covariant canonical general relativity with differential forms ed

some GR calculations for field theory with differential forms

Lagrangian ed

Definition ed

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 ed

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 ed

\[\lambda\]

\[ d\theta^A + {\omega^A}_b \wedge \theta^b = 0 \]

or

\[ \mathcal T = 0\]

\[\theta\]