Oh God...I just knew someone would eventually ask me that question!
Seriously, I remember from a graduate-level course in general relativity what the Weyl tensor is, but I will have to rummage through my notes to find an answer to this question, which sounds like a legitimate one. Bare with me!
The most complete way to define curvature is to use the so-called Riemann Curvature Tensor
R abcd
where a,b,c and d are its subscripts each running over the 4 different dimensions to space-time. Numerically, there can be 20 independent terms to define it in our space-time. As it turns out, the Riemann Tensor can be mathematically re-written as a combination of a new tensor, C, and the metric and Ricci Tensors
g and R ab ab
Now, the metric and Ricci tensors are the ones upon which general relativity were built by Einstein, and they can be related to each other in Einstein's relativistic equation for gravity:
1 8 pi G
R - --- R g = - ---------- T
ab 2 ab c^2 ab
It is solutions to this equation, and specifications of the metric tensor, that give us the famous 'black hole' and 'Big Bang' theories. However, they only give us half of the information we need to specify the complete curvature of space-time. The most complete specification is by the 20 quantities in 4-D space-time that are specified by the full Riemann Tensor with its four indices. The Ricci and metric tensors only specify 10 of these 20. What happened to the remaining 10 terms in the Riemann Tensor? We can recover them by breaking the Riemann Tensor into one part which we DEFINE as the Weyl tensor, and one part which we can write as combinations of the metric and Ricci tensors. This leads to the interesting result that in perfectly empty space where
T = 0 ab
the solution to Einstein's equation says that the Ricci curvature tensor vanishes everywhere in space-time, so you would think that space-time is 'flat' with no curvature. However, this only constrains 10 of the 20 curvature terms in the full Riemann Tensor, leaving the other 10 terms defining the Weyl Tensor to be non-zero. In other words, even though there is no local source for gravity because the local Ricci Curvature is zero, space-time can nonetheless be curved in at least 10 additional ways because of the non- vanishing of the full Riemann and Weyl tensors. Even in the empty space between the Earth and the Sun, there can still be curvature in space-time, which is why the gravitational force can exist in 'empty space' and propagate through a vacuum. The Weyl tensor is that part of the curvature that is NOT determined by the local distribution of matter and energy in space-time. Without it, gravity would never be able to reach beyond the confines of the massive bodies that are producing it.
Now, the relevance of the Weyl Tensor to Big Bang cosmology is something I am not too familiar with, but I do know that some papers on cosmology do spend time discussing the Weyl Tensor in this way. When we talk of the Big Bang solutions in which the universe expands from an 'initial' singularity, and re- collapses into a 'final' singularity, we have to examine the full Riemann tensor to properly 'classify' the kinds of cosmological solutions that can be possible. This is done by using the so-called 'Bianchi Identity' which is a simple mathematical formula that defines how the full Riemann tensor must behave. There are 9 distinct types of cosmological models that satisfy the Bianchi Identities and they differ in the various space-time symmetries that can arise. This means that the Weyl 'conformal' tensor, the Ricci Tensor and the Metric tensor all contribute to the full specification of these cosmologies with the Weyl Tensor accounting for the so-called 'trace-free' components, and the metric and Ricci tensors accounting for the 'non-zero trace' portion of Riemann curvature.
At the Big Bang, and the Big Crunch, although the Ricci Tensor predicts a singularity of infinite curvature, the Weyl Tensor need not also be infinite, but can be zero or anything in between. We know that the initial singularity was extremely smooth and contained a high degree of symmetry, but that need not, and will not be the case for the Final Singularity which will be quite asymmetric and 'lumpy'. This means that although the Weyl Tensor may have had a value close to zero at the Big Bang singularity, the merging of numerous black holes at the future 'Big Crunch' singularity will almost certainly mean that the Weyl tensor will not vanish at this future time.
i think this answer is more or less accurate, but I am not a relativist, so I wouldn't be surprised if I got something wrong!
Copyright 1997 Dr. Sten Odenwald
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