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How much Information can you Store on the Surface of a Black Hole?

In the currently best understood theory of gravity, spacetime is a physical entity which curves, bends, dips and expands. In some regions where gravity is so strong, light can't escape and the region has formed a black hole. When one just skims through any physics journal there are a plethora of papers published on the "entropy" of black holes, on Hawking radiation and on "holograms". But why do so many cosmologists believe in the holographic principle? The arguments in favor are really rather convincing, so lets try and follow the logic behind it. The radius of a black hole is

The energy of any given photon that cuts across the even horizon is given as

Normally, the formula includes the Greek symbol Lambda for wavelength but in our case, we are throwing in photons with a wavelength equal to the radius of the black hole. Such that

We use this equivalence because we only want one bit of information to be associated with each photon, a photon with much smaller wavelength will have a position. It's then simple to rearrange some of the equations, to find the change in wavelength of the photon

One small step for a mathematician is sometimes a big step for physics.

What we have now on the left side of the equation is the change in area, which is about equal to the Planck area. 

Planck's area is derived from Planck's constant, the Newton constant and the speed of light. Which tells us that quantum gravitational effects become important at that scale. But also the equation tells us that when a bit of information falls into the black hole and the entropy increases, it increases the area of the black hole. So that the entropy of a black hole is equal to the information of one bit multiplied by the number of bits. 

The entropy then, is the area of a black hole divided by the Planck area. Thus obviously, the entropy of a black hole is proportional to the area, not its volume. We can go further and find the universal upper limit of entropy that can be contained in a black hole. Which was done famously by Jacob Bekenstein. 

Through indirect arguments you can show an equivalent relationship with any region of space, by imagining that region engulfed in a black hole. The amount of entropy in any given region of space cannot be bigger than the area of that region in Planck units. This we call the "Holographic principle" because it suggests that spacetime is a hologram (if it wasn't information would be proportional to volume, not area). This relationship has been beautifully worked out in the AdS/CFT correspondence but that's for another time. 


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