# Matter collineation

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Vector field

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A **matter collineation** (sometimes **matter symmetry** and abbreviated to **MC**) is a [vector field](/source/Vector_field) that satisfies the condition,

- L X T a b = 0 {\displaystyle {\mathcal {L}}_{X}T_{ab}=0}

where T a b {\displaystyle T_{ab}} are the [energy–momentum tensor](/source/Energy%E2%80%93momentum_tensor) components.

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

The intimate relation between geometry and physics may be highlighted here, as the vector field X {\displaystyle X} is regarded as preserving certain physical quantities along the flow lines of X {\displaystyle X} , this being true for any two observers. In connection with this, it may be shown that every [Killing vector field](/source/Killing_vector_field) is a matter collineation (by the [Einstein field equations](/source/Einstein_field_equations) (EFE), with or without [cosmological constant](/source/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](/source/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](/source/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](https://en.wikipedia.org/w/index.php?title=Homothetic_vector&action=edit&redlink=1).[3]

## See also

- [Affine vector field](/source/Affine_vector_field)

- [Conformal vector field](/source/Conformal_vector_field)

- [Curvature collineation](/source/Curvature_collineation)

- [Homothetic vector field](/source/Homothetic_vector_field)

- [Spacetime symmetries](/source/Spacetime_symmetries)

## References

1. **[^](#cite_ref-1)** Aslam, M. Jamil (2007). [*Mathematical Physics: Proceedings of the 12th Regional Conference, Islamabad, Pakistan, 27 March - 1 April 2006*](https://books.google.com/books?id=wTVhDQAAQBAJ&pg=PA409&dq=%22Matter+collineation%22+-wikipedia&hl=en&newbks=1&newbks_redir=0&source=). World Scientific. p. 409. Retrieved November 14, 2025.

1. **[^](#cite_ref-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*](https://books.google.com/books?id=n3_BBAAAQBAJ&pg=PA301&dq=%22Matter+collineation%22+-wikipedia&hl=en&newbks=1&newbks_redir=0&source=). pp. 301–304. Retrieved November 14, 2025.

1. **[^](#cite_ref-3)** Corot, Jaume; da Costa, Jose (1967). [*Symmetries of Matter Distributions, in The Sixth Canadian Conference on General Relativity and Relativistic Astrophysics*](https://books.google.com/books?id=t-7QDgAAQBAJ&pg=PA22&dq=%22Matter+collineation%22+-wikipedia&hl=en&newbks=1&newbks_redir=0&source=). American Mattematical Society. pp. 19–22. [ISBN](/source/ISBN_(identifier)) [9780821805237](https://en.wikipedia.org/wiki/Special:BookSources/9780821805237). Retrieved November 14, 2025.

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