What is the Orthogonality Relation for the Energy-Momentum Tensor in Relativity?

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Homework Help Overview

The discussion revolves around deriving the orthogonality relation for the energy-momentum tensor in the context of special-relativistic electrodynamics. The original poster attempts to establish the relation {T^{\mu}}_{\alpha}{T^{\alpha}}_{\nu} = K{\delta^{\mu}}_{\nu} and determine the constant K.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore the structure of the energy-momentum tensor and its components, with one participant questioning the relevance of Einstein's equations in this context. Others suggest expressing the tensor in terms of the electromagnetic field tensor F to facilitate the derivation.

Discussion Status

The discussion is active, with participants providing guidance on how to approach the problem without reaching a consensus. There is a focus on understanding the relationship between the energy-momentum tensor and the electromagnetic field tensor.

Contextual Notes

Participants note that the context is limited to special-relativistic electrodynamics, and there is a mention of the original poster's uncertainty about how to proceed with the derivation.

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Homework Statement



Arrive at the orthogonality relation [itex]{T^{\mu}}_{\alpha}{T^{\alpha}}_{\nu} = K{\delta^{\mu}}_{\nu}[/itex]
and determine K.

Homework Equations


[itex]T_{ij}=T_ji}[/itex]

The Attempt at a Solution


[itex]{T^{\mu}}_{\alpha}{T^{\alpha}}_{\nu} = {T^{\mu}}_0{T^0}_{\nu}+ {T^{\mu}}_i{T^i}_{\nu}[/itex]
I am not sure how to continue from here, in which direction I should go...
Thanks!
 
Last edited:
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Is T appearing in the Einstein equations ? If so, use them.
 
No, we haven't even studied yet Einstein's equations (the context is special-relativistic electrodynamics. T is the energy-momentum (stress) tensor)
 
Aaaa, you should have said that. Ok, then you know how T looks like in terms of F. Then just express the contraction in the LHS in terms of F and regroup it so that you'll get a scalar times unit tensor.
 

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