Conformal Killing equation

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) flow defines conformal transformations, 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 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 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

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.
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 or pseudo-Euclidean space, 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 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 of pseudo-Euclidean space.

See also

* Affine vector field
* Curvature collineation
* Einstein manifold
* Homothetic vector field
* Invariant differential operator
* Killing vector field
* Matter collineation
* Spacetime symmetries

References

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

Differential geometry
Mathematical methods in general relativity