Information content in black holes

Question

Bekenstein-Hawking formula for entropy of a black hole tells us that information content in a black hole is proportional to its area which is in fact proportional to the mass^2 of the black hole. The information content before the formation of the black hole can be different which has nothing to do with the mass. Is there any kind of information loss during the formation so that we get two black holes with an equal amount of information. Is the information due to the fields lost in the process? So we get just the bare information which comes from the mass.

Answer 1

Is there any kind of information loss during the formation so that we get two black holes with an equal amount of information.

If I'm understanding correctly, this sentence is describing two different sets of initial conditions leading to two black holes with the same area, and therefore the same entropy.

You're using "information" as if it were a synonym for "entropy," but they aren't the same thing. For example, information is always conserved in quantum mechanics, in the sense that the time evolution is unitary, but entropy increases. Entropy can be interpreted as a measure of the amount of information in a system, but only when we take into account the coarse-graining of the phase space.

Gravitational collapse to a black hole doesn't involve a loss of entropy, it involves a gain in entropy. It's not particularly mysterious per se for two different initial systems, with two different amounts of entropy, to evolve into final states with the same amount of entropy. For example, suppose we take two different samples of an ideal gas, in two different nonequilibrium states, but both with the same number of molecules and both confined to the same volume. They will both gain entropy, and will both end up in the same maximum-entropy macrostate.

We don't have a theory of quantum gravity, but the rough picture is probably that after the black hole has formed, there is a large amount of energy present in its microscopic degrees of freedom, and this is where the entropy resides.

Answer 2

Infomation Content in Black Holes

Your question is essentially (if i interpret it right) can stars with differing sets of initial detailed information--and thus differing information entropy-- converge to identical black holes?

The short answer is Yes --provided the no hair hypothesis is correct.

The longer answer has to take account of the measurement process applied to a Black Hole. By definition, measurements cannot be performed by an external observer to elements within the interior of a black hole, beyond the event horizon. We thus do our best to characterise the Black Hole by partitioning the event horizon of the Black Hole with elements each of minimal area (following Loop Quantum Gravity) proportional to the square of the Planck length $l_P^2$ and let N be the total finite number of partitions.

The ‘no hair’ hypothesis states that a Black Hole can be completely characterised by its classical spin, mass and total electric charge. Thus there is no preferred location on the event horizon, so that each partition element must have the same weighting. The von Neumann 'entanglement ' entropy of this partition is thus given by;

$- \sum\limits_1^N {\frac{1}{N}\log \left( {\frac{1}{N}} \right)} = \log N = \log \frac{S}{{l_p^2}} = \log \frac{{{c^3}S}}{{\rlap{--} hG}}$

with S the surface area of the black hole, which is similar to the Beckenstein-Hawking formula. This relates to classical relativity (ignoring the cosmological constant) via the macro- level Clausius definition of entropy E;

$\oint {\frac{{\delta Q}}{T}} = \Delta E$

where, integrated over a Carnot cycle, a change in the energy tensor results in a change in entropy corresponding to a change in space-time curvature as measured by the Ricci Tensor.

Answer 3

Let's say we have an egg and a broken egg. As we know, the broken egg has more entropy.

Now we put the two eggs in two bowls and whip them hard.

Voilà, the eggs become macroscopically the same. One of the macroscopic properties is entropy.

So we can conclude that gravity is very good at beating up matter.

And matter has a limit of how beat up it can be. (Bekenstein bound)

So, the two aphorisms above are the answer to the question of why two black holes with the same mass have the same entropy.