Matter collineation

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


 * $$\mathcal{L}_X T_{ab}=0$$

where $$T_{ab}$$ are the energy–momentum tensor components. The intimate relation between geometry and physics may be highlighted here, as the vector field $$X$$ is regarded as preserving certain physical quantities along the flow lines of $$X$$, 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.