A projective vector field (projective) is a smooth
vector field on a semi
Riemannian manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ...
(p.ex.
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
)
whose
flow
Flow may refer to:
Science and technology
* Fluid flow, the motion of a gas or liquid
* Flow (geomorphology), a type of mass wasting or slope movement in geomorphology
* Flow (mathematics), a group action of the real numbers on a set
* Flow (psyc ...
preserves the
geodesic
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connecti ...
structure of
without necessarily preserving the
affine parameter
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connectio ...
of any geodesic. More intuitively, the flow of the projective maps geodesics smoothly into geodesics without preserving the affine parameter.
Decomposition
In dealing with a vector field
on a semi
Riemannian manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ...
(p.ex. 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 ...
), it is often useful to decompose the
covariant derivative
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differe ...
into its symmetric and skew-symmetric parts:
:
where
:
and
:
Note that
are the covariant components of
.
Equivalent conditions
Mathematically, the condition for a vector field
to be projective is equivalent to the existence of a
one-form
In differential geometry, a one-form on a differentiable manifold is a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a smooth mapping of the total space of the tangent bundle of M to \R whose restriction to e ...
satisfying
:
which is equivalent to
:
The set of all global projective vector fields over a connected or compact manifold forms a finite-dimensional
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
denoted by
(the projective algebra) and satisfies for connected manifolds the condition:
. Here a projective vector field is uniquely determined by specifying the values of
,
and
(equivalently, specifying
,
,
and
) at any point of
. (For non-connected manifolds you need to specify these 3 in one point per connected component.) Projectives also satisfy the properties:
:
:
Subalgebras
Several important special cases of projective vector fields can occur and they form Lie subalgebras of
. These subalgebras are useful, for example, in classifying spacetimes in general relativity.
Affine algebra
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
...
s (affines) satisfy
(equivalently,
) and hence every affine is a projective. Affines preserve the geodesic structure of the semi Riem. manifold (read spacetime) whilst also preserving the affine parameter. The set of all affines on
forms a
Lie subalgebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
of
denoted by
(the affine algebra) and satisfies for connected ''M'',
. An affine vector is uniquely determined by specifying the values of the vector field and its first covariant derivative (equivalently, specifying
,
and
) at any point of
. Affines also preserve the Riemann, Ricci and Weyl tensors, i.e.
:
,
,
Homothetic algebra
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 ...
s (homotheties) preserve the metric up to a constant factor, i.e.
. As
, every homothety is an affine and the set of all homotheties on
forms a Lie subalgebra of
denoted by
(the homothetic algebra) and satisfies for connected ''M''
:
.
A homothetic vector field is uniquely determined by specifying the values of the vector field and its first covariant derivative (equivalently, specifying
,
and
) at any point of the manifold.
Killing algebra
Killing vector fields (Killings) preserve the metric, i.e.
. Taking
in the defining property of a homothety, it is seen that every Killing is a homothety (and hence an affine) and the set of all Killing vector fields on
forms a Lie subalgebra of
denoted by
(the Killing algebra) and satisfies for connected ''M''
:
.
A Killing vector field is uniquely determined by specifying the values of the vector field and its first covariant derivative (equivalently, specifying
and
) at any point (for every connected component) of
.
Applications
In general relativity, many spacetimes possess certain symmetries that can be characterised by vector fields on the spacetime. For example,
Minkowski space
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the iner ...
admits the maximal projective algebra, i.e.
.
Many other applications of symmetry vector fields in general relativity may be found in Hall (2004) which also contains an extensive bibliography including many research papers in the field of
symmetries in general relativity.
References
*
*
* {{cite book , author = Hall, Graham , title=Symmetries and Curvature Structure in General Relativity (World Scientific Lecture Notes in Physics) , location= Singapore , publisher=World Scientific Pub. , year=2004 , isbn=981-02-1051-5
Differential geometry