# Weyl Curvature in Interior Spacetimes

Tags:
1. Jan 20, 2014

### Markus Hanke

I am just wondering - is space-time curvature in the presence of energy-momentum ( i.e. in interior solutions to the EFEs ) always pure Ricci in nature ? I had a discussion recently with someone who claimed that, but personally I would suspect that not to be the case in general, since I see no reason why gravitational radiation from distant sources couldn't penetrate into such regions, so that the Riemann tensor contains both Ricci and Weyl contributions. I am not completely sure though, so any input will be appreciated.

I have heard of the Petrov classification scheme for space-times, which is done via Weyl scalars, but to be honest it is a little over my head.

2. Jan 20, 2014

### Mentz114

Some pefect fluid solutions have zero Weyl curvature and some do not. I'm basing this on two solutions I have to hand. I don't know of any general criterion that separates these types of solutions but I'm sure someone will add to this.

3. Jan 20, 2014

### Markus Hanke

Ok, thank you. So the answer is basically that Weyl curvature does not always vanish for interior spacetimes - that's what I wanted to know.

4. Jan 20, 2014

### Bill_K

The Weyl tensor is the conformal curvature tensor. If it's zero then the spacetime is conformally flat.

The Petrov classification is based on the eigenvectors of the Weyl tensor. You can often deduce the class from symmetry considerations alone. For example the Schwarzschild solution has only one preferred direction, namely the radial direction, plus there is a reflection in time. Together they imply that the Petrov class for Schwarzschild must be type D.

Most cosmologies have no preferred direction, implying that the Weyl tensor must be zero.

5. Jan 20, 2014

### Markus Hanke

That's really handy, thanks.

Another question : as far as I know ( and please correct me on this if I am wrong ), the contraction of the Weyl tensor across two indices always vanishes :

$$\displaystyle{C{^{\alpha }}_{\mu \alpha \nu }=0}$$

Is there any physical or geometric significance or meaning to this ? What does this equation actually mean - if anything -, geometrically ?