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A '''projective vector field''' ('''projective''') is a smooth [[vector field]] on a semi [[Riemannian manifold]] (p.ex. [[spacetime]]) <math>M</math> whose flow preserves the [[geodesic]] structure of <math>M</math> without necessarily preserving the [[geodesic (general relativity)|affine parameter]] 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 <math>X</math> on a semi [[Riemannian manifold]] (p.ex. in [[general relativity]]), it is often useful to decompose the [[covariant derivative]] into its symmetric and skew-symmetric parts:

:<math>X_{a;b}=\frac{1}{2}h_{ab}+ F_{ab}</math>

where 

:<math>h_{ab}=(\mathcal{L}_X g)_{ab}=X_{a;b}+X_{b;a}</math> 

and 

:<math>F_{ab}=\frac{1}{2}(X_{a;b}-X_{b;a})</math>

Note that <math>X_a</math> are the covariant components of <math>X</math>.

===Equivalent conditions===

Mathematically, the condition for a vector field <math>X</math> to be projective is equivalent to the existence of a [[one-form]] <math>\psi</math> satisfying

:<math>X_{a;bc}\, =R_{abcd}X^d+2g_{a(b}\psi_{c)}</math>

which is equivalent to 

:<math>h_{ab;c}\, =2g_{ab}\psi_c+g_{ac}\psi_b+g_{bc}\psi_a</math>

The set of all global projective vector fields over a connected or compact manifold forms a finite-dimensional [[Lie algebra]] denoted by <math>P(M)</math> (the '''projective algebra''') and satisfies for connected manifolds the condition: <math>\dim P(M) \le n(n+2)</math>.  Here a projective vector field is uniquely determined by specifying the values of <math>X</math>, <math>\nabla X</math> and <math>\nabla \nabla X</math> (equivalently, specifying <math>X</math>, <math>h</math>, <math>F</math> and <math>\psi</math>) at any point of <math>M</math>. (For non-connected manifolds you need to specify these 3 in one point per connected component.)  Projectives also satisfy the properties:

:<math>\mathcal{L}_X R^a{}_{bcd} = \delta ^a{}_d \psi_{b;c} - \delta ^a{}_c \psi_{b;d}</math>
:<math>\mathcal{L}_X R_{ab}= -3 \psi_{a;b}</math>

==Subalgebras==

Several important special cases of projective vector fields can occur and they form Lie subalgebras of <math>P(M)</math>. These subalgebras are useful, for example, in classifying spacetimes in general relativity.

===Affine algebra===

[[Affine vector field]]s (affines) satisfy <math>\nabla h=0</math> (equivalently, <math>\psi=0</math>) 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 <math>M</math> forms a [[Lie subalgebra]] of <math>P(M)</math> denoted by <math>A(M)</math> (the '''affine algebra''') and satisfies for connected ''M'', <math>\dim A(M) \le n(n+1)</math>. An affine vector is uniquely determined by specifying the values of the vector field and its first covariant derivative (equivalently, specifying <math>X</math>, <math>h</math> and <math>F</math>) at any point of <math>M</math>. Affines also preserve the Riemann, Ricci and Weyl tensors, i.e. 

:<math>\mathcal{L}_X R^a{}_{bcd}=0</math>, <math>\mathcal{L}_X R_{ab}=0</math>, <math>\mathcal{L}_X C^a{}_{bcd}=0</math>

===Homothetic algebra===

[[Homothetic vector field]]s (homotheties) preserve the metric up to a constant factor, i.e. <math>h = \mathcal{L}_X g = 2c g</math>. As <math>\nabla h=0</math>, every homothety is an affine and the set of all homotheties on <math>M</math> forms a Lie subalgebra of <math>A(M)</math> denoted by <math>H(M)</math> (the '''homothetic algebra''') and satisfies for connected ''M''

:<math>\dim H(M) \le \frac{1}{2}n(n+1)+1</math>. 

A homothetic vector field is uniquely determined by specifying the values of the vector field and its first covariant derivative (equivalently, specifying <math>X</math>, <math>F</math> and <math>c</math>) at any point of the manifold.

===Killing algebra===

[[Killing vector fields]] (Killings) preserve the metric, i.e. <math>h = \mathcal{L}_X g = 0</math>. Taking <math>c=0</math> 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 <math>M</math> forms a Lie subalgebra of <math>H(M)</math> denoted by <math>K(M)</math> (the '''Killing algebra''') and satisfies for connected ''M''

:<math>\dim K(M) \le \frac{1}{2}n(n+1)</math>. 

A Killing vector field is uniquely determined by specifying the values of the vector field and its first covariant derivative (equivalently, specifying <math>X</math> and <math>F</math>) at any point (for every connected component) of <math>M</math>.

==Applications==
In general relativity, many spacetimes possess certain symmetries that can be characterised by vector fields on the spacetime. For example, [[Minkowski space]] <math>{\Bbb M}</math> admits the maximal projective algebra, i.e. <math>\dim P({\Bbb M}) = 24</math>. 

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 [[spacetime symmetries|symmetries in general relativity]].

==References==
* {{cite book | author = Poor, W. | title=Differential Geometric Structures| location= New York | publisher=McGraw Hill | year=1981 | isbn=0-07-050435-0}}
* {{cite book | author = Yano, K. | title=Integral Formulas in Riemannian Geometry| location= New York | publisher=Marcel Dekker | year=1970 | id=ISBN ???}}
* {{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}}

[[Category:Differential geometry]]