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
Famous quotes containing the words matter and/or symmetry:
“It seems to me that upbringings have themes. The parents set the theme, either explicitly or implicitly, and the children pick it up, sometimes accurately and sometimes not so accurately.... The theme may be “Our family has a distinguished heritage that you must live up to” or “No matter what happens, we are fortunate to be together in this lovely corner of the earth” or “We have worked hard so that you can have the opportunities we didn’t have.””
—Calvin Trillin (20th century)
“What makes a regiment of soldiers a more noble object of view than the same mass of mob? Their arms, their dresses, their banners, and the art and artificial symmetry of their position and movements.”
—George Gordon Noel Byron (1788–1824)