covariant canonical general relativity with differential forms
some GR calculations for field theory with differential forms
Lagrangian
Definition
bla bla
\[ {\Omega^a}_i \wedge \theta^c \wedge \theta^d \, \epsilon_{abcd} \, \eta^{ib} = 4 \star S \]
- Lagrangian
\[ \mathcal L = {\Omega^a}_i \wedge \theta^c \wedge \theta^d \, \epsilon_{abcd} \, \eta^{ib} + \lambda_a \wedge \mathcal T^a \]\[ \mathcal L(\theta, \omega, \lambda, d\dots) = \left( d{\omega^a}_i + {\omega^a}_k \wedge {\omega^k}_i \right) \wedge \theta^c \wedge \theta^d \, \epsilon_{abcd} \, \eta^{ib} + \lambda_a \wedge \left( d\theta^a + {\omega^a}_b \wedge \theta^b \right) \]
Derivatives
- fields
\[ \frac{\partial \mathcal L}{\partial \theta^A} = 2 \, {\Omega^a}_i \wedge \theta^c \, \epsilon_{abcA} \, \eta^{ib} + \lambda_a \wedge {\omega^a}_A \]\[ \frac{\partial \mathcal L}{\partial {\omega^A}_B}
= \left( {\omega^a}_A \, \epsilon_{abcd} \, \eta^{Bb}
- {\omega^B}_i \, \epsilon_{Abcd} \, \eta^{ib} \right) \wedge \theta^c \wedge \theta^d
- \lambda_A \wedge \theta^B \]\[ \frac{\partial \mathcal L}{\partial \lambda_A} = d\theta^A + {\omega^A}_b \wedge \theta^b \]
- momenta
\[ \pi_A = \frac{\partial \mathcal L}{\partial d\theta^A} = \lambda_A \]\[ {\pi_A}^B = \frac{\partial \mathcal L}{\partial d{\omega^A}_B} = \theta^c \wedge \theta^d \, \epsilon_{Abcd} \, \eta^{Bb} \]\[ \pi^A = \frac{\partial \mathcal L}{\partial \lambda_A} = 0 \]
Euler Lagrange equations
- \(\lambda\)
\[ d\theta^A + {\omega^A}_b \wedge \theta^b = 0 \]- or
\[ \mathcal T = 0 \]
- \(\theta\)