Matter collineation

A matter collineation (sometimes matter symmetry and abbreviated to MC) is a vector field that satisfies the condition,

[math]\displaystyle{ \mathcal{L}_X T_{ab}=0 }[/math]

where [math]\displaystyle{ 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]\displaystyle{ X }[/math] is regarded as preserving certain physical quantities along the flow lines of [math]\displaystyle{ 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

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