Computing an Energy-Momentum tensor given a Lagrangian

Thread starter JD_PM
Start date Mar 10, 2020
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Computing Energy-momentum Energy-momentum tensor Lagrangian Tensor

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The discussion focuses on computing the energy-momentum tensor from a given Lagrangian, specifically a simplified version related to Maxwell's equations. The participants explore the invariance of the Lagrangian under Lorentz transformations and apply Noether's theorem to derive the conserved current associated with translations. They detail the calculation of the current and the energy-momentum tensor, highlighting challenges in correctly applying derivatives and indices. The conversation emphasizes the need to clarify the relationship between different Lagrangians and the correct formula for the canonical stress-energy tensor. Ultimately, the participants seek to resolve discrepancies in their calculations and understand the derivation of the energy-momentum tensor.

Mar 20, 2020

JD_PM

nrqed said:

Even worse, some of the components are zero.

Ahhh you're right.

This equation

$$\partial_{\mu} \partial^{\mu} A^{\nu} = \eta^{\nu \rho} \partial_{\mu} \partial^{\mu} A_{\rho} = 0$$

It is equivalent to

$$\partial_{\mu} \partial^{\mu} A_{\nu} = \eta_{\nu \rho} \partial_{\mu} \partial^{\mu} A^{\rho} = \partial_{\mu} \partial^{\mu} A_{\nu} = 0$$

Which is what we want.