Does Gravity Really Exist?--Dark Energy's Equivalence Principle

According to the equivalence principle, when you drop a pen to the floor, the pen can be considered at rest while the floor moves to the pen. Because the pen is allegedly at rest, no force is needed to act on it and it can be any mass. Now here's the tricky part, because the pen has mass, it has a tiny bit of gravity of its own. It attracts the earth, or rather, the earth is at rest and the pen moves to the earth, so the earth can be any mass and no force needs to act on it. So one has to wonder: Is the earth moving to the pen or vice versa? How can they both be at rest and be moving? Is gravity really a force? It has been argued that it is not. As we shall see, one could also argue that gravity does not exist, especially when one takes a closer look at dark energy.

Galaxies are accelerating apart as the universe expands. Allegedly they are not moving through space, but rather, they are moving with space--and because they are moving with space, they can be any mass. No force is needed to push them apart? Kind of reminds you of gravity and the equivalence principle, doesn't it? It seems that dark energy is a kind of negative gravity, or, maybe gravity is positive dark energy? So many questions! But these questions will be addressed in this post. First, we define the variables we need:

Both gravity and dark energy have something in common: spacetime. To understand why galaxies move apart and why pens fall to the floor, we need to examine spacetime. What is it exactly? On the surface, it's a vacuum with dimensions of space and time. If we were put in charge of building a universe, how would we go about creating this thing called spacetime?

We could start by taking all forces (strong, weak, Higgs, EM, and unkown), add them together to get an energy field (see equation 1 below). We use that energy to derive equation 5: the Planck length.

At equation 4, notice how the energy field cancels itself. It could be any magnitude and any type of energy or combination of energies. It doesn't matter. In any case we get the Planck length unit. We can use this unit to create an arbitrary length (r) at equation 6.

Normally when we think of dimensions of space we think of fixed Cartesian coordinates (x,y,z), but our universe is not static; it is expanding--so how fast does our coordinate system move? We derive the answer to this question below (see equation 12).

According to equation 12, our universe's coordinate system is expanding at light speed, but this seems to contradict equation 13 below, which shows the expansion velocity (vd) depends on the distance r. Then again, the following equations show the velocity depends on which clock you use. If you use Hubble's constant, you get a velocity of Hr. If you use time t, you get light speed.

At equation 15, note that velocity Hct can go to infinity. This does not violate the light-speed limit if we assume galaxies are moving with space instead of through space. Moving with space means they are at rest. If they are at rest then no force or pressure is moving them. The math below demonstrates that the galaxies move the way they do regardless of how much or how little pressure vacuum energy has. The implication is "dark energy" isn't energy or vacuum pressure.

At equations 20 and 21 we see that mass (m) cancels itself no matter how big or small it is. Whether spacetime has a little or a lot of energy (mass), it moves at the rate of Hr or light speed (c), depending on the clock used.

To prove that dark energy and gravity are the same phenomenon we set up a series of Minkowski diagrams. The first one below breaks velocity Hr into a time component (Hrt) and a space component (Hrs).

At 23 we integrate over all the varying velocities at each point in spacetime to get the Pythagorean relation between Hr and it's components.

In the next diagram we express the distances (r's) in terms of spacetime (ct) and make some substitutions:

In the next diagram we measure velocity using time (1/t) instead of Hubble's constant:

Time t cancels itself and that brings us to the familiar Minkowski diagram that can be used to calculate gravitational velocity (vo):

At 28 we once again integrate over all velocities at all points in spacetime and get the Pythagorean relation between time velocity (u), space velocity (vo) and light speed (c).

We extract dark-energy equation 29 below from equation 23. From 29 we derive Newton's gravity (see equation 32).

Taking a few more steps and making a few more substitutions we derive Einstein's gravity (see equation 39):

We can also take equation 31 (restated at 40 below) and derive the Lorentz factor (equation 45):

Bottom line? We can start with dark energy and derive gravity! Or vice versa!

Now, the following 1D diagrams show how the equivalence principle works in terms of dark-energy. We use time t rather than Hubble's constant. That means all masses are coupled with c^2--hence the famous equation: E = mc^2. However, our first diagram is a universe with pure spacetime, no masses, so we just have c^2 in all directions:

Notice points A and B. They are static, i.e., going nowhere. At equation 46 we see why. The c^2's are equal and opposite and sum to zero.

Suppose we add a mass at point B. Doing so causes time to slow at that point. As a result, velocity c is reduced to velocity u. Equation 47 confirms this causes point A to move to point B.

Let's add a mass (m') at point A. Time is reduced, reducing c to w. We can see from the equations and diagram below that point B moves to point A and vice versa. The masses, no matter how large or small, located at those points move at the same rates as those points. When we add masses m' and m to points A and B, c^2 becomes m'c^2 and mc^2 respectively.

Now, it is commonly believed that gravity and dark energy are two opposing forces. One pulls everything together and the other pushes them apart. The following diagrams show this is not the case. The universe will expand at the rate of c or Hr no matter how much mass there is. For comparison, let's start with a universe with no mass and measure its expansion rate:

In the above diagram we interpret one direction as positive and the opposite direction as negative, so the universe expands at +/- c or +/- Hr (taking the square root of equation 54). (If we used a 3D diagram, each velocity would have an equal and opposite velocity; however, we only need 1D to illustrate the point.)

In the next diagram we introduce a mass and this appears to reduce the overall expansion rate, but the following math suggests this is relative, it depends on where the observer is located. An observer some distance from the mass will sum up a total value of c^2. According to the above Minkowski diagram, if space expands less, time expands more or vice versa. The total expansion rate is always Hr.

Note that equation 57 agrees with the corresponding Minkowski diagram.

The final diagram shows how the thing we call "dark energy" causes the thing we call "gravity" when matter is present. The red arrows are pens falling on opposite sides (pens moving with spacetime). the blue and red curves are satellites or photons moving along geodesic paths. Objects not moving through space move with space at the same rate, since they are technically at rest (mc^2).

Now you might wonder why mc^2 is rest mass energy and not light-speed energy. Well, you might think you are at rest, but to an observer a cosmological-horizon distance away, you are moving away at light speed! In fact, every point in the universe is moving at light speed relative to such an observer. So "at rest" really means you are moving with space and not through it. Oddly enough, if time is measured in units of r/c instead of Hubble's constant's reciprocal (1/H), points near and far (and their respective masses) are always moving at the speed of light! So rest-mass energy truly is E = mc^2.

Below we derive the dark-energy Lagrangian and equations of motion:

At 64 we recognize that the Ricci tensor and scalar have the same units as quantum wave numbers (k). As we shall see, it doesn't seem to matter what kind of wave numbers we use.

At 72 we have the Lagrangian (L). At 73 we have the speed of mass-less particles in a gravitational field (light speed c or Hr at the cosmological horizon). At 74 we have the speed of mass particles in a gravitational field--or is it a dark-energy field?

Final thoughts: Assuming objects can move with space and need no force to act on them, does this imply that no force-carrying particle is needed? The above math and diagrams show that there is no attractive force, per se. Gravity appears to be a by-product of expanding spacetime in different reference frames. Could this be the reason the "graviton" has eluded particle physicists?

A New Mathematical Dark Matter Discovery

While fooling around with some gravity equations, I discovered some intriguing results. First, let's define the variables we will use:

Dark matter and dark energy are currently treated as two different entities, but the diagram below shows they could be two sides of the same coin:

The above diagram shows galaxies (circles) being pushed apart by dark energy (arrows). But notice how the dark-energy arrows are pushing in on the center galaxy. Any galaxy in the universe could be deemed the center and experience inward pressure from dark energy. Of course there is also bound to be outward pressure. If the two are equal and opposite, nothing interesting happens. But if the two are unequal, things get mighty interesting!

The next diagram takes a rectangular volume of a spiral galaxy and some surrounding space. The volume is divided into section A, a volume of space, and section B which includes space and baryonic matter (stars, gases, etc.). Each section has a length of distance r. The arrows represent the directions of expanding dark energy.

Since the galaxy is rotating, it has angular momentum q. The angle of this momentum is approximately 90 degrees relative to r. In the next slide we zoom in closer and see more details:

Section A has dark energy, mass m and little or no baryonic matter m' to slow down the expansion momentum (p) along r. Section B has baryonic matter m' and mass m. To conserve energy, outgoing momentum p' is less than incoming momentum p. Mass m normally is attributed to dark matter, but below we redefine it and define the other variables in more detail:

Equation 1 describes the total momentum at the cosmological horizon(light speed * vacuum mass or mass equivalent). Equation 2 shows that dark energy momentum is a function of distance r and Hubble's constant. As r increases, the speed of expansion increases; i.e., dark energy is more kinetic relative to the observer. So momentum p increases. Equation three shows that the converse is true: when r is relatively short, expansion velocity is slower. When r is zero, expansion velocity is zero. In that case dark energy is less kinetic and behaves more like mass or potential energy. So variable m becomes dark-energy rest mass (or something equivalent if it is mass-less). Equation 4 shows variable p' is equal to itself plus an angular momentum q term.

At equation 5 (Einstein's energy equation) below, the left side represents section A, the right--section B. Both sections have distance r which is short compared to the distance to the cosmological horizon. Applying equations 2 and 3 we get a dark energy that has nearly zero momentum (p) and nearly 100% mass (m). We start with the assumption that sections A and B are filled with nothing but dark energy.

At equation 6 we add some baryonic matter (m') and angular momentum (q) to the right side. Doing so reduces dark-energy momentum from p to p'. (Caveat: If momentum is too small, then dark-energy mass is reduced.) Equations 7 and 8 show that, at 90 degrees and on average, the momentum q term is zero. That leaves us with equation 9.

And from equation 9 we can derive equation 22:

You will note equation 22 has an extra mass term which is necessary if you are measuring the gravity of a galaxy. If you measure gravity on solar and planetary scales, the extra mass term becomes insignificant--the equation reduces to Newton's law of gravitation. Also, at the cosmological horizon, the extra term vanishes. Equations 23 and 24 demonstrate these contingencies.

Now let's have some fun! Below are some typical numbers you will find on the web. However, we are going to assume there is no dark matter and substitute dark energy.

We want to calculate percentages and proportions, so we divide the above numbers by 4. That way baryonic matter is equal to 1. We divide dark-energy rest mass again by 2, so sections A and B can have equal portions. We plug and chug below:

As you can see, the result is fairly consistent with dark matter estimates. Now, let's do our galaxy. Here's some more numbers pulled off the web:

We plug and chug again:

According to our calculations, our galaxy consists of only 12% baryonic matter. The extra mass is called dark matter, but at least some of it could be dark-energy rest mass (or mass equivalent).

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: