only get a result that is mathematically consistent, but also consistent with observations.

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.

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: