Matter collineation

{{Short description|Vector field}} {{refimprove|date=November 2025}} A '''matter collineation''' (sometimes '''matter symmetry''' and abbreviated to '''MC''') is a [[vector field]] that satisfies the condition,

:<math>\mathcal{L}_X T_{ab}=0</math>

where <math>T_{ab}</math> are the [[energy–momentum tensor]] components.

There is a "general plain symmetric metric" and 10 "equations for plane symmetric [[spacetime]]".<ref>{{cite book|page=409|url=https://books.google.com/books?id=wTVhDQAAQBAJ&pg=PA409&dq=%22Matter+collineation%22+-wikipedia&hl=en&newbks=1&newbks_redir=0&source=|title=Mathematical Physics: Proceedings of the 12th Regional Conference, Islamabad, Pakistan, 27 March - 1 April 2006|first=M. Jamil |last=Aslam|year=2007|publisher=World Scientific|access-date=November 14, 2025}}</ref> The connections between symmetries and [[General Relativity]] has been studied extensively since 1993.<ref>{{cite book|url=https://books.google.com/books?id=n3_BBAAAQBAJ&pg=PA301&dq=%22Matter+collineation%22+-wikipedia&hl=en&newbks=1&newbks_redir=0&source=|title=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|year=2013|first=Josep|last=Llosa|pages=301–304|access-date=November 14, 2025}}</ref>

The intimate relation between geometry and physics may be highlighted here, as the vector field <math>X</math> is regarded as preserving certain physical quantities along the flow lines of <math>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. Likewise, a matter collineation is ''not necessarily'' a [[homothetic vector]].<ref>{{cite book|pages=19–22|url=https://books.google.com/books?id=t-7QDgAAQBAJ&pg=PA22&dq=%22Matter+collineation%22+-wikipedia&hl=en&newbks=1&newbks_redir=0&source=|title=Symmetries of Matter Distributions, in The Sixth Canadian Conference on General Relativity and Relativistic Astrophysics |first1=Jaume|last1=Corot|first2=Jose|last2=da Costa|year=1967|isbn=9780821805237|publisher=American Mattematical Society |access-date=November 14, 2025}}</ref>

==See also== * [[Affine vector field]] * [[Conformal vector field]] * [[Curvature collineation]] * [[Homothetic vector field]] * [[Spacetime symmetries]]

==References== {{Reflist}}

[[Category:Mathematics of general relativity]]