Recent content by shichao116

Hey bro, thanks a lot. That clear things up. I used to get where you showed in the first equation in your reply but did not proceed because I never tried to use a Riemann tensor on a tensor of rank 2 or higher. Now I get some new experience. :)

Hi WannabeNewton, I now have the same problem as you did in this thread, can you show how you "solve it quickly" from 1/2\xi^\nu\nabla_\nu R = \xi^\nu\nabla^\mu R_{\mu\nu} ? I'm stuck exactly here. Thanks!

Hi WannabeNewton, I now have the same problem as you did in this thread, can you show how you "solve it quickly" from 1/2\xi^\nu\nabla_\nu R = \xi^\nu\nabla^\mu R_{\mu\nu} ? I'm stuck exactly here. Thanks!

Homework Statement (This is problem 12 of Chapter 3 in Sean Carroll's book: Spacetime and Geometry: An Introduction to General Relativity.) Show that any Killing vector K^\mu satisfies the following relations: \nabla_\mu\nabla_\sigma K^\rho = R^\rho_{\sigma\mu\nu}K^\nu K^\lambda\nabla_\lambda R...

I'm now stuck in the second part of problem 12 in Chapter 3. The problem is " Show that any Killing vector K^\mu satisfies the following relations: \nabla_\mu\nabla_\sigma K^\rho = R^\rho_{\sigma\mu\nu}K^\nu K^\lambda\nabla_\lambda R = 0 Where R is Riemann tensor. I can prove the first one by...

Hi TSny, thanks for sharing. This seems to be a more elegant way, I like it :)

Hi TSny, thanks very much, I think that's the kind of answer I'm looking for.

Hi Peter, this is not Einstein tensor. You obtain Einstein tensor by contracting two indices of "this" G. This is exercise 13.11 of book "Gravitation".

Hi Muphrid, I tried to use brute force, but I don't know how to deal with the two Levi-Civita symbol, because as you might see, the divergence actually sum with one of the Levi-Civita's subscript on the RHS. Would mind show me how to do it if it's not too lengthy? Thanks very much.

Hi all, I encounter a technical problem about tensor calculation when studying general relativity. I think it should be proper to post it here. Riemann curvature tensor has Bianchi identity: R^\alpha_{[\beta\gamma\delta;\epsilon]}=0 Now given double (Hodge)dual of Riemann tensor: G = *R*, in...

Hi all, I'm now reading Chap 11 of Gravitation by Wheeler, etc. In exercise 11.7, by introducing Jacobi curvature tensor, which contains exactly the same information content as Riemann curvature tensor, we are asked to show that we can actually measure ALL components of Jacobi curvature tensor...

Hi, I'm working on exercise 9.13 of the "bible" Gravitation. The problem I have is how to derive the following equation: R_x(t)R_z(\psi)R_x(\theta)R_z(\phi) = R_z(\psi-tsin\psi cot\theta)R_x(\theta+tcos\psi)R_z(\phi+tsin\psi/sin\theta) Where R_x(t) denotes an infinitesimal rotation about...

Hi all, I'm reading Sean Carroll's Space Time and Geometry and haven't figure out how equation 4.64 is derived, where he is in the process of deriving Einstein's equation from Hilbert action. Given there is a variation of the metric, g_{\mu\nu} \rightarrow g_{\mu\nu} + \delta g_{\mu\nu}, The...