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In
conformal geometry In mathematics, conformal geometry is the study of the set of angle-preserving ( conformal) transformations on a space. In a real two dimensional space, conformal geometry is precisely the geometry of Riemann surfaces. In space higher than two di ...
, a conformal Killing vector field on a manifold of
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
''n'' with (pseudo) Riemannian metric g (also called a conformal Killing vector, CKV, or conformal colineation), is a vector field X whose (locally defined) flow defines
conformal transformation In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths. More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-preserving) at a point u_0\in ...
s, that is, preserve g up to scale and preserve the conformal structure. Several equivalent formulations, called the conformal Killing equation, exist in terms of the
Lie derivative In differential geometry, the Lie derivative ( ), named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector fi ...
of the flow e.g. \mathcal_g = \lambda g for some function \lambda on the manifold. For n \ne 2 there are a finite number of solutions, specifying the
conformal symmetry In mathematical physics, the conformal symmetry of spacetime is expressed by an extension of the Poincaré group. The extension includes special conformal transformations and dilations. In three spatial plus one time dimensions, conformal symmetry ...
of that space, but in two dimensions, there is an infinity of solutions. The name Killing refers to
Wilhelm Killing Wilhelm Karl Joseph Killing (10 May 1847 – 11 February 1923) was a German mathematician who made important contributions to the theories of Lie algebras, Lie groups, and non-Euclidean geometry. Life Killing studied at the University of Mü ...
, who first investigated Killing vector fields.


Densitized metric tensor and Conformal Killing vectors

A vector field X is a Killing vector field if and only if its flow preserves the metric tensor g (strictly speaking for each compact subsets of the manifold, the flow need only be defined for finite time). Formulated mathematically, X is Killing if and only if it satisfies :\mathcal_X g = 0. where \mathcal_X is the Lie derivative. More generally, define a ''w''-Killing vector field X as a vector field whose (local) flow preserves the densitized metric g\mu_g^w, where \mu_g is the volume density defined by g (i.e. locally \mu_g = \sqrt \, dx^1\cdots dx^n ) and w \in \mathbf is its weight. Note that a Killing vector field preserves \mu_g and so automatically also satisfies this more general equation. Also note that w = -2/n is the unique weight that makes the combination g \mu_g^w invariant under scaling of the metric. Therefore, in this case, the condition depends only on the
conformal structure In mathematics, conformal geometry is the study of the set of angle-preserving ( conformal) transformations on a space. In a real two dimensional space, conformal geometry is precisely the geometry of Riemann surfaces. In space higher than two d ...
. Now X is a ''w''-Killing vector field if and only if :\mathcal_X \left(g\mu_g^\right) = (\mathcal_X g) \mu_g^ + w g \mu_g^ \mathcal_X \mu_g = 0. Since \mathcal_X \mu_g = \operatorname(X) \mu_g this is equivalent to : \mathcal_X g = - w\operatorname(X) g. Taking traces of both sides, we conclude 2\mathop(X) = -w n \operatorname(X). Hence for w \ne -2/n, necessarily \operatorname(X) = 0 and a ''w''-Killing vector field is just a normal Killing vector field whose flow preserves the metric. However, for w = -2/n, the flow of X has to only preserve the conformal structure and is, by definition, a ''conformal Killing vector field''.


Equivalent formulations

The following are equivalent # X is a conformal Killing vector field, # The (locally defined) flow of X preserves the conformal structure, # \mathcal_X (g\mu_g^) = 0, # \mathcal_X g = \frac \operatorname(X) g, # \mathcal_X g = \lambda g for some function \lambda. 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 \lambda = (2/n) \operatorname(X). The last form makes it clear that any Killing vector is also a conformal Killing vector, with \lambda \cong 0.


The conformal Killing equation

Using that \mathcal_X g = 2 \left(\nabla X^\flat \right)^ where \nabla is the Levi Civita derivative of g (aka covariant derivative), and X^=g(X,\cdot) is the dual 1 form of X (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 :\nabla_a X_b + \nabla_b X_a = \fracg_\nabla_X^c. Another index notation to write the conformal Killing equations is : X_+X_ = \fracg_ X^c_.


Examples


Flat space

In n-dimensional flat space, that is
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
or
pseudo-Euclidean space In mathematics and theoretical physics, a pseudo-Euclidean space is a finite-dimensional real -space together with a non- degenerate quadratic form . Such a quadratic form can, given a suitable choice of basis , be applied to a vector , giving q(x ...
, there exist globally flat coordinates in which we have a constant metric g_ = \eta_ where in space with signature (p,q), we have components (\eta_) = \text(+1,\cdots,+1,-1,\cdots,-1). In these coordinates, the connection components vanish, so the covariant derivative is the coordinate derivative. The conformal Killing equation in flat space is :\partial_\mu X_\nu + \partial_\nu X_\mu = \frac\eta_ \partial_\rho X^\rho. 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 The Poincaré group, named after Henri Poincaré (1906), was first defined by Hermann Minkowski (1908) as the group of Minkowski spacetime isometries. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our und ...
of isometries of flat space. Considering the ansatz X^\mu = M^x_\nu,, we remove the antisymmetric part of M^ as this corresponds to known solutions, and we're looking for new solutions. Then M^ is symmetric. It follows that this is a dilatation, with M^\mu_\nu = \lambda\delta^\mu_\nu for real \lambda, and corresponding Killing vector X^\mu = \lambda x^\mu. From the general solution there are n more generators, known as special conformal transformations, given by :X_\mu = c_x^\nu x^\rho, where the traceless part of c_ over \mu,\nu vanishes, hence can be parametrised by c^\mu_ = b_\nu. We Taylor expand X_\mu in x^\mu to obtain an (infinite) linear combination of terms of the form :X_\mu = a_x^\cdots x^, where the tensor \mathbf is symmetric under exchange of \mu_i,\mu_j but not necessarily \mu with \mu_i. For simplicity, we restrict to n = 2, which will be informative for higher order terms later. The conformal Killing equation gives :a_ + a_ - \frac\eta_a^\sigma_ = 0. We now project a_ into two independent tensors: a traceless and pure trace part over its first two indices. The pure trace automatically satisfies the equation and is the c_ in the answer. The traceless part \tilde a_ satisfies the regular Killing equation, showing \tilde\mathbf is antisymmetric on the first two indices. It is symmetric on the second two indices. This shows that under a cyclic permutation of indices, \tilde\mathbf picks up a minus sign. After three cyclic permutations, we learn \tilde\mathbf = 0. Higher order terms vanish (to be completed) Together, the n translations, n(n-1)/2 Lorentz transformations, 1 dilatation and n special conformal transformations comprise the conformal algebra, which generate the
conformal group In mathematics, the conformal group of an inner product space is the group of transformations from the space to itself that preserve angles. More formally, it is the group of transformations that preserve the conformal geometry of the space. Se ...
of pseudo-Euclidean space.


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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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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Einstein manifold In differential geometry and mathematical physics, an Einstein manifold is a Riemannian or pseudo-Riemannian differentiable manifold whose Ricci tensor is proportional to the metric. They are named after Albert Einstein because this condition i ...
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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 ...
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Invariant differential operator In mathematics and theoretical physics, an invariant differential operator is a kind of mathematical map from some objects to an object of similar type. These objects are typically functions on \mathbb^n, functions on a manifold, vector valued fun ...
* Killing vector field *
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

* Wald, R. M. (1984). General Relativity. The University of Chicago Press. Differential geometry Mathematical methods in general relativity