A matter collineation (sometimes matter symmetry and abbreviated to MC) is a
vector field that satisfies the condition,
:
where
are the
energy–momentum tensor Energy–momentum may refer to:
*Four-momentum
*Stress–energy tensor
*Energy–momentum relation
In physics, the energy–momentum relation, or relativistic dispersion relation, is the relativistic equation relating total energy (which is also ...
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
In mathematics, a Killing vector field (often called a Killing field), named after Wilhelm Killing, is a vector field on a Riemannian manifold (or pseudo-Riemannian manifold) that preserves the metric. Killing fields are the infinitesimal gene ...
is a matter collineation (by the
Einstein field equations
In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it.
The equations were published by Einstein in 1915 in the form ...
(EFE), with or without
cosmological constant
In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: ), alternatively called Einstein's cosmological constant,
is the constant coefficient of a term that Albert Einstein temporarily added to his field equ ...
). 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
In physics, a perfect fluid is a fluid that can be completely characterized by its rest frame mass density \rho_m and ''isotropic'' pressure ''p''. Real fluids are "sticky" and contain (and conduct) heat. Perfect fluids are idealized models in whi ...
, every Killing vector field preserves the energy density, pressure and the fluid flow vector field. When the energy-momentum tensor represents an
electromagnetic field
An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical c ...
, a Killing vector field does ''not necessarily'' preserve the electric and magnetic fields.
See also
*
Affine vector field
An affine vector field (sometimes affine collineation or affine) is a projective vector field preserving geodesics and preserving the affine parameter. Mathematically, this is expressed by the following condition:
:(\mathcal_X g_)_=0
See also
...
*
Conformal vector field
In conformal geometry, a conformal Killing vector field on a manifold of dimension ''n'' with (pseudo) Riemannian metric g (also called a conformal Killing vector, CKV, or conformal colineation), is a vector field X whose (locally defined) flo ...
*
Curvature collineation
A curvature collineation (often abbreviated to CC) is vector field which preserves the Riemann tensor in the sense that,
:\mathcal_X R^a_=0
where R^a_ are the components of the Riemann tensor. The set of all smooth curvature collineations form ...
*
Homothetic vector field In physics, a homothetic vector field (sometimes homothetic collineation or homothety) is a projective vector field which satisfies the condition:
:\mathcal_X g_=2c g_
where c is a real constant. Homothetic vector fields find application in the s ...
*
Spacetime symmetries
Spacetime symmetries are features of spacetime that can be described as exhibiting some form of symmetry. The role of symmetry in physics is important in simplifying solutions to many problems. Spacetime symmetries are used in the study of exact ...
Mathematical methods in general relativity
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