Matter Symmetry
A less well-known form of symmetry concerns vector fields that preserve the energy-momentum tensor. These are variously referred to as matter collineations or matter symmetries and are defined by:
where are the energy-momentum tensor components. The intimate relation between geometry and physics may be highlighted here, as the vector field is regarded as preserving certain physical quantities along the flow lines of, 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, 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.
Read more about this topic: Spacetime Symmetries
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