Resolving the Black Hole Information Paradox: How Information is Lost and Conserved

According to the current paradigm, quantum information is conserved. With perfect knowledge of the current universe it should be possible to trace the universe backwards and forwards in time. This principle would be violated if information were lost. When information enters a black hole we might assume the information is inside, but then black holes evaporate due to Hawking radiation, and the black hole's temperature is as follows:

As the black hole evaporates, its mass shrinks and its temperature increases. Take note that equation 1 fails to tell us what information went into the black hole, so looking at the final information (remaining mass, momentum, charge) pursuant to the no-hair theorem we can't extrapolate that data backwards and determine what information went into the black hole. This is known as the Black Hole Information Paradox.

Many hypotheses have been set forth that attempt to resolve this paradox. One popular one is the holographic principle (T'Hooft and Susskind). Unfortunately, it is easy to punch holes in this one. You can read about it by clicking here.

Another common proposal is the information goes inside the black hole, then through a wormhole into another universe. Personally, I don't care for this one, since it requires the establishment of another universe (good luck!). Then there's the explanation that begins with a shrug and ends with a sigh: the information is lost.

Of course I'm not without a brainstorm of my own, which is why I'm now scribbling. It occurred to me that maybe there's at least two kinds of information: information that is conserved and information that is not. This random thought popped into my head when I was working on the following math proof:

The proof starts with the absurd claim that if 'a' doesn't equal 'b' then 'a' is equal to 'b.' Let's suppose 'a' is information. At equation 2 it is defined. However, by the time we get to equation 4, 'a' becomes undefined. Zero times infinity can equal any number, so the definite information we started with appears to be lost. Although, unlike black-hole information, by the time we get to equation 7, 'a' is defined again, but this time it is defined as 'b.'

What we can take away from the proof above is specific information is not conserved, but information overall is conserved. The information changed from 'a' to undefined to 'b.' Unfortunately, even though the information is conserved, we can't tell by looking at 'b,' that it was once 'a.'

Below is another example of what I'm scribbling about. Start with two distinct binary numbers. Let's pretend they enter a fictitious binary black hole and come out identical (zeros on the left, ones on the right). Now we put them into a cosmic hat. You reach in and pull one out. Can you tell whether it used to be 1010 or 0101? I don't see how. The information is conserved however--there's still the same number of ones and zeros.

Rather than say information is lost, perhaps it is more prudent to say it is undefined. In the case of a black hole, most of the information becomes undefined. Having perfect knowledge of it doesn't help us trace it back to its defined state prior to entering a black hole.

There are many examples in everyday life where we can observe information evolving from defined to undefined. Write a message on a blackboard. Erase the message. The chock that made up the message is now smeared onto the eraser. Give the eraser to a physicist and see if he/she can tell you what your unique message was. At this point, the chalk has mass, for instance, but chances are excellent that physicist won't be able to know your unique message. That information is lost. It was defined, now it is undefined (except for some basic properties like mass, etc.)

The no-hair theorem reminds me of brown paint. Imagine some masterpiece paintings, each with a unique set of information. The paint on each painting is scraped off the canvass and mixed in a bucket of paint thinner. At the end you have several buckets of brown paint. If they are mixed up and you choose one at random, can you tell which painting it came from? Probably not. It's another case where defined information becomes less defined--so it may also be true even at the quantum scale. For example, according to quantum field theory, particles and their unique, well-defined properties are excitations of fields where the information is kind of blurry or undefined.

Imagine an electron-positron pair popping into existence. The electron is spin up, the positron is spin down. They annihilate. Is it possible to look at the resulting photon and know it was previously a spin-up electron and a spin-down positron? For all you know the electron was spin-down and the positron was spin up. Yet another case of information evolving into something where you can't know its previous state. So why should we be surprised there's an information paradox if we believe perfect knowledge of the current state of information allows us to trace it backwards and forwards in time?

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.

Debunking Bosonic String Theory's 26 Dimensions

About one hour into the above video, Leonard Susskind (one of my favorite professors) shows how string theorists mathematically derived 26 dimensions. Are you thinking what I'm thinking? If there are really 26 dimensions, then why do most string theories have fewer dimensions? This question made me watch the video with a CSI-investigator disposition.

The original goal was to get a minus-one ground state, so when a creation operator is applied, the photon will have zero mass. Imagine an x-y plane or system going down the z axis with momentum p. The frequency of the ground-state oscillator is n/2. Here is the desired equation and result:

If the result is infinity plus -1, that's OK, according to Susskind, it's just absorbed into the momentum along z and can be ignored. After some fancy math, including a Taylor expansion, the final result is ...

According to Susskind, if we drop the infinity (that lop-sided 8) we end up with -1/24. We need -1, so multiply by 24--that gives us 24 dimensions. Add the z axis and time, and that brings us to 26 dimensions. Voila!

But here's the rub: What if the result was the desired -1? How many dimensions would there be then? Let's see ... multiply by 1. That gives us one dimension. Add the z-axis and time--that brings us to two space dimensions and one time dimension!

Obviously the math and/or logic is seriously flawed.

Since it is OK to do away with infinities, we could take another approach. If you stop and think about it, we ignore infinities whenever we measure anything. How long is a meter? It's an infinite number of points. How big is a circle? It contains and infinite number of points. Everything we measure would be infinite if we didn't invent arbitrary units of measurement. We can do that here:

So instead of having 1/2 of an infinity, why not have 1/2 of a unit of ground state? And, while we're making changes, why do the frequencies of the ground state oscillators have to be a positive number when the ground state is negative? Why not be consistent? You'll note I made both sides of the above equation negative. The result is -1/2. Multiply by 2, add z and time--and voila! We live in a universe with three space and one time dimension after all!

Update: Here is another way to arrive at four dimensions instead of 26. It's based on the premise that the desired result is infinity plus -1/2. This ensures that within the infinity, there is a negative ground state.