Solving Q1 of MathIII Paper60: Ricci Identity & Killing Vectors

[latentcorpse]

Hi I'm trying Q1 of this paper:
http://www.maths.cam.ac.uk/postgrad/mathiii/pastpapers/2005/Paper60.pdf
and have got to the bit where I need to show that $\xi_{b;ca}=-R_{bca}{}^d \xi_d$

Now I know that $R_{bca}{}^d \xi_d=R_{bcad} \xi^d = R_{adbc} \xi^d = \nabla_b \nabla_c \xi_a - \nabla_c \nabla_b \xi_a$

where I got that last equality by rearranging the Ricci identity $\nabla_c \nabla_d Z^a - \nabla_d \nabla_c Z^a = R^a{}_{bcd}Z^b$

So then we have $-R_{bca}{}^d \xi_d = \nabla_c \nabla_b \xi_a - \nabla_b \nabla_c \xi_a = -\nabla_c \nabla_a \xi_b + \nabla_b \nabla_a \xi_c$
where I used the Killing property on the first term to get a $\xi_b$ term like we are looking for but I can't quite manipulate it into the final answer. Can anybody see what I am doing wrong?And then in the last bit, can somebody help me to show that if the Killing vector and the first derivative vanish at a point then they vanish everywhere?

And what about the final part about how many linearly independent Killing vectors can there be? My notes say that a n dimensional spacetime is maximally symmetric if there exist $\frac{n(n+1)}{2}$ linearly independent Killing vectors. But I don't actually know whether this is relevant to the question at hand or not?

Thanks.

 

[dextercioby]

I would start with

$$R_{[ij|k]l} = 0$$

from which you get based on Ricci's identity in the absence of torsion

$$\nabla_{[k}\nabla_{j}\xi_{i]} = 0$$

from which you should get by expansion into the 6 possible terms and regrouping based on

$$\nabla_{(i}\xi_{j)} = 0$$

an expression showing you the right way to get the final identity.

 

[latentcorpse]

I would start with

$$R_{[ij|k]l} = 0$$

from which you get based on Ricci's identity in the absence of torsion

$$\nabla_{[k}\nabla_{j}\xi_{i]} = 0$$

from which you should get by expansion into the 6 possible terms and regrouping based on

$$\nabla_{(i}\xi_{j)} = 0$$

an expression showing you the right way to get the final identity.

Thanks for your reply. What do you mean by $$R_{[ij|k]l} = 0$$? Is there supposed to be another vertical line in there somewhere to indicate one of the indices is not antisymmetrised?

 

[dextercioby]

No, sorry, the vertical line comes from the Youg tableau. So disregard it. I put it there from discussions on such tensors outside GR that have not left my mind yet.

 

[paranormal]

I am not familiar with the specific details of the mathematics involved in solving Q1 of MathIII Paper60. However, I can offer some general guidance and suggestions.

Firstly, it seems like you are on the right track with your approach using the Ricci identity. It is important to carefully manipulate the equations and apply the appropriate identities to arrive at the desired result. One suggestion would be to check your calculations step by step to ensure that you are not making any mistakes or overlooking any terms.

Secondly, in order to show that the Killing vector and its first derivative vanish at a point, you may need to use the definition of a Killing vector and its properties. It may also be helpful to consider the physical meaning of a Killing vector and how it relates to the geometry of the spacetime.

Finally, the relevance of the concept of maximally symmetric spacetime to the question at hand may depend on the specific context and assumptions of the problem. It may be worth considering how the existence of Killing vectors relates to the symmetry of a spacetime and how this may affect the number of linearly independent Killing vectors.

In summary, my advice would be to carefully review your calculations, use the definitions and properties of Killing vectors, and consider the relevance of the concept of maximally symmetric spacetime to the problem. It may also be helpful to consult with other experts in the field for further guidance and clarification. Good luck with your work!