## Monday, May 27, 2013

### Leonard Susskind: Holograms and Indestructible bits

For those of you who haven't watched this video by Leonard Susskind on Holograms and Black Holes, I suggest watching it (a few times.)
Below are some of my favorite lines from the lecture, as well as some comments on the ideas.

(1) Dr. Susskind has created what he calls the Negative First Law of Physics:
"Bits are indestructible"

By this, he means that information can't be destroyed. To paraphrase how he puts it, "Can you erase a bit?"   Answer: "From the computer..yes. But you eject it out of the computer into the environment, the bit does not get destroyed." This means that information can be spread out or brought together, but it can't be destroyed.

While I completely agree with Susskind's statement that information can't be destroyed, I think that this is just a partial restatement of the 2nd Law of Thermodynamics (as stated by Joe Rosen.) Dr. Rosen stated that "The symmetry group of the cause is a subgroup of the symmetry group of the effect. Or less precisely: The effect is at least as symmetric as the cause." According to Dr. Rosen, the symmetry group of the universe increases or remains the same. In fact, the symmetry group of the past is embedded into the symmetry group of the future, and hence you can't destroy information. This is why I think that Susskind's Negative First Law is just part of the Second Law of Thermodynamics...which is really just the Symmetry Principle, as stated by Joe Rosen.

But there's more to the 2nd Law of Thermodynamics than what is stated in Susskind's "Negative First Law." The 2nd Law states that the symmetry state of the universe can increase. This means that we can increase the amount of information. Second Law of Thermodynamics is saying that the information in the universe is non-decreasing with time. The negative first law states that information can't be destroyed, so when you combine the two (as in Joe Rosen's formulation of the 2nd Law), you get the following: "Information in the universe can't decrease with time and can't be destroyed." New information can be generated, but old information can't be destroyed. Information might be hidden from us (such as inside of a Black Hole.) But the black hole eventually will evaporate. When it evaporates, it releases the information it grabbed when it "ate my homework" plus it also releases more information about what happened inside of the black holes during its irreversible lifetime. Just like the information in the thermal energy ejected from your computer screen into the environment when you go to a new website, the information ejected from a black hole is in a form such that it is virtually impossible for us to "recreate our homework after the black hole eats it."

(2) So, what is the maximum amount of information that can be hidden inside of a blackhole?

"Bekenstein found that the number of hidden bits of information in a black hole is equal to the AREA of the horizon measured in Planck units...where one Planck unit =  G/(hc3) = 10-66 cm."

It may be odd to think about the area of a black hole because a black hole is a singularity in 4D space-time. But a black hole can't be a real singularity because there is only a finite amount of mass in a black hole. This means that there is only a finite amount of curvature associated with a black hole. And since there is less mass in a black hole than in the mass of the universe during the Big Bang, then the radius of curvature of the black hole must be still much larger than the radius of curvature during the first few moments after the Big Bang. Just think of a black hole as a dimple in the 3D surface of our 4D space-time sphere. A larger mass black hole is a deeper dimple than a smaller mass black hole and is even deeper than the dimple caused by a planet, but none of these objects is an infinite dimple. And more importantly, the dimple can't go deeper than the current average radius of our 4D space-time sphere. For example, the dimple on a golf ball may look deep for bacteria staying near the edge of the dimple, but the dimple's depth is small compared with the radius of the golf ball. But a golf ball is a 3D object, and the area of the dimple is not exactly analogous to the "surface area of a black hole." Area is a weird term to describe a dimple on a 3D surface of a 4D sphere. As such, it's probably better to just speak in terms of the mass (rather than the area) of a black holes (ignoring its charge and spin), and to say that the amount of hidden information (entropy) is proportional to the mass of the black hole.

(3) But all of the information in Susskind's lecture leaves me with nagging questions. If you can't destroy information, if the 2nd Law of Thermodynamics says information increases, and if the laws of physics are time reversible, how do black holes increase the entropy of the universe? (or how do any other processes increase entropy?
I'm left with the still speculative feeling that it's the weak nuclear force of nature that increases the information in the universe.
The laws of gravity, E&M and SNF don't seem to be able to generate new information, i.e. you don't gain new bits of information by colliding particles when there's only gravity, E&M and SNF. For example, entropy doesn't seem to increase when we only have these forces in action (such as in superfluid helium, photons from distant galaxies, and superconductivity.) We need a force of nature that can increase information without destroying old information.

What this would look like is the following:
Imagine that the laws of physics work like a computer code with rules on how to change the code.

Starting information looks like the following strings:  00101010, 01010, 110, 1110001010101101000010101010010

Rules for changing the information:
(1)  01 must be exchanged for 10
(2)  001 must be exchanged for 001
(3)  000 must be exchanged for 0000

If you start with a complicated enough string of information and follow the rules above, then the string length will increase with "time."

Now, let's get more complicated. Let's assume that each line in the code actually represents a symmetry group (i.e. a permutation between exchangeable particles.) The code would now look the following  (where each number N represents the permutation group S(N).)  For example, N =2 means the permutation group of between 2 particles. The size of S(N) is N! (N factorial.)

Starting information:    12123311     (the actual number of symmetry group is 1+2+1+2+6+6+1+1 = 20)

Rules for changing the information:
(1)  12 must be exchanged for 21
(2)  23 must be exchanged for 32
(3)  123 must be exchanged for 124, 132 must be exchanged for 142, 312 must be exchanged for  412, etc...  (i.e. this rule is non-abelian.)
(4) 124 must be exchanged for 12345 (or some other rule like this)

Since the symmetric group of 4 permutations is larger than the symmetric group for 3 permutations, the total number of symmetries increases, and the information about the present state contains information about the past state.

Gravity, E&M, and the strong nuclear force do not have the capability of increasing the number of symmetries (i.e. increasing the information) in the universe. The questions really is: how does the weak nuclear force increase the amount of information in the universe (i.e. increase the permutation symmetries in the universe.) Ultimately, my guess is that gravity, E&M and the strong nuclear force boil down to rules such as (1) and (2) in the examples above, whereas the weak nuclear force is like rule (3) in the examples above.

Since every symmetry group is a subgroup of one of the permutation groups S(N), then we have the capability to produce lots of different looking codes if we follow the rules above (and if you can always generate N+1 from some combination of N, N-1, N-2, ...,2, and 1.) This means that we could eventually make a code that contains the Monster or Baby Monster symmetry group just by following rules like the ones above.)

So. to summarize, Dr. Susskind raises a lot of interesting questions about black hole entropy in this lecture, but I think that it's important to highlight the fact that the 2nd Law of Thermodynamics is more than "Information can't be destroyed." The 2nd Law of Thermodynamics ultimately is about how information increases, and how that increase in information makes it impossible to turn thermal energy into work with 100% efficiency. We still need to figure out why and how the weak nuclear force is capable on increasing information.