Conformal and Homothetic Killing Fields

Conformal Killing fields preserve angles and null cones rather than the metric itself. Their contraction with an affinely parametrized null geodesic is conserved, while the same contraction along a timelike geodesic generally changes. The difference follows from a single factor of the tangent norm and cleanly separates exact, homothetic, and conformal symmetry.

Required background. Tensor Fields and Natural Operations supplies pointwise contractions and symmetry operations; Flows and Lie Derivatives supplies the Lie derivative. Affine Connections and Covariant Derivatives supplies the general connection framework, and The Levi-Civita Connection supplies metric-compatible differentiation.

Helpful background. Calculus and Linear Algebra for Relativity supplies differentiation and bilinear forms; the infinitesimal conformal rescaling is developed here. Affine and Variational Geodesics supplies affine geodesics and causal character.

Three levels of metric symmetry

A vector field $X$ is conformal Killing when

$$\mathcal L_Xg_{ab}=2\psi g_{ab}, \qquad \nabla_{(a}X_{b)}=\psi g_{ab}.$$

Taking the trace in $n$ dimensions fixes

$$\psi=\frac{1}{n}\nabla_aX^a.$$

The important special cases are exact Killing symmetry, $\psi=0$, and homothety, $\psi$ constant. A general conformal factor varies from point to point. For the local flow $\Phi_s$, integration of $d(\Phi_s^*g)/ds=2(\psi\circ\Phi_s)\Phi_s^*g$ gives

$$\Phi_s^*g=\Omega_s^2g, \qquad \Omega_s(p)=\exp\!\left(\int_0^s\psi(\Phi_\tau(p))\,d\tau\right)\gt0.$$

Therefore the sign of $g(v,v)$ and the null cone are preserved. The conformal Killing equation and its null geodesic integral are given in Frolov, Krtouš, and Kubizňák 2017, §2.2, eqs. (2.32)–(2.33), p. 19, PDF.

The three cases form a strict hierarchy. Every Killing field is homothetic and every homothetic field is conformal, but neither converse holds in general. A nonzero constant $\psi$ changes every squared length at the same infinitesimal fractional rate. A variable $\psi$ changes that rate from point to point. In all three cases the angle between non-null vectors is preserved after normalization, while exact lengths survive only in the Killing case.

Because $\psi$ is fixed by the divergence, it is not an additional freely chosen field once $X$ and $g$ are given. Substituting the trace value back into the equation produces the trace-free conformal Killing equation, a useful direct test for candidate generators.

The null conservation law

Let $u^a$ be the tangent to an affinely parametrized geodesic and set $J_X=X_au^a$. Exactly as for a Killing field,

$$\frac{dJ_X}{d\lambda} =u^au^b\nabla_{(a}X_{b)} =\psi\,g_{ab}u^au^b.$$

The squared norm $\kappa=g(u,u)$ is constant along an affine geodesic. Therefore

$$\frac{dJ_X}{d\lambda}=\psi\kappa.$$

For a null geodesic, $\kappa=0$, so $J_X$ is conserved for every conformal Killing field. For a unit-speed timelike geodesic, $\kappa=-1$, so $dJ_X/d\lambda=-\psi$ and no local constant follows in general. The conservation law is therefore conformal for lightlike motion but metric for massive motion.

The algebra behind the null result is pointwise. A null tangent satisfies $g(u,u)=0$, so the trace part $\psi g_{ab}$ cannot be detected by contracting twice with $u$. Timelike and spacelike tangents do detect it. Conversely, if a smooth vector field produces a constant $X\cdot u$ for all affinely parametrized null geodesics, its symmetric derivative is constrained only up to a multiple of the metric; under the usual Lorentzian dimension assumptions, that is the conformal Killing equation.

Homotheties and nonlocal corrections

When $\psi=c$ is constant, integration gives a simple affine correction:

$$J_X-c\kappa\lambda=\text{constant}.$$

This corrected quantity depends explicitly on the affine origin and is not the ordinary Killing integral. When $\psi$ varies, one can formally subtract $\kappa\int\psi(\gamma(\lambda))d\lambda$, but the result depends on the history of the curve. It is not a local polynomial first integral on the tangent bundle.

The null statement also requires attention to parametrization. A conformal rescaling maps null geodesics to null geodesics as unparameterized curves, but generally changes their affine parameters. If $u^b\nabla_bu^a=f u^a$, then $dJ_X/d\lambda=\psi\kappa+fJ_X$; for a null curve, the remaining $fJ_X$ term vanishes only after an affine choice or a compensating rescaling.

Dilation in flat spacetime

In Minkowski space, the dilation field

$$D=t\partial_t+x^i\partial_i$$

satisfies $\mathcal L_Dg=2g$, so $\psi=1$. Along an affine straight line $x^a=x_0^a+u^a\lambda$,

$$J_D=x_au^a=x_{0a}u^a+\kappa\lambda.$$

For a null line the second term vanishes, and $J_D$ is constant. For a unit timelike line it decreases as $-\lambda$. This one example rules out the common but incorrect claim that every conformal Killing field generates a constant for every geodesic.

For an example with nonconstant $\psi$, take a constant covector $b_a$ and the special conformal field

$$X^a=2(b_cx^c)x^a-(x_cx^c)b^a.$$

Differentiating in Cartesian coordinates gives $\partial_{(a}X_{b)}=2(b_cx^c)\eta_{ab}$, so $\psi=2b_cx^c$ varies with position. The same null conservation law applies even though this field is not homothetic when $b\ne0$.

Exact Killing, homothetic, and general conformal fields also differ globally. A local solution need not be complete, and a conformal flow need not extend to a global conformal group. Nothing in the differential equation alone defines a physical charge or fixes the normalization of a generator.

Nor does conformal invariance make massive geodesics invariant as unparameterized curves. Under a nonconstant metric rescaling, the connection changes by terms involving the gradient of the conformal factor. For a null tangent those extra terms are tangent to the curve and can be absorbed into reparametrization; for a non-null tangent there is also a transverse term. That is the geometric reason the distinction persists beyond the conserved-scalar calculation.

Exercises

Suppose $X$ is homothetic with $\mathcal L_Xg=2c\,g$. Prove that $J_X-c\kappa\lambda$ is conserved on an affine geodesic and check that the formula is unchanged when the affine parameter is shifted by a constant.

Solution

Metric compatibility and the affine geodesic equation give $dJ_X/d\lambda=c\kappa$. Since $\kappa$ is constant, differentiation of $J_X-c\kappa\lambda$ gives zero. Under $\lambda'=\lambda+\lambda_0$, the expression becomes $J_X-c\kappa\lambda'+c\kappa\lambda_0$. It differs only by an additive constant, so its conservation is unaffected, although its numerical zero depends on the chosen affine origin.

References

• Valeri P. Frolov, Pavel Krtouš, and David Kubizňák, “Black Holes, Hidden Symmetries, and Complete Integrability,” Living Reviews in Relativity 20, 6 (2017). DOI. Open PDF.