Phonon scattering: Difference between revisions
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A '''matter collineation''' (sometimes '''matter symmetry''' and abbreviated to '''MC''') is a [[vector field]] that satisfies the condition, | |||
< | :<math>\mathcal{L}_X T_{ab}=0</math> | ||
< | |||
where <math>T_{ab}</math> are the [[energy-momentum tensor]] components. The intimate relation between geometry and physics may be highlighted here, as the vector field <math>X</math> is regarded as preserving certain physical quantities along the flow lines of <math>X</math>, this being true for any two observers. In connection with this, it may be shown that ''every Killing vector field is a matter collineation'' (by the [[Einstein field equations]] (EFE), with or without [[cosmological constant]]). Thus, given a solution of the EFE, ''a vector field that preserves the metric necessarily preserves the corresponding energy-momentum tensor''. When the energy-momentum tensor represents a perfect fluid, every [[Killing vector field]] preserves the energy density, pressure and the fluid flow vector field. When the energy-momentum tensor represents an [[electromagnetic field]], a Killing vector field does ''not necessarily'' preserve the electric and magnetic fields. | |||
==See also== | |||
* [[Affine vector field]] | |||
* [[Conformal vector field]] | |||
* [[Curvature collineation]] | |||
* [[Homothetic vector field]] | |||
* [[Spacetime symmetries]] | |||
{{relativity-stub}} | |||
[[Category:Mathematical methods in general relativity]] | |||
Revision as of 19:00, 20 March 2013
A matter collineation (sometimes matter symmetry and abbreviated to MC) is a vector field that satisfies the condition,
where are the energy-momentum tensor components. The intimate relation between geometry and physics may be highlighted here, as the vector field is regarded as preserving certain physical quantities along the flow lines of , this being true for any two observers. In connection with this, it may be shown that every Killing vector field is a matter collineation (by the Einstein field equations (EFE), with or without cosmological constant). Thus, given a solution of the EFE, a vector field that preserves the metric necessarily preserves the corresponding energy-momentum tensor. When the energy-momentum tensor represents a perfect fluid, every Killing vector field preserves the energy density, pressure and the fluid flow vector field. When the energy-momentum tensor represents an electromagnetic field, a Killing vector field does not necessarily preserve the electric and magnetic fields.
See also
- Affine vector field
- Conformal vector field
- Curvature collineation
- Homothetic vector field
- Spacetime symmetries
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