In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, a homothetic vector field (sometimes homothetic collineation or homothety) is a
projective vector field A projective vector field (projective) is a smooth vector field on a semi Riemannian manifold (p.ex. spacetime) M whose flow preserves the geodesic structure of M without necessarily preserving the affine parameter of any geodesic. More intuitivel ...
which satisfies the condition:
:
where c is a real constant. Homothetic vector fields find application in the study of
singularities in
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
. They can also be used to generate new solutions for Einstein equations by similarity reduction.
See also
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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
...
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Conformal Killing 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) fl ...
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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 ...
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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 ...
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Matter collineation
A matter collineation (sometimes matter symmetry and abbreviated to MC) is a vector field that satisfies the condition,
:\mathcal_X T_=0
where T_ are the energy–momentum tensor components. The intimate relation between geometry and physics ma ...
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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 ...
References
Mathematical methods in general relativity
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