- #1
Steve Rogers
- 9
- 2
- Homework Statement
- Derive the following transformation rules for vielbein and spin connection:
$$\delta e_a^\mu=(\lambda^\nu\partial_\nu e_a^\mu-e_a^\nu\partial_\nu\lambda^\mu)+\lambda_a^b e_b^\mu$$
$$\delta\omega_a^{bc}=\lambda^\mu\partial_\mu\omega_a^{bc}+(-e_a^\mu\partial_\mu\lambda^{bc}+\omega_a^{d[b}\lambda_d^{c]}+\lambda_a^d\omega_d^{bc})$$
- Relevant Equations
- $$[M_{ab},X_c]=X_{[a}\eta_{b]c}$$
$$[M_{ab},M^{cd}]=-\delta_{[a}^{[c}M_{b]}^{d]}=-\delta_a^c M_b^d+\delta_b^c M_a^d+\delta_a^d M_b^c-\delta_b^d M_a^c$$
I am taking a course on General Relativity. Recently, I was given the following homework assignment, which reads
> Derive the following transformation rules for vielbein and spin connection:
$$\delta e_a^\mu=(\lambda^\nu\partial_\nu e_a^\mu-e_a^\nu\partial_\nu\lambda^\mu)+\lambda_a^b e_b^\mu$$
$$\delta\omega_a^{bc}=\lambda^\mu\partial_\mu\omega_a^{bc}+(-e_a^\mu\partial_\mu\lambda^{bc}+\omega_a^{d[b}\lambda_d^{c]}+\lambda_a^d\omega_d^{bc})$$
I was instructed to use:
$$[M_{ab},X_c]=X_{[a}\eta_{b]c}$$
and
$$[M_{ab},M^{cd}]=-\delta_{[a}^{[c}M_{b]}^{d]}=-\delta_a^c M_b^d+\delta_b^c M_a^d+\delta_a^d M_b^c-\delta_b^d M_a^c.$$
Also, the professor told us to consider the covariant derivative
$$\nabla_a=e_a^\mu\partial_\mu+\frac{1}{2}\omega_a^{bc}M_{cb}$$
To be honest, I have no idea what these symbols are (after examining my GR lecture note carefully). And most frustratingly, even if I have taken a one-year course on differential geometry (mathematical rigor), I still know nothing about the covariant derivative above. What on Earth do these symbols stand for? Is there any standard textbook that can help a GR beginner like me? I came here for some advice, please. Thank you very much.
> Derive the following transformation rules for vielbein and spin connection:
$$\delta e_a^\mu=(\lambda^\nu\partial_\nu e_a^\mu-e_a^\nu\partial_\nu\lambda^\mu)+\lambda_a^b e_b^\mu$$
$$\delta\omega_a^{bc}=\lambda^\mu\partial_\mu\omega_a^{bc}+(-e_a^\mu\partial_\mu\lambda^{bc}+\omega_a^{d[b}\lambda_d^{c]}+\lambda_a^d\omega_d^{bc})$$
I was instructed to use:
$$[M_{ab},X_c]=X_{[a}\eta_{b]c}$$
and
$$[M_{ab},M^{cd}]=-\delta_{[a}^{[c}M_{b]}^{d]}=-\delta_a^c M_b^d+\delta_b^c M_a^d+\delta_a^d M_b^c-\delta_b^d M_a^c.$$
Also, the professor told us to consider the covariant derivative
$$\nabla_a=e_a^\mu\partial_\mu+\frac{1}{2}\omega_a^{bc}M_{cb}$$
To be honest, I have no idea what these symbols are (after examining my GR lecture note carefully). And most frustratingly, even if I have taken a one-year course on differential geometry (mathematical rigor), I still know nothing about the covariant derivative above. What on Earth do these symbols stand for? Is there any standard textbook that can help a GR beginner like me? I came here for some advice, please. Thank you very much.
