Naked Science Forum
Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: lean bean on 24/03/2013 18:18:33

In figuring out the Schwarzschild solution to Einstein’s field equation did Schwarzschild use the Weyl tensor also ?
When we are in truly empty space, there's no Ricci curvature, so actually our ball of coffee grounds doesn't change volume. But there can be Weyl curvature due to gravitational waves, tidal forces, and the like. Gravitational waves and tidal forces tend to stretch things out in one direction while squashing them in the other. So these would correspond to our ball changing into an ellipsoid! Just as we hoped.
Similarly, when a ball of coffee grounds falls freely through outer space in the earth's gravitational field, it feels no Ricci curvature, only Weyl curvature. So the "tidal forces" due to some coffee grounds being near to the earth than others may stretch the ball into an ellipsoid, but not change its volume.
From…
http://math.ucr.edu/home/baez/gr/ricci.weyl.html (http://math.ucr.edu/home/baez/gr/ricci.weyl.html)
I'm not an expert, so go easy :)

Here is the original paper. On the Gravitational Field of a Mass Point according to Einstein’s Theory. (http://arxiv.org/pdf/physics/9905030) And no, I don't think so, but I'm not entirely sure?
"Weyl, as a major figure in the Göttingen school, was fully apprised of Einstein's work from its early days. He tracked the development of relativity physics in his Raum, Zeit, Materie (Space, Time, Matter) from 1918, reaching a 4th edition in 1922. In 1918, he introduced the notion of gauge, and gave the first example of what is now known as a gauge theory. Weyl's gauge theory was an unsuccessful attempt to model the electromagnetic field and the gravitational field as geometrical properties of spacetime. The Weyl tensor in Riemannian geometry is of major importance in understanding the nature of conformal geometry. In 1929, Weyl introduced the concept of the vierbein into general relativity."
Schwarzschild wrote it 1916, and it seems as if Weyl started thinking of it 1918, maybe 1917? Then again, look through the paper and see if you find anything resembling it. Weyl seem to have been very productive mathematically, through the whole of his life. Hermann Klaus Hugo Weyl. (http://en.wikipedia.org/wiki/Hermann_Weyl) The best question may be if they were friends, sharing their thoughts with each other?

The Schwarzschild "black hole" has zero spin and zero electrical charge. It's also not experiencing any other external forces. I don't think the Weyl tensor would apply.

From Baez's site, I don't think it is used in the Schwarzschild solution. I don't understand GR that well, but I do understand his analogy to electromagnetism. Basically the Ricci tensor describes spacetime's curvature due to sources (energy, momenta), while the Weyl tensor describes the possible gravitational waves that could be moving around without any sources present. If you know the sources in your region of space (as in the Schwarzschild solution), you can compute the Ricci tensor. To get the Weyl tensor, you need to know about the background fields that would be present without your sources of gravity.

Thanks to all so far.
I can see now (thanks to your posts) he probably didn't use the weyl tensor.
To get the Weyl tensor, you need to know about the background fields that would be present without your sources of gravity.
I'm a little confused here JP.
Doesn't the weyl tensor describe the tidal forces experienced by the 'coffee ground ball" falling to earth, and so we do know the source in this example, the Earth.
when a ball of coffee grounds falls freely through outer space in the earth's gravitational field, it feels no Ricci curvature, only Weyl curvature. So the "tidal forces" due to some coffee grounds being near to the earth than others may stretch the ball into an ellipsoid, but not change its volume.
From link in first post.

Yeah, rereading Baez I'm not so sure. One part of the page says it's akin to background solutions away from sources and another says it describes tidal forces. Maybe Pmb can clarify it.

Here’s something from Roger Penrose describing Weyl curvature…notice he also use’s the Earth in the example.
The Weyl curvature effectively measures the tidal effect. What is the ‘tidal’ effect?
Recall that, from the astronaut’s point of view, it seems that gravity has been abolished but that is not quite true.
Imagine that the astronaut is surrounded by a sphere of particles. Which are initially at rest with respect to the astronaut. Now, initially they will just hover there but soon they will start to accelerate because of the slight differences in the gravitational attraction of the Earth at different points on the sphere. (Notice that I am describing the effect in Newtonian language, but that is quite adequate.) these slight differences cause the original sphere of particles to become distorted into an elliptical arrangement,
From Roger Penrose (http://books.google.co.uk/books?id=jWHqlijAjyMC&pg=PA19&lpg=PA23&dq=from+the+astronaut%E2%80%99s+point+of+view,+it+seems+that+gravity+has+been+abolished+but+that+is+not+quite+true.&source=bl&ots=bbWCk8JcIe&sig=R1mRHAzhFFZRpdk7cipSVO_wVA0&hl=en&sa=X&ei=73ZZUePmG4PO0QX1hIH4Cg&ved=0CCMQ6AEwAA)