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

${\displaystyle {\mathcal {L}}_{X}T_{ab}=0}$

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

See also

• Affine vector field
• Conformal vector field
• Curvature collineation
• Homothetic vector field
• Spacetime symmetries

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