The Edge question for 2012 is “What is your favorite deep, elegant or beautiful explanation?” There are lots of interesting answers given but my favorite is not included. That’s partly because they did not have the foresight to ask me, but never mind I can post it here.

My favorite explanation is Stephen Hawking’s argument for why the area of a black hole horizon increases. This is a very non-trivial result from gravitational dynamics and yet the explanation can be summarized almost rigorously in words without any equations. Not only is it elegant but it is also very deep since it leads to the idea that the area of a black hole is related to entropy, a hunch that Hawking later clarified with his theory of black hole radiation and thermodynamics. This in turn led to the information loss paradox which was explained by the holographic principle. It is a remarkably persuasive train of thought that takes us on a journey far beyond anything that experiment or observation can currently reach, an amazing demonstration of the power of the human mind. I am sure I do not need to describe the details of these ideas to most readers of this blog. Any of you that are not familiar can read about them in Wikipedia when it comes back from its anti-anti-piracy sulk.

Anyway, here is Hawking’s beautiful explanation quoted from “a Brief History of Time”

“I had already discussed with Roger Penrose the idea of defining a black hole as the set of events from which it was not possible to escape to a large distance, which is now the generally accepted definition. It means that the boundary of the black hole, the event horizon, is formed by the light rays that just fail to escape from the black hole, hovering forever just on the edge. It is a bit like running away from the police and just managing to keep one step ahead but not being able to get clear away!

Suddenly I realized that the paths of these light rays could never approach one another. If they did, they must eventually run into one another. It would be like meeting someone else running away from the police in the opposite direction – you would both be caught! ~ But if these light rays were swallowed up by the black hole, then they could not have been on the boundary of the black hole. So the paths of light rays in the event horizon had always to be moving parallel to, or away from, each other. ~ If the rays of light that form the event horizon, the boundary of the black hole, can never approach each other, the area of the event horizon might stay the same or increase with time but it could never decrease because that would mean that at least some of the rays of light in the boundary would have to be approaching each other. In fact, the area would increase whenever matter or radiation fell into the black hole. ~ This nondecreasing property of the event horizon’s area placed an important restriction on the possible behaviour of black holes.”

If I try to imagine what happens if a photonic planar wave front enters the black hole area,from one side,

Then there is

A: Photon energy absorbed by the black hole,

B. Photon energy not absorbed but curved ore better lensed by the black hole

C: Photon energy which is forced to stay and accumulate at the Black hole horizon

Now we may assume that from all sides such photonic wave fronts will enter this area and accumulate at the horizon.

This accumulating energy wil be a sort of photon smasher and perhaps even create particles like positrons and electrons.

So: Entropy decrease and annihilation. and radiation and even perhaps quarks

Regrettable that they did not ask me either about deep and fundamental and elegant ideas: I would have had a lot of them and all of my own;-). In any case the geometric form of second law is this kind of idea. In the following I provide an explanation of its generalization and guarantee that no one understands it;-).

In my own alarmingly private Universe;-) black-hole horizons are replaced with light-like 3-surfaces as orbits of what I call partonic 2-surfaces and defined as the surfaces at which the signature of the induced metric of space-time changes from Minkowskian to Euclidian. Hawking’s argument applies more or less as such to them and the analogs of black-hole interiors have Euclidian metric signature in TGD Universe.

For elementary particles p-adic length scales characterize the sizes of wormhole throats. Also the hierarchy of Planck constants gives a hierarchy of sizes. These hierarchies seem to be closely related. There are good reasons to believe that the p-adic prime characterizing the quantum arithmetics assigned with the particle divides the integer characterizing the (effective) Planck constant.

In the case of ordinary elementary particles the braids at wormhole throats carrying fermions and antifermions have only one or two braid strands. When the number is higher one obtains anyonlike states. As the number of strands becomes large these correspond to what one might call black-hole like states but also ordinary many-anyonstates might be example of this kind of states. If the size of wormhole throats and also the braid strand number increases during cosmic expansion, the geometric form of second law holds true.

Dear Phil,

you must check some of the talks in Madrid.

http://motls.blogspot.com/2012/01/madrid-ift-inaugural-conference-slides.html

For example, David Gross showed your combination chart – without any negative extra comments (except that the graph wasn’t authorized). ;-)

LM

Excellent, thanks for the heads up.

Wow, Phil, you’re up there with the big boys if David Gross has a transparency with “Philip Gibbs” on ;)

So, the really cool guys like Your plots and You can forget about the other sourballs grumping about Your analyzes … ;-)

Unfortunately I have to remind myself that it is just an approximate Higgs combination that any theorist could so. When Gross recognizes me for pointing out the way to find the right formulation of string theory it will be worth crowing about. :)

Spinoza takes the main prize here with his principle of possibility. I paraphrase: Let mathematics and non mathematics do what they do. No cosmic censor or enforcer is needed.

The Bianchi identity should definitely be mentioned in any such discussion. Oh why are neither Spinoza nor this mentioned on Edge?

Start from Gibbs classical expression for an equilibrium thermodynamic system with area A

dS = (gamma/T) dA

where gamma is surface tension. Apply the second law and obtain

dA >= 0

This is a result known for close a century.

In fact, the phenomenon of capillary rise is due to that increasing the area of liquid-glass interface lowers the free energy.

The surprise, at least for me, would be if Hawking had proposed the contrary for black holes…

Recall that dF =< 0 is a consequence of the Second law.