Weyl Tensor Gravity propagation

[superbat]

I read Weyl tensor helps on propagating gravitational effects. Ricci is local depending on mass energy at that point and would vanish at other points. Weyl propogates the gravity effects (for example gravity at any point between Earth Moon is due to Weyl Tensor). I didn't quite get it mathematically. Why is Ricci local and Weyl somehow propagates gravity effects? I know basics like we can breaqk down Riemann curvature tensor in Ricci and Weyl and Einstein's Field equations have Ricci tensor.

 

[ShayanJ]

The Einstein's field equations in a vacuum are $R_{ab}-\frac 1 2 R g_{ab}=0$. Now if I take the trace of this equation(i.e. multiply it by $g^{cd}$ and contract indices and use $g^{ab}g_{ab}=n$ where n is the number of dimensions of the space-time, in this case n=4), I will get $R-2R=0 \Rightarrow R=0$. So the Einstein's field equations become $R_{ab}=0$. So in a vacuum, the Ricci curvature is zero but space-time may be still curved because the Riemann tensor may be non-zero.
Now if you take a look at this page, at the bottom you can find a decomposition of Riemann tensor into three components, one of which is the Weyl tensor and the other two are constructed somehow that become zero if $R_{ab}=0$ which is exactly the case in a vacuum. So the part that is responsible for keeping the Riemann tensor non-zero in a vacuum, seems to be the Weyl tensor.

 

[superbat]

Hey Thanks a lot for your reply.
I was thinking in terms of Rab(x0,y0,z0,t0) -1/2R(x0,y0,z0,t0)gab(x0,y0,z0,t0) = Tab(x0,y0,z0,t0) . So basically even if energy momentum tensor exists at x0,y0,z0,t0 it will create curvature at that point (sat Earth) but Weyl curvature coming out of this (since above equation will give us metric at x0,y0,z0,t0 which will give us riemann curvature tensor at x0,y0,z0,t0 which will give us weyl tensor) won't disappear even at other x,y,z,t between Earth moon or actually even at other far away places in universe. So what i see is Ricci(x0,y0,z0,t0) at that point produces curvature but disappears at other places in universe but Weyl doesn't disappear at other places in universe is that right? If yes can you show me how the equations predict the same as above

Thanks a lot

 

[ShayanJ]

can you show me how the equations predict the same as above

You surely know that $R_{ab}=0$ and $R_{ab}-\frac 1 2 R g_{ab} =\kappa T_{ab}$ are two different sets of differential equations and so have different solutions. So its not like finding the geometry of the spacetime in one go, unless you're dealing with a completely empty spacetime or a spacetime filled with matter somehow that you can associate a single SEM tensor field to it. In case the spacetime is partly empty and partly filled with matter with a definite SEM tensor field, people solve Einstein's equations for the two regions separately and attach them in the boundaries.

 

[superbat]

Ok,
Let me ask this.
Hypothetically spacetime is not empty. Only one planet exists and a vast empty universe.
So at that those points where planet exists we have Rab−1/2Rgab=κTab.
I want to know does the Ricci Tensor become zero at places other than where planet exists?
And does Weyl tensor not become zero at other points in spacetime? and so ends up curving other areas in universe as a consequence of that one planet.
is that fair to say?
I am sorry I am new to GR please bear with me.

Thank You

 

[ShayanJ]

Ok,
Let me ask this.
Hypothetically spacetime is not empty. Only one planet exists and a vast empty universe.
So at that those points where planet exists we have Rab−1/2Rgab=κTab.
I want to know does the Ricci Tensor become zero at places other than where planet exists?
And does Weyl tensor not become zero at other points in spacetime? and so ends up curving other areas in universe as a consequence of that one planet.
is that fair to say?
I am sorry I am new to GR please bear with me.

Thank You

That's what I said in my first reply!
In a vacuum, the Einstein's Field Equations become $R_{ab}=0$ whether or not there is any matter somewhere else. But if there is some matter somewhere else, then the Weyl tensor will be non-zero.
So inside the planet, both Ricci and Weyl tensors are non-zero. But outside it, in the vacuum, only the Weyl tensor is non-zero.

EDIT: There is at least one situation where a vacuum part of spacetime is flat although there is some matter somewhere else. Inside a spherical shell of mass, spacetime is flat!

 

[superbat]

Thanks a lot man!
So now if we take a very simple single planet hypothetical situation.
How difficult will it be to come up with Riemann Ricci and Weyl tensors?
if we assume simplest of conditions can we come up to a solution?
Ricci Tensor - I want to see equations which are function of space-time coordinates which become 0 outside planet .
Weyl Tensor - I want to see equations which are function of space-time coordinates which are nonzero outside planet .

We can assume things which will make it very simple to solve.

Can you guide me through this process mathematically. Will be very kind of you.

Thanks a lot!

 

[ShayanJ]

Any introductory textbook on GR will definitely contain the calculations you want. Simply look up for Schwarzschild solution!

 

[superbat]

Ok
Thanks a lot!