Understanding EM Stress Energy Tensor: Sources and Current Density

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SUMMARY

The electromagnetic (EM) stress-energy tensor is defined as T^{\mu\nu} = -\frac{1}{4}g^{\mu\nu} F^{\alpha\beta}F_{\alpha\beta} + F^{\mu}\,\!_{\lambda}F^{\nu\lambda}. This formulation does not include sources, as confirmed by referencing 't Hooft's notes, specifically Equation 10.9. To incorporate current density, an additional term proportional to the product of the four-potential A and the current density J must be included. This clarification is essential for accurate calculations in general relativity and field theory.

PREREQUISITES
  • Understanding of the electromagnetic field tensor F^{\mu\nu}
  • Familiarity with general relativity concepts, particularly the stress-energy tensor
  • Knowledge of Lagrangian mechanics in field theory
  • Basic understanding of current density in electromagnetism
NEXT STEPS
  • Study 't Hooft's notes on general relativity, focusing on Equation 10.9
  • Research the derivation of the electromagnetic stress-energy tensor in various contexts
  • Learn about the role of current density in modifying the stress-energy tensor
  • Explore applications of the stress-energy tensor in cosmology and astrophysics
USEFUL FOR

Physicists, graduate students in theoretical physics, and researchers working on electromagnetism and general relativity will benefit from this discussion.

tim_lou
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I've been trying to obtain the EM stress energy tensor... but I'm not sure if what I got is correct.

In all the websites I googled, I found
T^{\mu\nu} =- \frac{1}{4}g^{\mu\nu} F^{\alpha\beta}F_{\alpha\beta} + F^{\mu}\,\!_{\lambda}F^{\nu\lambda}

However, is this with the sources or without the sources? Seems to me that this is what I would get for a Lagrangian without current. what happens if I add a current density?

I just need a quick answer... I can't seem to find anything online.
 
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Try 't Hooft's notes, Eq. 10.9
http://www.phys.uu.nl/~thooft/lectures/genrel.pdf
 
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thanks, got my answer, indeed that form above is without any source. A term proportional to the product of AJ are included, as expected.
 

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