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.
