Solving the Dark Energy Mystery

From our point of view, the universe is expanding at an exponential rate. We would have this perception no matter where we are in the universe. It's as if we are a raisin on a raisin cake baking in the oven. Energy from the oven makes the cake expand and the raisins move apart. (See illustration below.)

If raisins could see, each raisin would see the other raisins moving away. In the case of the raisin cake, outside energy is needed to heat the oven so the cake can expand. If the universe is expanding the same way, where is the outside energy coming from? What kind of energy is making the universe expand? One popular hypothesis says the universe is filled with this mysterious stuff called dark energy. Sounds like something out of a sci-fi novel, doesn't it?

If energy in our universe is conserved, it seems inconceivable that a fixed amount of energy could cause the universe to expand at an exponential rate. Perhaps this "dark energy" or vacuum energy is being piped in from from a higher dimension or another universe through a wormhole. Let's do the math and see where it leads us.

We begin our investigation with a metric that includes an exponent scale factor to the power of Hubble's constant (H) times t (time). The x's are dimensions, c is light speed and v is velocity. From here we derive something that resembles the Lorentz factor:

Velocity squared is equivalent to Gm/r. (G is Newton's constant; m is the universe's mass and r is the universe's radius.) We make a substitution below:

Next, calculate the derivative of a sphere's surface area. That gives us 8pi times r. Multiply both sides of the above equation by that figure:

We know that mass is the universe's energy (E) divided by c squared. We make a substitution below and get an expression that resembles the right side of Einstein's field equations.

The 8pi's cancel and we end up with a formula that allows us to calculate the radius of the universe as a function of time:

Notice we can hold energy (E) constant and the universe's radius will increase anyway. The radius increases when time increases. No added outside energy is needed. Say bye bye to the wormhole pipe and the hidden universe or dimension.

So what the heck is going on? This is a real mystery. Time goes by and the universe simply expands at an exponential rate! Let's delve a little deeper and see if we can unravel this conundrum. To make things easier let's set all the constants to one:

Let's take a closer look at energy (E). What is it exactly? It has a mass aspect; it also has a spacetime aspect: the v^2 is a distance squared over a time squared. Let's assume the distance is the universe's radius. Let's assume the time is the proper time or the universe's time rather than the observer's time (t). Set 1/2m to one, and we get the revised equation below:

Now we can solve for the universe's proper time:

Notice we dispensed with r by substituting e^t (e^t*e^t=e^2t). If you plug in some numbers you will discover the universe's proper time changes faster than the observer's time. This gives the observer the impression the universe is expanding at an exponential rate when in fact it is expanding at a steady rate. If the universe's radius grows proportionately with its proper time, energy is conserved.

Since the radius equals the exponent, we can derive the velocity of the expansion:

When the observer's time increases, the velocity of the expansion increases, i.e., the expansion is accelerating from the observer's point of view. Also notice if we substitute a shorter radius, we get a slower velocity. This is consistent with the universe expanding at a slower rate when the observer looks a short distance, and the universe expanding at a faster rate when the observer looks farther. (Another common equation used is v = Hr.)

If we assume relativity is the culprit, our expanding universe no longer seems mysterious.

Update: Here is a mathematical proof showing how the universe's energy is conserved:

Equations 1 and 2 below show the universe has no choice but to expand. Vacuum pressure is never relieved as the universe expands, so it keeps expanding. The remaining equations show how the overall energy is conserved. As volume and entropy increase, temperature decreases.