What can be derived from a vanishing Weyl tensor

[Airsteve0]

I was just wondering what physical conclusions could be made about a spacetime which possesses a vanishing Weyl curvature tensor, aside from the spacetime being conformally flat. By this question I am simply interested if any inferences can be made about the metric describing the spacetime, or just general properties of the spacetime itself. Thanks for any insight!

 

[TrickyDicky]

I was just wondering what physical conclusions could be made about a spacetime which possesses a vanishing Weyl curvature tensor, aside from the spacetime being conformally flat. By this question I am simply interested if any inferences can be made about the metric describing the spacetime, or just general properties of the spacetime itself.

Basically no vacuum and no tidal forces.

 

[Airsteve0]

Vanishing of the Weyl tensor implies the spacetime cannot be a vacuum?

 

[PAllen]

Well, the one possibility for vacuum + Weyl=0 is flat Minkowski space; a universe with no matter or fields. This can be part of a bigger solution. For example, you have this inside a spherical matter shell containing vacuum.

 

[Airsteve0]

Oh ok, so for example if the interior of a spherical region was vacuum and outside the sphere was a Robertson-Walker type spacetime? Are there any other interesting properties that the vanishing Weyl tensor can give rise to?

 

[Naty1]

from Roger Penrose THE ROAD TO REALITY...PG 766

Let us now think of a universe evolving so that an initially uniform distribution of material [with some density fluctuations] gradually clumps gravitionally, so that eventually parts of it collapse into black holes. The initial uniformity corresponds to a mainly Ricci-curvature [matter] distribution, but as more and more material collects gravitationally, we get increasing amounts of Weyl curvature...The Weyl curvature finally diverges to infinity as the black-hole singularities are reached. If we think of the material as having been originally spewed out from the Big Bang in an almost completely uniform way, then we start with a Weyle curvature that is...[essentially] zero. Indeed, a feature of the FLRW models is that the Weyl curvature vanishes completely. ...For a universe to start out closely FLRW we we expect the Weyl curvature to be extremely small, as compared with the Ricci curvature, the latter actually diverging at the Big Bang. This picture strongly suggests what the geometrical difference is between the initial Big Bang singularity- of exceedingly low entropy- and the generic black hole singularities, of very high entropy.

 

[Naty1]

Good general discussion by John Baez here:

http://math.ucr.edu/home/baez/gr/ricci.weyl.html

 

[PAllen]

If you Weyl away your time, you won't get fat (grow in volume).

 

[atyy]

If you Weyl away your time, you won't get fat (grow in volume).

 

[PAllen]

But if you eat Ricci food, you will get fat.

 

[TrickyDicky]

The Baez expalanation is interesting but one thing that I don't see addressed and I find important is that there are no (isotropic) solutions in GR that allow you to obtain both types of curvature, they are mutually exclusive, as if everything is set up by the EFE so that we can never get the whole picture of curvature (the 20 components), either you have to conform with only the Weyl ones or the Ricci ones. One cannot have both vacuum and matter like it seems the case physically.

 

[Bill_K]

What about a Bianchi cosmology, i.e. a FRW cosmology with anisotropic expansion rates. I think in that case both the Ricci and Weyl tensors will be nonzero.

 

[TrickyDicky]

What about a Bianchi cosmology, i.e. a FRW cosmology with anisotropic expansion rates. I think in that case both the Ricci and Weyl tensors will be nonzero.

Which one specifically? AFAIK all FRW cosmologies are isotropic by definition. The Krasner and the Taub-NUT cosmologies come to mind but they are vacuum solutions (vanishing Ricci tensor).

 

[TrickyDicky]

You might be referring to the Mixmaster universe, but I don't see how it would have Weyl curvature.

 

[Mentz114]

You might be referring to the Mixmaster universe, but I don't see how it would have Weyl curvature.

The Bianchi II dust with nil hyperslices has both Weyl and Ricci curvature. Came as a surprise to me too. I'll post more details later - busy now.

The metric is
<br /> ds^2=-{dt}^{2}+{e}^{-2\,c-2\,b}\,{dy}^{2}\,\left( {e}^{2\,b}\,{x}^{2}+{e}^{2\,c}\right) +2\,{e}^{-2\,c}\,dy\,dz\,x+{e}^{-2\,c}\,{dz}^{2}+{e}^{-2\,a}\,{dx}^{2}<br />
where a,b,c depend on t only.

 

[TrickyDicky]

I was associating non-vacuum to vanishing Weyl because all cosmologies considered viable (at least for our current universe epoch) generally are spatially isotropic. But yes, there are GR solutions like certain Bianchi types (i.e. type IX special case:Misner's Mixmaster, I don't know about type II) that seem to have both Ricci and Weyl curvature. One must note that usually isotropy is considered a requirement for physically plausible universes. For instance the Mixmaster is a cyclic and highly chaotic universe that was considered once for the near BB singularity stage only.
Anyway my comment in the above post I'd say holds only for isotropic solutions.

 

[Mentz114]

I was associating non-vacuum to vanishing Weyl because all cosmologies considered viable (at least for our current universe epoch) generally are spatially isotropic.
...
Anyway my comment in the above post I'd say holds only for isotropic solutions.

Agreed. I don't see how Weyl and Ricci curvature can coexist. Like about 99% of EFE solutions, these anisotropic models are interesting but probably unphysical.

 

[PeterDonis]

But yes, there are GR solutions like certain Bianchi types (i.e. type IX special case:Misner's Mixmaster, I don't know about type II) that seem to have both Ricci and Weyl curvature. One must note that usually isotropy is considered a requirement for physically plausible universes.

In most phases of the evolution of the universe, that's true. However, I believe one of the original motivations for developing the BKL singularity model (the Mixmaster is one version of this) was to see if a singularity still appeared when the assumptions of exact isotropy and spherical symmetry were dropped in cases like the Big Bang and gravitational collapse to a BH (answer: yes, it does). I don't think anyone believes that an exactly isotropic and spherically symmetric collapse to a BH, or an exactly isotropic and spherically symmetric Big Bang, is physically realistic.

Living Reviews has a good (long!) review article on spacelike singularities that talks about BKL here:

http://relativity.livingreviews.org/open?pubNo=lrr-2008-1&amp;page=articlesu11.html

 

[Naty1]

I thought I had a rough conceptual, not a detailed mathematical, understanding of at least some of 'curvature' in GR. But TrickyDicky's post #11 still has me scratching my head.

It sounds like he is saying we don't usually have non zero values for both the Ricci and Weyl curvature, say at the same point at the same time. Since they measure different types of curvature, that sure seems curious.

There may be some hints why that may be so here:
[for other discussions on curvature in these forums]

The Ricci tensor measures the kind of curvature that is produced by masses that are right there in that region of space. The Riemann tensor R measures all kinds of curvature, including curvature that is produced by tidal forces of distant masses.

Here's a more detailed explanation: http://www.lightandmatter.com/html_b...tml#Section5.1 [The Ricci tensor is the contraction of the Riemann tensor, the scalar curvature is the contraction of the Ricci tensor. Our usual measure of curvature, the Riemann tensor, depends on second derivatives of the metric. ]

If anyone can comment how and whether the lightandmatter explanation sheds any light on the issue of simultaneous Ricci and Weyl curvature, I'd sure find that interesting.

[Meantime, I have to continue removing more of the 21 trees Hurricane Sandy tipped in my yard...got to get them under control before a new Nor'easter comes roaring up the east coast..due here tomorrow night.]

 

[PAllen]

Actually, I thought the discussion had reached a clear conclusion.

In a region of mass/energy (SET nonzero), with an-isotropic distribution, you will have both types of curvature present. Otherwise, only Ricci.

In a vacuum region, you will have only Weyl; if Weyl=0, you have flat spacetime (but not necessarily a Minkowski universe, as my shell example showed).

In the large, it is considered that the universe is isotropic. However, there are various sized regions where this would not be true; in such regions, you would have both types of curvature.