How to Falsify the Holographic Principle

Does the surface of a black hole contain its information? Is our universe really a hologram? In the video above, Leonard Susskind makes the case, but Karl Popper would no doubt scream, "Where's the evidence!" So to keep Karl Popper from rolling over in his grave, we will attempt to falsify the holographic principle.

According to the holographic principle, the maximum amount of information in a region of space is proportional to the area of the region--not the volume! That seems counter-intuitive. It's as if a black hole's surface area, for instance, is like a holographic plate storing all the black hole's information. The black hole entropy equation is based on this assumption or vice versa:

However, we could make up a new principle that states the following: The maximum amount of information in a region of space is proportional to the area of Cleveland Ohio. We could postulate that all information from each black hole ends up in Cleveland. Here's the equation:

Granted, using the area of the black hole seems more convenient, but as we shall demonstrate, the area used in the equation is arbitrary. Let's assume for five milliseconds that the black hole's surface area is not really a holographic plate. Why is the information proportionate to area rather than volume? To answer this question we need to review some basic laws of motion. Consider an acceleration vector. It has units of distance per time squared:

Now, take note that the distance D is not any particular distance; it's just a unit or dimension. In fact it's one dimension, not two, not three. OK, suppose there's a particle accelerating along one dimension of space. It propagates a distance of x. That gives us the following:

On the right side of equation 4 above, we have x times D--that makes an area:

What exactly is this area A? Is it the area of a black hole or Cleveland? It's not the area of anything, but if we want, we can pretend it is the area of a black hole. The choice is completely arbitrary. Now let's take this non-existent area A and place it into an entropy equation:

The area in equation six is not any particular area, including a black hole's. However, equation six could be used to model the entropy of a black hole notwithstanding. So it's true that a black hole's entropy or information is proportionate to an area, but it is also proportionate to a volume:

Using a little algebra we derive 11 below:

Looking at 11 we can infer that the maximum amount of information in a region of space is proportional to area or volume. However, to calculate the information using the volume requires we know the pressure (or energy density) and the black hole's temperature as well as the volume. If we know the area, then we know all we need to know to calculate the information. Therefore, a principle involving the area instead of the volume is more convenient--and most likely has nothing to do with holograms.

How Entropy, Temperature and Chemical Reactions Impact Time Dilation

How do we measure time? Normally we take some event that happens over and over again, and, we assign it a time unit. For example, the complete rotation of the earth we call a day. We can take that time unit (or any fraction thereof) and assign it to other events--that may or may not be periodic--such as a chemical reaction, for instance. We say the chemical reaction happened in time t.

What if that chemical reaction were to slow down? Can we say its rate of time has slowed? If we define time as the rate of change, then the change in the rate of chemical reactions, entropy, or the earth's rotation would indeed impact the rate of time.

Consider the famous twin paradox, where one twin boards a rocket that hurls him into outer space close to light speed. According to Einstein, his time slows, he ages more slowly than his twin back on earth. His body's biochemical reactions are slower, entropy is reduced--at least that is the implication.

When he arrives back on earth, he will be younger than his twin brother. But his twin has a plan: while he's out in space, his twin cryogenically freezes himself. His twin will be thawed out by the time he gets back to earth. If his twin's plan works, they will both be the same age. Now, did the twin on earth reduce his time rate? Can the rate of time be reduced by means other than high velocities and mass density? This post shall address these questions, but first, let's define the variables we will use:

Let's start with entropy. If we somehow reduce the rate of entropy, will the time rate also be reduced?

If we take the Boltzmann constant (which has entropy units) and multiply it by the unit-less Lorentz factor's squared reciprocal, we derive equation 2 below:

Equation 2 shows that entropy is reduced when velocity (v) is increased. The time rate is also reduced. So far it appears we have a correlation between entropy and time. If equation 2 is valid, we should be able to use it to derive a more standard entropy equation:

Equation 6 above confirms the validity of equation 2. So we can say the twin in outer space, traveling near light speed, has reduced his entropy. Now the twin on earth wants to freeze himself, i.e., lower his body's temperature. Will this reduce his entropy? Equation 8 below confirms that it will. From 8 we derive equation 12 which shows lowering the temperature (T) reduces the time rate.

According to equation 12, the twin on earth will age more slowly. Does this imply the rate of his body's biochemical reactions will slow down?

Equation 13 is the Arrhenius equation, where k is the rate of a chemical reaction. When temperature T is reduced, so is the rate of the chemical reaction. From 13 we derive a new entropy equation at 19:

Equation 19 tells us that when the rate of a chemical reaction is reduced, so is entropy. And the reduced chemical-reaction rate reduces the rate of time:

So the age difference between the twins could be nil when the space twin arrives back on earth. If the earth twin cryogenically freezes himself, he may be younger than the space twin. Both twins have found a way to reduce their respective time rates, they have found ways to time travel into the future. Here are four ways to reduce the time rate:

Why Entropy Happens

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.

What We Didn't Know About Time Travel Could Surprise Us

According to Hollywood and even some scientists, going back in time means seeing history repeat itself. We worry that if we interfere with the past, the future might unfold differently, unexpectedly and even disastrously.

Imagine Hitler winning World War II or your parents never getting together, so you are never born--all because you spit on the sidewalk in the year 1933, which set off a chain of events that changed the future.

What I described above is rather nonsensical if we take entropy into account (s=change in entropy; k=Boltzmann's constant; ln is the natural log; omega=number of ways particles can arrange themselves):

If our most common belief about time travel is correct, then entropy would have to shut down, so history could repeat itself exactly as before. But then we might ask, why should history repeat itself exactly as before (assuming we don't interfere)?

To satisfy our curiosity, what would happen if entropy remains intact during our time-travel excursion? Imagine a universe with one particle in it. It can go either right or left as it passes through time. The probability (p) is 0.5 for each move just like a coin toss:

The particle makes three moves, following a path that is one out of eight possible paths. The path it takes is outlined in red:

The red path represents its history. The entropy starts with a single possibility and evolves into eight possible paths.

Now let's find a wormhole and travel back in time to the particle's starting point and watch its history unfold again:

Oh wow! History failed to repeat itself. We didn't interfere. We were good. We watched from afar ensconced in our time-travel VW bug. Relax, it's not our fault. Entropy was just doing its job. There was only a one-in-eight chance history would repeat itself and a seven-out-of-eight chance it would not. I mentioned earlier that we didn't interfere, but we did go back in time and that in itself is a kind of interference that could possibly alter history.

OK, so maybe going through a wormhole is not the best way to go back in time. Sometimes going back in time is described as a movie going in reverse. If we flip a cosmic projector switch and make the particle go back in time, it should follow the red path back to its starting point:

Oh no! What happened? The particle didn't behave like a backward-running film. It took a different path back. And why not? It can go left or right at any node along the path. To get the equivalent of a backward-running flick, the particle's probability distribution would have to exit the theater.

Backward time travel, as we imagine it, requires an entropy of zero: one path the particle can take and not the usual eight. If we treat the particle's present as a single starting point, the change in entropy would evolve as follows:

Just for fun, let's assume the particle's probabilities of going left or right stay intact when time is reversed. Its past would most likely be different than expected. The past it remembers is only one of eight possible paths to its present time.

Now apply what we've learned to your past. Imagine the implications! You go back in time, but you see a different past than you remember. You think to yourself, "This is not my life!" Your past self looks different, like a brother or sister--because you may not be the same person or sex! A different sperm cell beat all the others to your mother's egg. Your parents still gave you the same name (assuming your sex is the same) but you're watching the life of a different person who took your place because you just had to try out that bloody time machine!

Since your past is messed up anyway, you might as well interfere with it, i.e., take advantage. You read in a history book that the racehorse, Fools-errand, won the Kentucky Derby, so you decide to place a bet, but don't place that bet--it's a fool's errand.

For more on this topic, click here or here or here.

Entropy Succeeds Where the Ideal Gas Law Fails

The universe is getting bigger and colder. We are told that one day it could suffer a heat death. By contrast, the early universe was smaller and hotter. All this implies an inverse correlation between the universe's average temperature and its size.

We might be tempted to model it with the Ideal Gas equation. We could think of the universe as a big balloon filled with gas. As the volume of the balloon increases, the pressure decreases along with the temperature. Below is the equation (P=pressure; T=temperature; V=volume; R=universal gas constant; n=kilomoles of gas):

Right away the equation bursts our balloon. When volume (V) increases, pressure (P) decreases but the temperature can remain constant. So why isn't our universe the same temperature as it was in its early years? Its volume is increasing; its pressure is decreasing; yet, the temperature isn't remaining constant.

So what's the problem with the Ideal Gas Law? We know that pressure is equivalent to energy density, so let's plug energy density (E/V) into the equation in lieu of pressure (P):

The second equation above reveals the problem: The volume (V) in the denominator cancels the volume in the numerator. Change in volume in this case has no effect on the temperature.

We know that entropy increases as the universe expands. Perhaps lower temperature is a function of increased entropy? Let's take a look at the entropy equations (E=heat or energy; s=change in entropy; T=temperature; k=Boltzmann's constant; omega=number of different arrangements of particles):

If we equate the two equations above, we can derive a temperature equation:

Temperature appears to be a function of energy (E) over the number of ways particles can arrange themselves (omega). Once again, temperature is not a function of volume, but it seems like it should be. We know from experience that a candle can heat up and maintain the temperature of pickle jar, but a candle in an aircraft hanger can be a much colder space.

Let's explore the omega variable and see if we can make it a function of volume. Imagine a single particle with one state in a single space (see first diagram below). In that case, omega equals one. Now double the space. The particle will have two spaces it can occupy (see second diagram below). Omega now equals two.

Entropy has increased due to increased space. We can define omega in the following way:

Below are some examples that include the above diagrams, a coin toss, and dice:

Consider the coin-toss example above. Normally we think of a coin as having two states: heads or tails, so omega equals two, right? Well that's only true if the coin lands on the same spot on your floor because there is no other spot it can land. But suppose you have a big floor and you divide it into a grid of 20 spaces where the coin can land. Omega would then be 2^2 * 20 = 40 states.

Now suppose you have the same floor space, but you now have two coins. Omega becomes 1520 states! We are now ready to put together a temperature equation:

Notice the temperature is now a function of volume because entropy is now a function of volume. If the volume increases, the temperature decreases and so does the pressure (Boyle's Law) if the other omega variables and E are held constant. Remember pressure is E/V. The difference here is that E/V is no longer canceled by V*E/V.

Taking the temperature of the Universe or any other system can be done by this system of equations. The first equation is for systems that have more space than particles. The second is for singularities, high pressure systems, where the energy density is high. The third equation allows you to make a quick-and-dirty calculation. It is similar to the Ideal Gas equation albeit it includes entropy.