In the beginning, there was order, a very hot singularity, but as time progressed the universe expanded and cooled--and became more disorderly. Scientists predict a "big freeze." It's all due to entropy. As you read this blog, entropy continues. Why? That's what we will explore below. First, let's define the variables we will use:
We begin with the partition function, which has the Boltzmann factor, an exponent with a thermodynamic beta power over the base e:
If we want to determine the probabilities of the energies in a system, we make sure the probabilities add up to 1, so we normalize the partition function by dividing it by itself (Z):
However, if we want to model the universe's evolution, we need to make a slight change to the partition function. Instead of using the thermodynamic beta, we use its reciprocal. We also change the i index to a time (t) index:
Also, we want the universe's total energy to be conserved. We know dark energy is increasing and radiation energy is decreasing, so we put together an energy-conservation equation:
At equation 5, notice how an increase in the universe's volume (V) reduces the radiation energy (Er) but increases the dark energy (pV). Multiply the two energies, add a little dark and baryonic matter, and take the square root and we get a constant energy (E).
We define temperature as follows:
As volume (V) increases, the universe's temperature decreases. Equation 7 below gives us the probability of the temperature at a given time t:
A high temperature has a low probability. A low temperature has a high probability. So there is a high probability the universe's temperature will continue to decrease, and a low probability the temperature will increase. Thus, an expanding universe has a higher probability.
Now, let's take a look at entropy. We define it as follows:
We see that entropy increases as temperature decreases--so it has the same probability as temperature:
So why does entropy happen? Greater entropy has a higher probability than lower entropy. We can also say that reverse entropy is possible but less probable. A good example is the one Tyson discussed in the above video. There are pockets of order caused by star energy, so life is possible.
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.
Update: Here's another take on this topic: Is dark energy adding energy to our universe? If so, where is the extra energy coming from? How about our universe? The equations below show dark energy increasing at the expense of radiation energy. Overall, energy is conserved.
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:
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.
