Quadropole Formalism: Necessary Condition?

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Discussion Overview

The discussion revolves around the necessary conditions for using the quadrupole formalism in the context of radiation, particularly focusing on the velocity regime where v << c. Participants explore the assumptions and limitations of this formalism in relation to the quadrupole moment and the conditions under which it applies.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants propose that v << c is the only necessary condition for using the quadrupole formalism.
  • Others argue that contributions from higher moments become negligible at the v << c range, allowing for their exclusion.
  • It is suggested that the radiation must travel radially, which simplifies the retarded potential and allows for the standard definition of the quadrupole tensor.
  • Participants note the importance of ignoring r^-2 and r^-1 terms that are not dominant in the slow motion approximation.
  • Some contributions emphasize the need for r >> d (where d is the size of the source) and r >> lambda, linking these conditions to the applicability of the quadrupole formalism.
  • A later reply questions whether a system can radiate without a quadrupole moment, especially at higher velocities, suggesting that the formalism may not be suitable in those cases.

Areas of Agreement / Disagreement

Participants express various viewpoints on the necessary conditions for the quadrupole formalism, indicating that multiple competing views remain regarding the applicability and limitations of the formalism in different scenarios.

Contextual Notes

Limitations include the dependence on specific assumptions about the motion of the source and the nature of the radiation, as well as unresolved mathematical steps related to the conditions for applying the quadrupole approximation.

mersecske
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v << c is the only necessary condition to use the formalism?
 
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Contributions from higher moments fall off at 1/c2 so for the v << c range you can forget them.

If you mean the quadrupole moment.
 
You also have to assume that the radiation being generated is pretty much traveling radially so that the space intervals in the retarded potential can simple be replaced by the radius and the quadrupole tensor can be defined in the normal way under the quadrupole approximation. Also, you ignore any r^-2 terms and the r^-1 terms that are not dominant in the slow motion approximation you mentioned.
 
You also need r>>d, where d is the size of the source,
and r>>lambda.
v<<c is also equivalent to lambda>>d.
 
For v << c the system must have quadrupole to radiate.
For large velocities the quadrupole formalism is not good,
but this statement is still true?
Or it is possible to radiate without quadrupole moment?
 

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