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?