If we take bigger and bigger volumes

[Dmitry67]

(Holographic principle)

Volume of a sphere is proportional to R^3
However, max amount of information inside is proportional to it's surface, to R^2
So information density is proportional to 1/R

It means that if we take bigger volumes, the content inside appears to be correlated with the outside, so some of the chaos we see inside (and interpret as information) in fact is predetermined by the environment.

It looks logical, but...
What is bugging me, if we take R --> INF, we find that density of information = 0.

 

[mrspeedybob]

However, max amount of information inside is proportional to it's surface, to R^2

What is the justification for this statement?

 

[Dmitry67]

It is well known:

http://en.wikipedia.org/wiki/Holographic_principle

 

[mrspeedybob]

I'm not saying it's untrue but the derivation they have on wikipedia doesn't hold water.

In the case of the black hole anything that falls across the event horizon has an effect on the horizon as it crosses. Information about that matter is carried away from the black hole as gravity waves and hawking radiation, there is no need to suppose that the information about everything that ever crossed the event horizon is permanently encoded thereon.

In other words everything that crosses the event horizon is happening in 2 dimensions over time, the information about it is radiated away from a 2 dimensional surface over time.

As for the information contained in the matter that initialy collapsees to form the black hole... We don't need a black hole to demonstrate that gravity is incompatible with the 2'nd law of thermodynamics. Imagine 2 rigid bodies floating freely in space. There are twice as many possible configurations of that system then there would be if there were only 1 body. If gravitational attraction causes the 2 bodies to stick together then the 1'st scenario has become the 2'nd.

I admittedly don't have a perfect grasp of the behavior of entropy so if I'm typing nonsense please educate me.

 

[Dmitry67]

Frankly, I don't know how Holographic principle works is non-eucledean spacetimes. Especially, BH