# Maximum Density that We Can Store Information at?

I was informed that:

There is a maximum density at which we can store information. For a sphere with surface area A, the maximum information that can be contained within is equivalent to the maximum entropy of a sphere of size A, which is given by $$S_{max} = \dfrac{A}{4l_p^2}$$ where $l_p$ is the Planck length and Boltzmann constant is set to $1$. Incidentally, that's the equation for the entropy of a black hole.

Is this true? If so, why or how does it work? Why is the Boltzmann constant set to 1, and how does that relate to the Planck length?

## 2 Answers

Yes, it is true.

The Planck length is defined as $$L = \sqrt{\frac{G\hbar}{c^3}}$$ which, in the real world, happens to be equal to $1.616\times 10^{-35}\,{\rm m}$ (meters). In everyday life, we use units like the SI units – based on kilograms, meters, second, kelvins etc.

But adult theoretical physicists often use smarter, more natural units chosen so that the numerical value of several universal constants, namely those below, is equal to one: $$c = \hbar = k = \epsilon_0 = 1$$ and sometimes $G=1$, too. There is no obvious relationship between the Planck length and the Boltzmann constant – the usual formulae for the former don't even include the latter because the former is a non-thermal concept. The only relationship is that both of them like to be set to one by adult physicists.

At any rate, if one tries to compress too much information (imagine memory chips) to too small space, the information has to be carried by matter which is massive and gravitationally attract. If the density increases above the density where a star would collapse to a black hole, any piece of matter will collapse, too.

The black hole carries a huge entropy which is – because the black hole is the ultimate stage of macroscopic evolution and because the entropy never goes down (the second law) – the maximum entropy that a localized object of the same mass or the same size may have. The black hole entropy (information it carries in the invisible "atoms" that the black hole is composed of) is equal to $$S = k \frac{A}{4L^2} = k\frac{Ac^3}{4 G\hbar}$$ which, by the choice of units I mentioned, physicists often simplify as $S=A/4G$ or even $A/4$.

The black holes just can't be beaten in the amount of information, assuming that something else is kept fixed.

The fact that the information can't be any denser is also the basis of the holographic principle whose most explicit and mathematical incarnation (or proof) is the AdS/CFT correspondence.

k and c can be set to 1 in a more real sense than the other constants because they do not have any fundamental units like length/time, mass/energy, or charge. By relativity, c's length over time cancels, but you should always conserve the sqrt(-1) that goes with time so its unit is 1/i. In other words, whenever you see "seconds" in units, like G and h, replace it with meters/i/c. To be precise, Einstein's book "Relativity" in appendix 2 says meters = i c seconds.

k is units of heat energy divided by temperature which is a kinetic energy constrained to a strict distribution profile, so it is also unitless. Temperature is a measure of energy, period. Making these substitions into the equation given by the other answer and you get G~ m/kg/-1 and h~ -kg m/i, c^3~ -i, and k~ -1/-1. I converted energy to mass by E=mc^2 which has corrected modern units of E=-m due to the i^2 from c^2 (cosmology agrees), which gives the modern corrected units of a black hole to be

S ~ -1/4

or

S= - A/4

Since cosmology agrees with the equivalent unit derivation procedure for E = - m, I would assume it agrees black holes are a negative entropy, but I don't know what they say, I'm just using the correct units.

Making the assumption that the 4 refers to the dimensions of space-time, this states that entropy for all black holes of any radius is proportional to a negative dimension, whatever that means. You can have negative entropy by making use of -1=e^(i pi).

Our brain contains 6 layers of neurons in the cortex that might be what allows us to use 6 degrees of freedom in compressing reality, so it might be why we perceive 3D space and 1D time (6 variables are needed to specify movement for translation and rotation and this might force concepts of mass and spin, according to someone a lot smarter than me).

Therefore, the dimension would be a variable of compression. So a negative dimension would be a positive variable of decompression of data. Not the decompression or data itself.

The only other interesting thing I could find is someone saying that -1/4 is the sum of the negated infinite series -(1-2+3-4...)

I personally consider all integers above 1 in physics as symptoms of something we do not yet understand, even squares like E=mc^2. But notice that if you add this series up as you go it is alternating between 1 and -1, which I'll accept as something possibly important. It might be a way of describing the singularity of black holes. But infinite series math is tricky, and adding them up as you go is a statistical adjustment (data reduction of the details) that needs justification. Adding again like this would be 0, not -1/4. Addition of the whole is not equal to addition of the parts in an infinite series.