## Monday, August 23, 2021

### Susskind: horizon complementarity, second law ban bigly ripped, cyclic, and bottled universes

Bill has persuaded me to read the whole July 2021 hep-th paper
Three Impossible Theories
by Leonard Susskind. Let me say in advance that I saw the abstract last month, sort of agreed with it ("it is nothing new for me"), and I didn't read the body of the preprint which I did now. The body is visually funny because each page (and there are just 13 pages in the PDF file) contains a small number of letters, either because the "paper" is small or because the fonts are large.

OK, Susskind starts by articulating the "central dogma" which says
from the outside, a black hole may be described as a unitarily evolving quantum system operating in the Hilbert space of dimension $$\exp(A/4G)$$
which I modified a little bit (an improvement, I think). This central dogma is especially supported by the gauge-gravity cases of the AdS/CFT correspondence, by Matrix theory, and perhaps by some other arguments, Now, something special which allows us to count the degrees of freedom through the area happens at the event horizon.

An amusing observation that we considered potentially very deep from the 1990s (and I have discussed it on this website occasionally, for almost two decades) is that the cosmic horizons (the boundary of the faraway regions of a cosmological spacetime which can't affect you – an observer defined by a trajectory in the middle – for causal reasons; and which you can't affect, either) seem geometrically qualitatively equivalent to the black hole horizons. When it comes to the relationships between the regions,
the exterior of the Universe behind the cosmic horizon of an observer (the faraway world) behaves just like the interior of a black hole.
The bold face font was used to emphasize that the interior and the exterior are "exchanged" if you compare the single black hole scenario with the cosmological one. This qualitative "spherical inversion" has some consequences. In particular, the "cosmological interior", your causal patch, is bounded and lacks an asymptotic region. That is why the counterpart of the black hole, the "faraway world", is emitting thermal, Hawking-like radiation, just like a black hole does; but it is also absorbing the same amount of this (or analogous) radiation because all of this radiation (the former, emitted one) only moves through the patch for a finite amount of time. For this reason, the "faraway world" doesn't grow and doesn't shrink, unlike black holes which do shrink.

Great, the (less established, relatively to the black hole case) cosmological version of the "central dogma" says that the "faraway world" is similarly describable as a unitarily evolving Hilbert space of dimension $$\exp(A/4G)$$. Taking this principle seriously, Susskind argues that three cosmological scenarios are impossible, namely:
• big rip
• cyclic universe
• universe in a bottle
Concerning the big rip, it postulates a generalized cosmological constant with $$p\lt -1\cdot \rho$$ i.e. $$w\lt -1$$ (which is incidentally said to be suggested by some recent cosmological observations). In this scenario, the cosmological constant grows with time and so does the Hubble constant. The Hubble constant is inversely proportional to the radius of the cosmic horizon which means that the cosmic horizon shrinks! But if it shrinks, this area shrinks and so does the corresponding entropy. It is impossible for the entropy to shrink, by the second law of thermodynamics, so the big rip is impossible.

I like this argument (which has probably been around for decades as well, and I may have said it in the past as well) because it may potentially admit generalizations: Note that the big rip violates an energy condition and this violation is identified with a violation of the second law, through holography or complementarity. It is possible that all truly valid energy conditions may be derived from the second law of thermodynamics by a similar argument.

The second "impossible theory" is the cyclic universe. I have been saying that the cyclic universes contradict the second law since the 1990s, in this case I am certain. In fact, I think it is also written in a popular book by Brian Greene. The oscillations produce some friction, heat, dissipation, so the cycles cannot be quite the same. They are either getting shorter or they are getting longer. If the duration of the cycle goes up (exponentially), then there was a beginning, after all; if they are getting shorter, there would be an end (because the geometric series are convergent). The former situation negates the key attractive feature, the infinite longevity in the past; the second leads to some arguably impossible version of compression at the end.

On top of the old argument, Susskind says that in the cyclic universe, a cosmological horizon (and therefore an area and an entropy) is oscillating as well, up and down and up and down. It is only possible for the entropy to go up, permanent oscillations like that violate the second law for 50% of the time and they resemble a perpetuum mobile device.

The third impossible theory is the universe in a bottle, Farhi-Guth-Guven, the preparation of a new Universe in a laboratory. It's remotely possible within some vague frameworks that merge an informally quantized general relativity with the characteristic quantum effects such as the quantum tunneling (to justify the change of the spacetime topology, something that GR and especially its quantization seems to be tolerant about but you can't be quite sure whether a particular topology change has a nonzero probability in a given situation).

Here, the argumentation is a bit more complex but Susskind says that it was known to him or Guth or both in 2004. Again, the creation of the bottled universe may lead to a decrease of entropy but I think it is a "subjective entropy" of a cosmic horizon that is considered relevant by a feminist observer, Alice (at least I am pretty sure that Susskind wouldn't allow Alice to make it to his papers if she were not an obnoxious frigid feminist). She (her whole body? Is tunneling likely to preserve composite bodies, in the sense of conditional probabilities?) may tunnel from the bottle outside the bottle or vice versa, one of these processes would lead to the shrinking of the cosmic horizon around her, and it's therefore forbidden. I am not motivated enough to investigate which one.

This usage of the "cosmic central dogma" is even less reliable than the previous one because one is not talking about the area of a "fixed horizon" somewhere in the Universe but the area of a horizon attributed to someone "who can jump to faraway places just by being a victim of a tunneling event". It is less reliable but I personally think that even this usage of the principle could be justified.

But I feel uncomfortable about another issue, namely the low probability of the tunneling. He argues that the bottled universe must be banned because something contradicting the principles could happen, with a very low probability. Is it enough to rule out the bottled scenario? We could say that most of the principles of physics may be violated either with some insanely low probability or for a tiny amount of time. The tunneling effect itself may be described as a very short-lived violation of the energy conservation law. Like a leftist who wants to pay anything to prevent a scenario, even if it has a negligible probability (like a problem with the climate; or the extinction of a nation through Covid), Susskind thinks that the laws of Nature are "obliged" to do "prevention" and prevent even tiny probabilities of some events. I am not sure.

In other words, the reason why this argument could be wrong is that his calculation of the tunneling odds for Alice could be just an approximation and the precise one, in a full theory of quantum gravity, could yield precisely zero, and therefore no conflict with the principles. A tiny number and zero are very close which means that it is possible that his basically semiclassical calculation of the tunneling probability could be close to the exact result and there would be no material conflict with locality.

At any rate, these are three examples of the "principled thinking". Adopting some new principles in physics may lead to answers about seemingly very different questions. Theoretical physics has incorporated a massively profound and productive industry generating these clever thoughts at least since the golden years of Einstein and Susskind's playful preprint is a modern continuation of this approach.