I How to compute the expansion of this congruence?

[LCSphysicist]

$u^{a}\partial_{a} = (1-3M/r)^{-1/2}(\partial_{t} + (M/r^3)^{1/2} \partial_{\theta})$ in Schwarzschild metric. I need to compute the expansion of this vector field, but i am not sure how.
Adopting the fact that $u_{a};_{b} = B_{ab} (1)$, i want at first to construct the B matrix, but i am having trouble here! I mean, technically we can write the B matrix as the sum of three Matrix: One with trace, and two traceless, in which one is symmetric and the other is anti symmetric. But, using (1) we will have $
\begin{pmatrix}
0 & U_{1,2} &0 \\
0 &0 &0 \\
U_{3,1}& 0 & 0 \end{pmatrix}}$, this certainly can't be decompose in the way i cited above. I am really confused :S. Any help?

 

[PeterDonis]

Adopting the fact that $u_{a};_{b} = B_{ab}$, i want at first to construct the B matrix, but i am having trouble here!

The matrix you wrote down is wrong. Check your math. Bear in mind that you are computing a covariant derivative, which means you need to take into account the connection coefficients.

 

[PeterDonis]

I need to compute the expansion of this vector field

As John Wheeler once said, before computing anything you should first figure out the answer.

In this case, you should think about what the expansion (or more generally, the expansion, shear, and vorticity, since all three are components of the kinematic decomposition and all three have useful physical meanings) means, physically, and what, physically, the vector field you are describing means. What kind of orbits around the central body are the integral curves of the vector field you describe? If you imagine a fleet of such objects, each following its own integral curve, what would an object in the fleet see its neighboring objects doing? And how does that relate to the expansion, shear, and vorticity?

 

[PeterDonis]

the kinematic decomposition

This Wikipedia article might be a thought starter:

https://en.wikipedia.org/wiki/Congr...atical_decomposition_of_a_timelike_congruence

 

[PeterDonis]

This Wikipedia article

Note that the article mentions a fourth quantity, the proper acceleration. For this particular scenario, however, that is zero and can be ignored. Can you see why?

 

[LCSphysicist]

The matrix you wrote down is wrong. Check your math. You should end up with a pure antisymmetric matrix.

I edited the post and the matrix (I think we need actually a 3x3 matrix, since it is purely spatial). As you noted, i should end up with an antisymmetric matrix. But, the problem is that i didn't even started the math because i was puzzled with the derivatives. I mean, we have that the only terms that are nonvanishing are terms involving the derivative wrt r.

As John Wheeler once said, before computing anything you should first figure out the answer.

In this case, you should think about what the expansion (or more generally, the expansion, shear, and vorticity, since all three are components of the kinematic decomposition and all three have useful physical meanings) means, physically, and what, physically, the vector field you are describing means. What kind of orbits around the central body are the integral curves of the vector field you describe? If you imagine a fleet of such objects, each following its own integral curve, what would an object in the fleet see its neighboring objects doing? And how does that relate to the expansion, shear, and vorticity?

I think i can understand these terms when thinking about a fluid. It is confused because in theory i see that we can decompose in: $\delta_{ab} \theta /3 + \rho_{ab} + \omega _{ab}$, i am not being able to do it here. I am pretty sure my matrix is wrong, but i can't see what step was wrong.

I will try to do the things slowly here because probably i missed some part of the theory.
Taking just the spatial part of the metric:
\begin{pmatrix}
U_{r,r} &U_{r,\theta} &U_{r,\phi} \\
U_{\theta,r} &U_{\theta,\theta} &U_{\theta,\phi} \\
U_{\phi,r} &U_{\phi,\theta} & U_{\phi,\phi}
\end{pmatrix} =
\begin{pmatrix}
0&0 &0 \\
U_{\theta,r}& 0 & 0\\
0 & 0 & 0
\end{pmatrix}

Following the wikipedia article ${\dot {X}}^{a}={X^{a}}_{{;b}}X^{b}$ is zero here probably because we are dealing with geodesics, and i am assuming that we are using affine parameters.

 

[PeterDonis]

I edited the post and the matrix

Yes, I saw that, thanks!

(I think we need actually a 3x3 matrix, since it is purely spatial).

No, it isn't. It's transverse, but it's not purely spatial, because surfaces of constant time in the coordinates you are using are not orthogonal to the vector field.

As you noted, i should end up with an antisymmetric matrix.

No, actually you shouldn't. The matrix will be traceless (which is actually sufficient to answer the title question of this thread), but it won't be purely antisymmetric; it will have a symmetric part as well. (I originally thought it would be antisymmetric but then realized I was wrong, I was thinking of a different, though related, vector field.)

we have that the only terms that are nonvanishing are terms involving the derivative wrt r.

Not necessarily. Remember that, as I said, we are talking about covariant derivatives, so you have to take into account the connection coefficients.

I think i can understand these terms when thinking about a fluid.

Yes, at least a fair number of the intuitions associated with a fluid will work here; you can think of the integral curves of the vector field as "flow lines" of the fluid. But you still need to think about how the "speed of flow", so to speak, varies with $r$.

Following the wikipedia article ${\dot {X}}^{a}={X^{a}}_{{;b}}X^{b}$ is zero here probably because we are dealing with geodesics

Yes.

and i am assuming that we are using affine parameters.

Geodesics are geodesics whether you are using affine parameters or not. As it happens, we aren't going to have to worry about affine parameters at all for this particular problem; just cranking through the computation as described in the Wikipedia page will take care of it automatically.

 

[PeterDonis]

Taking just the spatial part of the metric

This is already wrong in at least two ways.

First, as I noted in my last post, the matrix you are trying to compute is not purely spatial.

Second, as I noted in my last post, you are taking covariant derivatives, not partial derivatives, so you have to take into account the connection coefficient terms.

There is also a third potential issue: the components of the vector field you wrote down in the OP are the components of $U^\alpha$, with an upper index; but what you need to compute the matrix are the components of $U_\alpha$, with a lower index--i.e., the covector field that corresponds to your vector field. So you have to lower an index on the field first before computing the matrix. I suspect you haven't done that.