 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2023

Paper 309

Since $de^0=0$ and $de^i=\tfrac12\epsilon^i{}_{jk}e^j\wedge e^k$, <Cartan's first structure equation> gives
$$
\boxed{\omega^0{}_\mu=0,
\qquad \omega^i{}_j=\frac12\epsilon^i{}_{jk}e^k}.
$$
<Cartan's second structure equation> then gives
$$
\boxed{\Theta^0{}_\mu=0,
\qquad \Theta^i{}_j=\frac14e^i\wedge e^j}.
$$
Using $\Theta^i{}_j=\tfrac12R^i{}_{jkl}e^k\wedge e^l$ yields
$$
\boxed{R_{ijkl}=\frac14(\delta_{ik}\delta_{jl}-\delta_{il}\delta_{jk})},
$$
with no time-index curvature components. Contraction gives
$$
R_{00}=0,
\qquad R_{ij}=\frac12\delta_{ij},
\qquad R=\frac32,
$$
and therefore
$$
\boxed{G_{00}=\frac34,
\qquad G_{ij}=-\frac14\delta_{ij},
\qquad G_{0i}=0}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 309 / 4 / c / Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 309 / 4 / c / Solution

 Ancestors (11)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 309 / 4 / c / Solution

 Ancestors (11)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)