Matter collineation

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.

There is a "general plain symmetric metric" and 10 "equations for plane symmetric spacetime".^[1] The connections between symmetries and General Relativity has been studied extensively since 1993.^[2]

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. Likewise, a matter collineation is not necessarily a homothetic vector.^[3]

See also

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

References

1. Aslam, M. Jamil (2007). Mathematical Physics: Proceedings of the 12th Regional Conference, Islamabad, Pakistan, 27 March - 1 April 2006. World Scientific. p. 409. Retrieved November 14, 2025.
2. Llosa, Josep (2013). Matter and Ricci Collineations, in Progress in Mathematical Relativity, Gravitation and Cosmology: Proceedings of the Spanish Relativity Meeting ERE2012, University of Minho, Guimarães, Portugal, September 3-7, 2012. pp. 301–304. Retrieved November 14, 2025.
3. Corot, Jaume; da Costa, Jose (1967). Symmetries of Matter Distributions, in The Sixth Canadian Conference on General Relativity and Relativistic Astrophysics. American Mattematical Society. pp. 19–22. ISBN 9780821805237. Retrieved November 14, 2025.