# Weyl Curvature Hypothesis

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1. Dec 4, 2014

### twistor

Hello guys. I was thinking about alternatives to inflation, especially old ones (such as the hawking-hartle state and imaginary time) and I remebered a theory put foward by Penrose, in which his relatively new CCC is based. Called the Weyl Curvature Hypothesis. No idea of what it is. Could you please clarify if it is a viable alternative to inflation, and how it works?
Thank you
Twistor

2. Dec 4, 2014

### marcus

This seems an exceptionally nice question. I hope well-versed people will respond because I would like to understand this better. Will respond in non-expert fashion merely to get the ball rolling, experts please correct any errors of fact or emphasis. :)

http://en.wikipedia.org/wiki/Weyl_tensor#Conformal_rescaling
"The Weyl tensor has the special property that it is invariant under conformal changes to the metric. That is, if g′ = f g for some positive scalar function f then the (1,3) valent Weyl tensor satisfies C′ = C. For this reason the Weyl tensor is also called the conformal tensor. It follows that a necessary condition for a Riemannian manifold to be conformally flat is that the Weyl tensor vanish. In dimensions ≥ 4 this condition is sufficient as well. ..."

Weyl curvature gives one no sense of either spatial or temporal scale, and is called the "conformal tensor" for that reason. If all curvature other than Weyl is zero then you can change the scale at any point arbitrarily and the metric is still equivalent.

You know (maybe from Penrose CCC book or from his video lectures on CCC) that ENTROPY of gravitational field is equated to structure. Uniform featurelessness is low entropy and as matter coagulates and clumps into structure the entropy increases. I think Penrose wiley curvature hypothesis and his see-see cosmology are suggesting that if somehow decay, expansion, and dissipation could eventually zero out any curvature besides Weyl, then we couldn't tell the difference between that end of a cosmic era and the low entropy uniform conditions (people imagine brought about by inflation) we associate with the beginning of a cosmic era. I'll stop here and see if someone can reply more definitively.

Last edited: Dec 4, 2014
3. Dec 4, 2014

### Chalnoth

I think my opinion of this idea of Penrose's will forever be tainted by the absolutely execrable job that he and Gurzadyan did in trying to use the CMB to produce evidence of their idea.

Basically, they mistook "random" for "uncorrelated" and then doubled-down when this basic math error was pointed out to them. It was severely disappointing.

4. Dec 6, 2014

### bcrowell

Staff Emeritus
The Weyl curvature hypothesis is separate from CCC and predates it by 30 years. CCC turned out not to be viable because it made predictions about particle physics that were falsified (and this is independent of the failure to find observational support for it). But the falsification of CCC doesn't imply that the Weyl curvature hypothesis is false.

5. Dec 6, 2014

### julcab12

.. Penrose reasoned that Weyl curvature might have something to do with entropy, given the fact that they both increase as the universe gets older. Penrose maintained that as the universe ages, more and more black holes will accumulate, so much so that at some point nearly all the mass of the universe will reside in black holes, with a corresponding large Weyl curvature. This will not quite mark the end, however, as the black holes themselves will all slowly evaporate via Hawking radiation. In the end, the universe will consist only of stray photons, and entropy will have reached its maximum extent (perhaps it will be infinite). In this sense, Penrose believes, Weyl curvature and entropy are related. Indeed, the Bekenstein-Hawking black hole entropy-area formula.

http://nonlocal.com/hbar/universe.gif [Broken]

S=1/4 . kc^3 A/Gℏ

(which can be derived a number of different ways) demonstrates the intimate relationship between entropy and the surface area A of the ultimate gravitational object, the black hole. Most interestingly, the entropy and all the information it contains exists on the surface of the event horizon!

...If Penrose is right, the Weyl conformal tensor Cαβγδ would have to be proportional to gravitational entropy. However, since the contracted trace of this tensor is zero, the only scalar they could form is CαβγδCαβγδ, which they call P2. Then begins an amusing search for a radial vector field whose divergence is proportional to the entropy. After laboriously trying out three alternatives for the proportionality term, they finally settle on the Kretschmann scalar, RαβγδRαβγδ. They then explore this choice in both Schwarzschild and de Sitter spacetimes. Conclusion: at the cosmological horizon (end of time), entropy must be of non-geometrical origin. Oh well. Maybe something else will work.

...It also introduces the idea that at the universe's end, only photons will exist, so that the geometry of the universe must be conformal. Weyl himself explored a strictly conformal space-time in his 1918 theory, but it failed, mostly because the line element ds of matter cannot be made scale-invariant.

Penrose now believes that at the end of the universe all information will have been destroyed via black hole evaporation. The universe, consisting of nothing but radiation, will then have no concept of time or history, and so will "reset" itself by forgetting about its large entropy content. At that point there will be nothing to distinguish the universe from its pre-Big Bang state. Thus, a new Big Bang will occur, possibly with a different set of fundamental physical constants; then another, and another after that, for all eternity. I believe this is more ordered than using the other principle -- "multiple points" to fill the constants problem. We have (1) universe "beating" and beating at different rate, each beats has different sets of constant which determines how fast the universe expand/contract. Constants will reshuffle on each beats until it reaches it maximum order and starts all over again.

Last edited by a moderator: May 7, 2017
6. Dec 6, 2014

### bcrowell

Staff Emeritus
It's not an alternative to inflation. It's independent of inflation. Inflation potentially explains things that WCH can't, and vice versa.

We observe the second law of thermodynamics to be true, and the second law can only be explained in one of two ways: (1) time-reversal asymmetry in the microscopic laws of physics, or (2) the imposition of special boundary conditions. (There is a common misconception that it can be explained without either of these ingredients.) Although the laws of physics do have time-reversal asymmetry, that asymmetry does not appear to be one that would be capable of causing the second law, so we have to look to the boundary conditions.

One way of handling this is to say that the boundary conditions were simply such that the early universe had a low entropy: of all possible big bangs, we just happened to have one that was extremely unusual in this regard. However, we can also try to get deeper insight from observation, by looking at what degrees of freedom were equilibrated in the early universe and which ones weren't. It appears that the matter degrees of freedom were equilibrated, e.g., because we observe that the CMB has very nearly a perfect black-body spectrum. Since entropy wasn't maximized, something must have not been equilibrated, and it seems that this was the gravitational degrees of freedom. In a generic, high-entropy early universe, nearly all the energy should have been in the form of gravitational waves.

The type of spacetime curvature that describes gravitational waves is called the Weyl curvature tensor, as opposed to the Ricci curvature, which is present even in an isotropic and homogeneous FRW model. Basically the Weyl tensor describes tidal forces, while the Ricci curvature describes non-tidal ones. Penrose proposed elevating this to a fundamental principle: that for some naturalistic reason, the big bang had to have a vanishing Weyl tensor. For example, it's conceivable that a theory of quantum gravity such as LQG could naturally lead to the prediction of a vanishing Weyl tensor at the Planck epoch.