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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SUMMARY

This discussion focuses on computing the Energy-Momentum tensor from a given Lagrangian, specifically the Lagrangian $$\mathcal{L} = -\frac{1}{2} (\partial_{\mu} A_{\nu})(\partial^{\mu} A^{\nu})$$. Participants analyze the invariance of the Lagrangian under Lorentz transformations and derive the conserved current using Noether's theorem. The final expression for the Energy-Momentum tensor is established as $$T^{\mu}_{\ \nu} = -\partial_{\mu} A_{\rho} \partial_{\nu} A^{\rho} + \frac{1}{2} \eta_{\mu \nu} \partial_{\rho} A_{\sigma} \partial^{\rho} A^{\sigma}$$, highlighting the importance of correctly applying the canonical stress-energy tensor formula.

PREREQUISITES

Understanding of Lagrangian mechanics
Familiarity with Noether's theorem
Knowledge of Lorentz transformations
Proficiency in tensor calculus

NEXT STEPS

Study the derivation of the canonical stress-energy tensor in field theory
Learn about the implications of Noether's theorem in various physical systems
Explore the relationship between Lagrangians and conserved quantities
Investigate the role of Lorentz invariance in classical field theories

USEFUL FOR

The discussion is beneficial for theoretical physicists, particularly those specializing in quantum field theory, as well as students and researchers interested in the mathematical foundations of physics and field theory applications.

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.