How is Weyl Tensor associated with tidal force ?

[yicong2011]

How is Weyl Tensor associated with tidal force?

I checked my book, the acceleration in tidal effect can be expressed as:

a^c=-R_abd^cZ^aw^bZ^d

Note: Z^a is the tangent of geodesics, w^b is the separation vector

I cannot see from this equation how Weyl Tensor affects tidal force.

It is said that in tidal effect Weyl Tensor contributes to distort the shape of the body without changing its volume? Why can Weyl Tensor affect like this?

Does it (changing the shape without changing the volume) result from traceless property of Weyl tensor? [This is my bold guess... :)]

Thanks...

 

[pervect]

The decomposition of the Riemann might help. http://en.wikipedia.org/w/index.php?title=Ricci_decomposition&oldid=386393911, for example. Or consult your textbook, the Wiki has a slightly different formulae and defintions than Wald does, I haven't taken the effort to confirm that everything "matches up".

The test case I'd imagine is being near a massive body in a vacuum region of space-time, so you know that the Ricci and its trace R is zero (because you're in a vacuum), but you also know that the tidal forces are non-zero (because your'e near a massive body).

Using the Wiki formulae, we conclude that since R and R_ab are 0, S_ab is 0. This leads to the conclusion that E_abcd and S_abcd are zero. Using Wald's formulae I get the same conclusion even though the details appear slightly different at a casual glance.

But, we know the Riemann isn't zero, because we do measure the tidal forces. So, we conclude that the tidal forces must be due to the Weyl part, because all the other parts in the decomposition of the Riemann are zero.

 

[atyy]

Here's another reference for what pervect says: http://math.ucr.edu/home/baez/gr/ricci.weyl.html

 

[yicong2011]

The decomposition of the Riemann might help. http://en.wikipedia.org/w/index.php?title=Ricci_decomposition&oldid=386393911, for example. Or consult your textbook, the Wiki has a slightly different formulae and defintions than Wald does, I haven't taken the effort to confirm that everything "matches up".

The test case I'd imagine is being near a massive body in a vacuum region of space-time, so you know that the Ricci and its trace R is zero (because you're in a vacuum), but you also know that the tidal forces are non-zero (because your'e near a massive body).

Using the Wiki formulae, we conclude that since R and R_ab are 0, S_ab is 0. This leads to the conclusion that E_abcd and S_abcd are zero. Using Wald's formulae I get the same conclusion even though the details appear slightly different at a casual glance.

But, we know the Riemann isn't zero, because we do measure the tidal forces. So, we conclude that the tidal forces must be due to the Weyl part, because all the other parts in the decomposition of the Riemann are zero.

Thanks.

But I am still puzzled at

Why Weyl Tensor can change the shape of the body without changing its volume and why Ricci can change the volume? I have read John Baez's tutorial, yet still puzzled...

 

[pervect]

Try looking up the Raychaudhuri equation http://en.wikipedia.org/w/index.php?title=Raychaudhuri_equation&oldid=426789699

 

[atyy]

Some explanation is given here http://arxiv.org/abs/1012.4869

It appears that it may describe tidal forces only for spatially separated observers.

 

[atyy]

Baez and Bunn give their mathematical details here http://math.ucr.edu/home/baez/einstein/node10.html

Eq 6 is the geodesic deviation equation. Immediately after that, they define the rate of volume change.

It appears that they don't define geodesic deviation to be the same thing as tidal forces, whereas most others do.

So maybe:
(i) in general Riemann describes geodesic deviation
(ii) in some circumstances Weyl describes geodesic deviation of some spatially separated particles http://arxiv.org/abs/1012.4869
(iii) Weyl describes Baez and Bunn's definition of tidal forces for small round objects, which is not geodesic deviation

?