The Dark Matter and Dark Energy Effects of Spacetime

Alice and Bob don't always agree. Alice is outside our universe, looking at the big picture. She sees the universe expanding at a steady rate, like a balloon attached to a helium tank. From her point of view, what we classically think of as energy is conserved. Here's what she sees at time one (t1):

Here's what she sees at time two (t2):

Bob has a different take. He's on earth looking outward. He sees the universe expanding at an accelerated rate--the red shift of distant galaxies is greater than that of closer galaxies. Energy is not conserved--it is increasing! Here's what he sees at t1 and t2:

In the diagram above, the blue dot represents Bob's galaxy. The red dot is a galaxy far far away. As far as Bob is concerned, that red dot is moving the fastest. Bob uses equation 1) below to model what he sees; whereas, Alice uses equation 2):

Equation 1) shows velocity increasing as the radius (r) increases. However, radius (r) fails to tell us the effect gravity has on time. Where gravity is strong, time runs more slowly. Where gravity is weak, time runs faster. Gravity is weaker where the radius is larger, and vice versa. So we can substitute ct' for r in equation 1).

Alice sees the universe expanding at a steady rate. However, we can change this to ct'/t' in equation 2) to show that time (t') cancels itself. By contrast, Bob, uses Hubble's constant, a fixed time--it doesn't cancel time (t'). As a result, Alice sees a short expansion over a short time where gravity is strong, and a long expansion over a long time where gravity is weak. She sees a steady expansion velocity. Bob sees a slower velocity where gravity is strong and a faster velocity where gravity is weak.

Both Alice and Bob notice that spacetime has energy density or pressure. They both use the following equations:

Normally when there is pressure, the volume increases and that relieves the pressure. In the case of space, increasing volume just adds more energy. This keeps the pressure constant, so expansion is continuous. (Equation 5) above shows the volume (V) cancelling itself.)

Notwithstanding the added energy, Alice sees conserved energy. Using Einstein's field equations, we can derive something that shows why.

At equation 14), when the spacetime mass (ms)in the numerator increases, so does V * bar-lambda in the denominator. The increase in spacetime curvature caused by spacetime mass(energy) cancels or is always proportionate to the spacetime mass. Also, when volume (V) increases, spacetime curvature (K) decreases, so V * K is constant. The remaining variables are also constant. Thus, energy (E) is conserved.

Bob, of course, disagrees. He says the energy is growing. If we perform an operation on the conserved-energy equation, we can see why:

According to equation 17), as the universe's radius (r) increases, so does energy (E'). So who's right? Alice or Bob? Answer: they both are. What we observe depends on our frame of reference and whether we use Hubble's constant or time (t'). What Alice and Bob observed can be conveniently labeled the dark energy effects of spacetime.

Equation 15) reveals the dark matter effect. We mentioned earlier, when volume (V) grows, so does spacetime mass (ms). Variable m, however, remains constant. This means spacetime mass becomes more significant at greater volumes and matter mass becomes less significant. When we crunch the numbers, spacetime makes up most of the mass and energy within the volume of the known universe. It follows that it would cause most of the universe's gravity.

One possible mechanism for the extra gravity we observe is spacetime's expansion pressure. Imagine a weightless environment. Imagine a transparent balloon being filled with gas. Floating in the middle of that balloon is a marble. The gas pressure presses outward against the balloon's inner surface. It expands the balloon, but the pressure goes inward against the marble's surface as well.

The arrows in the above diagram represent the pressure going outward and inward. If the marble is a metaphor for a galaxy, then, in addition to gravity caused by matter, the galaxy is receiving pressure from the outside. There is also counter-pressure from within. This could give the impression of additional gravity.

Another cause of additional gravity is time (t'). Since spacetime adds more mass, the rate of time must be slower, and slower time should produce some gravitational effects.

One has to wonder, though: if the galaxies had enough gravity to attract each other, would the universe still expand? Back to the balloon. Imagine the balloon has a bunch of marbles floating inside it. They are held together by strings. Hot gas is pumped into the balloon. The balloon expands anyway. The hot gas flows around the marbles' surfaces and creates pressure on them and between them. If the strings are strong enough, the marbles won't separate. If the strings are weak, the marbles may separate and go with the flow of the hot gas in the expanding balloon. Our universe may work the same way.

Update: Here are the complete equations that model the dark matter and dark energy effects of spacetime. First we introduce some new variables:

To conserve energy we assume expanding spacetime goes in equal and opposite directions. These equal and opposite directions cancel each other. We use +/- signs to indicate that.

In the diagram above we arbitrarily label one half the spacetime volume (V) on the left as "-" and the right half as "+." Now here are the equations:

The following equation was designed to show that energy is truly conserved. No matter how big or small spacetime energy (Es) gets, overall energy is conserved. Es appears in both the numerator and denominater; it is the energy of both space and time, so it cancels itself.

Although, Bob insists that energy overall is increasing, so his equation is as follows: