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?

How to Build a Dark Matter Galaxy and a Baryonic Matter Galaxy

According to the above video, an ultra-diffuse galaxy (far fewer stars than ours) contains little or no dark matter. Light passes through it in a seemingly straight line (little or no gravitational lensing). Then there's the dark matter galaxy, with far fewer stars than our galaxy. Most galaxies appear to have approximately five to ten times more dark matter than baryonic matter.

Using Einstein's field equations, we shall build these different types of galaxies and derive an equation that models each of them. However, to accomplish this task, we make the following assumptions: A dark matter particle is an excitation of a field, possibly dark energy or the vacuum. Like the vacuum of space, dark matter particles are diffuse. They have no electromagnetic force. They don't interact significantly with ordinary-matter particles or with themselves. There is, however, gravitational attraction. Finally, dark matter has something similar to the Pauli exclusion principle which prevents DM particles from gravitationally collapsing into a single point in space.

Because dark matter is diffuse, its gravity is insignificant near, say, a black hole, but becomes more significant as we increase the distance (r). Below is a crude illustration of a dark-matter galaxy. It has a black hole in the center. Inside the smaller circle, there's the big black dot (black hole) surrounded by only a few small dots (dark matter). By contrast, inside the large circle the small dots dominate (make up most of the mass).

Now that we've defined "dark matter," let's define the variables we will need:

We are ready to begin working with Einstein's field equations:

Let's change the cosmological constant term and make a substitution:

We make the substitution at equation 3. At equation 4 we put the term on the other side of the equals sign:

To make the math less clunky, let's change the terms into some simple variables:

After making more substitutions we get equation 12 which has a surprising implication!

Spacetime can be flat no matter how much mass (energy density, pressure, etc.) you have! Variables C and D are the energy-stress or mass terms. As long as they are equal, spacetime is flat--which implies no gravity! How can this be?

If there is no gravity between C and D, they will not move (or they will be pushed apart by dark energy). Imagine a red bowling ball and a blue one sitting apart from each other on a trampoline. Imagine the trampoline is expanding. It might look something like this:

Each bowling ball makes a dent in the trampoline, but they don't roll to each other. But around the region of each dent, there are some marbles that roll toward each bowling ball. The analogy demonstrates the following: Gravity is local and not global. If two or more masses have equal energy densities, there is no attraction. Where they are not equal, there is attraction.

The next diagram shows why galaxies move away from each other rather than towards each other:

The above diagram is a toy universe consisting of four multi-colored galaxies. They have equal masses and are the same distance apart. The arrows assume there is gravity between them, but the red galaxy is equally attracted to the green and blue galaxies. The other galaxies are in a similar predicament. The gravity of one galaxy is cancelled by the gravity of another galaxy. So there's no movement. This universe is static. Even though there is plenty of mass, spacetime may as well be flat and gravity may as well be zero. Add some dark energy and this universe will expand indefinitely.

So how do we get some gravity going in our toy universe? We break the symmetry. One way is to alter the distances between the galaxies (see first diagram below). Another way is to vary the masses (see second diagram below).

Now, you may be thinking, this is all fine and good, but what does this have to do with dark matter? Well, to understand dark matter, we have to have a better understanding of how gravity works and how it doesn't work. The main point you want to hang on to is equal masses that are equidistant aren't attracted to each other or that attraction is cancelled. Thus galaxies stay apart and move apart except when the symmetry is broken:

Now, what is true for massive galaxies must also be true for smaller stuff like dark matter/energy particles:

So all we have to do to make a dark matter galaxy or any galaxy containing dark matter is break the symmetry by introducing, say, a huge mass:

Let's assume the big mass is baryonic matter. Why baryonic matter? Baryonic matter is electromagnetic. It can form into molecules, planets and stars. It can come in large clumps. Whereas dark matter is non-electromagnetic and does not form clumps. It remains diffuse much like dark energy or the vacuum of space. When its particles are equally spaced, they are not gravitationally attracted to each other, so they need a catalyst, a symmetry breaker. Baryonic matter comes in different-sized clumps and is a perfect symmetry breaker.

In our toy universe we can imagine little dark matter particles falling into the spacetime dent created by the huge clump of baryonic matter. The dent, of course, gets deeper, allowing even more DM particles to enter:

So when does this process end? It looks like it could go on indefinitely! Dark matter pours into the dent, the dent gets deeper, more particles pour in, and so on. There must be an equilibrium where the mass of the dark matter is around five to ten times greater than the baryonic matter. One way to create such an equilibrium is to use the inverse-square law and Hooke's law. Below we use these laws to limit how deep the spacetime dent gets. We treat it like an elastic surface that springs back into place when mass is removed.

At equation 21 above, the two main variables are M (total galactic mass) and x (how deep is the dent). As dark matter is added, M and x grow proportionately (see 19), but as x adds to the dent's magnitude (equation 21's second term), it has an inverse-square effect (equation 21's first term) and the gravity is reduced until equation 21 equals zero--equilibrium is achieved (see 23 below). Using a pinch of algebra, we calculate the final mass of the galaxy (see equation 28).

We see the total mass is the dark-matter constant (we set to 10) plus 1 times the original mass (Mo) we used to break the symmetry. If we assume that baryonic matter is a catalyst for the build up of dark matter within a galaxy, then we can predict that galaxies with few stars, i.e., a small amount of baryonic matter will have little or no dark matter. And we can predict that galaxies that are nearly exclusively dark matter, have or have something equivalent to a massive black hole at their center. Finally, we predict that normal galaxies with lots of dark matter also have far more stars than galaxies lacking dark matter. These predictions lead to the final equation:

Where the radius (r) is small, equation 30 reduces to Newton's equation, but when r is large, dark matter rules!

Update and caveat: Click here to read a new article that shows there are galaxies that have no dark matter.

Resolving the Cosmological Constant Problem

Imagine you're a quantum physicist and you just got through calculating the energy density of the vacuum. To your horror you find that your calculation is around a hundred orders of magnitude greater than vacuum energy density measurements! This, in a nutshell, is the cosmological constant problem. According to your numbers, the cosmological constant should be huge. According to WMAP measuremments, the cosmological constant is too small to tweet about. In this post we shed some light on this problem. First, here are the variables we will be using:

Below we derive the cosmological constant value based on the WMAP vacuum energy density measurement of approximately 10^-10 Joules per cubic meter (see equations 3 and 4):

The next order of business is to consider the limitations caused by the Compton wavelength formula and the Heisenberg uncertainty principle--so we derive equation 8:

There are two ways we can interpret equation 8: 1. If delta-x grows smaller, energy (delta-E) grows bigger. This makes perfect sense when you consider that smaller spaces contain shorter wavelengths, and shorter wavelengths correspond to higher energies. 2. If delta-x grows smaller, delta-E becomes more uncertain. In any case, we use equation 8 as our means to make sense of the cosmological constant problem.

Equation 9 below is equation 8 modified. On the left side we have the vacuum energy measurement (E) and one meter (x). Of course the product of these values is far greater than Planck's constant, so the right side has the scale factor n:

OK, here's what we need to do: we need to reduce the vacuum's volume while maintaining a constant energy density. This can be done, but only up to a point. The question becomes, up to what point? Or, if you prefer, down to what point? How small can we make the energy and space before things get weird? Here's the answer:

Equation 10 above is the formula for finding the cutoff, or, the Heisenberg uncertainty limit. The product of the energy and space can't go below h-bar/2 without violating the uncertainty principle.

At 11 and 12 below, we plug-n-chug:

At the cutoff, the distance is around 10^-4 meters. The volume is 10^-12 cubic meters. The energy is approximately 10^-22 Joules. If the space is reduced further, the energy value will explode! With an upper limit of infinity! Obviously, if the energy skyrockets, or becomes more uncertain, the energy density won't be the same--and any calculation done at that level will yield ridiculous results.

The diagram below shows vacuum energy density matches observations at or above the cutoff. Any meaningful measurement is taken above that point. But if you prefer infinities and uncertainties, feel free to go smaller than the Heisenberg limit.

Thus, the cosmological constant problem appears to be a natural consequence of doing calculations and measurements below the cutoff limit.

How to Derive a Black Hole From Einstein's Field Equations

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.