Can an EM Stress-Energy Tensor Exist with Equal Sigma-Values?

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Discussion Overview

The discussion revolves around the possibility of an electromagnetic (EM) stress-energy tensor (SET) having equal sigma-values, specifically whether a configuration exists where \(\sigma_{xx} = \sigma_{yy}\). Participants explore theoretical implications and potential solutions within the context of electromagnetism and general relativity.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant questions the existence of an EM stress-energy tensor with equal sigma-values, suggesting that the Maxwell stress tensor implies the presence of off-diagonal components if both \(E_x\) and \(E_y\) are non-zero.
  • Another participant references electro-vacuum solutions of the Einstein field equations (EFE) that may allow for a diagonal SET, indicating a potential area for further investigation.
  • There is a suggestion regarding the Nordstrom-Reissner solution and the possibility of a perfect fluid solution with a specific diagonal form, although uncertainty remains about its viability.
  • One participant expresses skepticism about the existence of such an EMT, noting that fields typically have a \(zz\) component and that point particles would also introduce additional components.
  • Another participant mentions looking into domain walls and d2-branes as part of their exploration, indicating a broader context of theoretical frameworks being considered.

Areas of Agreement / Disagreement

Participants do not reach consensus on the existence of an EM stress-energy tensor with equal sigma-values. Multiple competing views and uncertainties remain regarding the implications of different theoretical models and configurations.

Contextual Notes

Participants note limitations in their discussions, such as the dependence on specific configurations and the presence of additional components in the stress-energy tensor that complicate the proposed scenarios.

Gatchaman
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Is it possible for an EM stress-energy tensor such as this:

[PLAIN]http://www3.telus.net/public/kots1906/emtensor.jpg

to exist, where \sigma_{xx} = \sigma_{yy} ?
 
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I don't think so, because the Maxwell stress tensor is

<br /> \sigma_{ij} = \epsilon_0 E_i E_j + \frac{1}<br /> {{\mu _0 }}B_i B_j - \frac{1}<br /> {2}\left( {\epsilon_0 E^2 + \frac{1}<br /> {{\mu _0 }}B^2 } \right)\delta _{ij} <br />

so if Ex and Ey are non-zero then the \sigma_{xy} must be present.

But, there are electro-vacuum solutions of the EFE where the SET is diagonal so it may be worth looking that up.
 
Nordstrom-Reissner solution? I looked, but couldn't figure it out.

Would a perfect fluid solution exist with diag(\rho, p, p, 0) you think?
 
Gatchaman said:
Nordstrom-Reissner solution? I looked, but couldn't figure it out.

Would a perfect fluid solution exist with diag(\rho, p, p, 0) you think?

Electrovac is here
http://en.wikipedia.org/wiki/Electrovacuum_solutionPerfect fluid is isotropic. You could try a scalar field where the Lagrangian depends only on \partial_x\phi and \partial_y\phi

[Edit]my suggestion won't help because there will be a Tzz term.
 
Last edited:
I don't think one exists.

How about an energy-momentum tensor? Perfect fluid with zz having no interaction with xx or yy components?
 
No, I don't think there is any EMT that corresponds to your example. A field will always have a zz component, and a point particle will have T0x and T0y components and Txy Tyx components.
 
Mentz114 said:
No, I don't think there is any EMT that corresponds to your example. A field will always have a zz component, and a point particle will have T0x and T0y components and Txy Tyx components.

I don't think you're correct. I've been looking into domain walls and d2-branes.
 
Gatchaman said:
I've been looking into domain walls and d2-branes.
That's beyond my ken so I have to leave it to you.
 

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