Conformal Killing vector field

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In conformal geometry, a conformal Killing vector field on a manifold of dimension n with (pseudo) Riemannian metric (also called a conformal Killing vector, CKV, or conformal colineation), is a vector field whose (locally defined) flow defines conformal transformations, that is, preserve up to scale and preserve the conformal structure. Several equivalent formulations, called the conformal Killing equation, exist in terms of the Lie derivative of the flow e.g. for some function on the manifold. For there are a finite number of solutions, specifying the conformal symmetry of that space, but in two dimensions, there is an infinity of solutions. The name Killing refers to Wilhelm Killing, who first investigated Killing vector fields.

Densitized metric tensor and Conformal Killing vectors

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A vector field   is a Killing vector field if and only if its flow preserves the metric tensor   (strictly speaking for each compact subsets of the manifold, the flow need only be defined for finite time). Formulated mathematically,   is Killing if and only if it satisfies

 

where   is the Lie derivative.

More generally, define a w-Killing vector field   as a vector field whose (local) flow preserves the densitized metric  , where   is the volume density defined by   (i.e. locally  ) and   is its weight. Note that a Killing vector field preserves   and so automatically also satisfies this more general equation. Also note that   is the unique weight that makes the combination   invariant under scaling of the metric. Therefore, in this case, the condition depends only on the conformal structure. Now   is a w-Killing vector field if and only if

 

Since   this is equivalent to

 

Taking traces of both sides, we conclude  . Hence for  , necessarily   and a w-Killing vector field is just a normal Killing vector field whose flow preserves the metric. However, for  , the flow of   has to only preserve the conformal structure and is, by definition, a conformal Killing vector field.

Equivalent formulations

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The following are equivalent

  1.   is a conformal Killing vector field,
  2. The (locally defined) flow of   preserves the conformal structure,
  3.  
  4.  
  5.   for some function  

The discussion above proves the equivalence of all but the seemingly more general last form. However, the last two forms are also equivalent: taking traces shows that necessarily  .

The last form makes it clear that any Killing vector is also a conformal Killing vector, with  

The conformal Killing equation

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Using that   where   is the Levi Civita derivative of   (aka covariant derivative), and   is the dual 1 form of   (aka associated covariant vector aka vector with lowered indices), and   is projection on the symmetric part, one can write the conformal Killing equation in abstract index notation as

 

Another index notation to write the conformal Killing equations is

 

Examples

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Flat space

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In  -dimensional flat space, that is Euclidean space or pseudo-Euclidean space, there exist globally flat coordinates in which we have a constant metric   where in space with signature  , we have components  . In these coordinates, the connection components vanish, so the covariant derivative is the coordinate derivative. The conformal Killing equation in flat space is   The solutions to the flat space conformal Killing equation includes the solutions to the flat space Killing equation discussed in the article on Killing vector fields. These generate the Poincaré group of isometries of flat space. Considering the ansatz  , we remove the antisymmetric part of   as this corresponds to known solutions, and we're looking for new solutions. Then   is symmetric. It follows that this is a dilatation, with   for real  , and corresponding Killing vector  .

From the general solution there are   more generators, known as special conformal transformations, given by

 

where the traceless part of   over   vanishes, hence can be parametrised by  .

Together, the   translations,   Lorentz transformations,   dilatation and   special conformal transformations comprise the conformal algebra, which generate the conformal group of pseudo-Euclidean space.

See also

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References

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  1. ^ P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory, 1997, ISBN 0-387-94785-X

Further reading

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  • Wald, R. M. (1984). General Relativity. The University of Chicago Press.