Conformal Metrics and Preserved Causal Structure

Two metrics are conformally related when $\widetilde g=\Omega^2g$ for a smooth positive function $\Omega$. They have exactly the same pointwise timelike, null, and spacelike directions, and their null geodesics trace the same unparametrized curves. They do not share proper time, affine parameters, volume, curvature, or timelike geodesics in general. A conformal class therefore remembers the cone field but forgets its local scale.

Required background. Tensor Fields and Natural Operations supplies tensor rescaling and curves.

Helpful background. Calculus and Linear Algebra for Relativity supplies quadratic-form scaling; Exterior Algebra and Differential Forms supplies forms and volume; Events, Intervals, and Minkowski Geometry supplies null trajectories in flat spacetime.

Cone data in a conformal class

Let $g$ be pseudo-Riemannian and let $\Omega:M\to(0,\infty)$ be smooth. Then

$$\widetilde g(v,v)=\Omega^2g(v,v).$$

Because $\Omega^2$ is positive, the sign of every squared norm is unchanged. Thus $g$ and $\widetilde g$ have the same null cone and the same timelike and spacelike regions at every point. If a time orientation has been chosen, it can be used for both metrics. Orthogonality is also preserved because $g(v,w)=0$ if and only if $\widetilde g(v,w)=0$.

The scale does change. For a timelike curve, $d\widetilde\tau=\Omega\,d\tau$; for a spacelike curve, $d\widetilde\ell=\Omega\,d\ell$. In $n$ dimensions,

$$\widetilde g^{\mu\nu}=\Omega^{-2}g^{\mu\nu}, \qquad \mathrm{vol}_{\widetilde g}=\Omega^n\mathrm{vol}_g.$$

For an oriented $n$-manifold of the same signature and orientation, the Hodge stars on degree-$p$ forms satisfy

$$\widetilde\star\alpha=\Omega^{n-2p}\star\alpha.$$

In four dimensions, the Hodge star on two-forms is conformally invariant, but the volume form still scales by $\Omega^4$.

The power $n-2p$ follows directly from the defining pairing $\alpha\wedge\star\beta=\langle\alpha,\beta\rangle_g\mathrm{vol}_g$. The volume contributes $\Omega^n$, and contracting $p$ covector indices uses $p$ inverse metrics, contributing $\Omega^{-2p}$. Their product gives the displayed scaling without choosing coordinates.

Why null paths survive

For the path statement we need one result from the next chapter: a metric determines its Levi-Civita connection, and an affinely parametrized geodesic has a tangent $k$ satisfying $\nabla_k k=0$. This means its tangent is parallel transported along itself. Substituting $\widetilde g=\Omega^2g$ and $\widetilde g^{-1}=\Omega^{-2}g^{-1}$ into the Christoffel formula gives

$$\widetilde\Gamma^\rho{}_{\mu\nu} =\Gamma^\rho{}_{\mu\nu} +\delta^\rho{}_\mu\,\partial_\nu\log\Omega +\delta^\rho{}_\nu\,\partial_\mu\log\Omega -g_{\mu\nu}\partial^\rho\log\Omega.$$

Let $k^\mu=dx^\mu/d\lambda$ be an affinely parametrized $g$-null geodesic. Using $g(k,k)=0$ and $\nabla_kk=0$ gives

$$k^\nu\widetilde\nabla_\nu k^\mu =2\,k(\log\Omega)\,k^\mu.$$

The acceleration is parallel to $k$, so the image of the curve is still a null geodesic for $\widetilde g$, but $\lambda$ is generally not affine. If $d\widetilde\lambda/d\lambda=\Omega^2$ along the curve, then $dx/d\widetilde\lambda=\Omega^{-2}k$ is affinely parametrized for $\widetilde g$.

For a timelike geodesic the omitted connection term $-g(k,k)\nabla^\mu\log\Omega$ is nonzero and generally transverse to $k$. Therefore conformal maps do not preserve timelike geodesic trajectories. Tong derives the connection transformation and the preservation of unparametrized null geodesics in Tong 2019, §4.4.1, pp. 168–170, PDF.

A conformally flat check

Consider

$$\widetilde g=a(\eta)^2 \left(-d\eta^2+dx^2+dy^2+dz^2\right), \qquad a(\eta)>0.$$

The coordinate curves $x=\pm\eta+\text{constant}$ with $y,z$ fixed remain null because $-d\eta^2+dx^2=0$. Hence their paths in the $(\eta,x)$ plane are the same as for Minkowski space. If $\eta$ is an affine Minkowski parameter, an affine parameter for $\widetilde g$ obeys

$$d\widetilde\lambda=a(\eta)^2\,d\eta.$$

The volume element is $a(\eta)^4\,d\eta\wedge dx\wedge dy\wedge dz$, so causal directions remain fixed while both affine scale and four-volume change. This single example tests all three claims separately.

Limits of conformal information

The function $\Omega$ must be smooth and nowhere zero for $g$ and $\widetilde g$ to be two regular metrics on the same manifold. If $\Omega$ vanishes at an added conformal boundary, the rescaling is no longer an equivalence there; statements at that boundary need separate hypotheses. A negative representative produces the same square $\Omega^2$, but choosing $\Omega>0$ gives an unambiguous smooth scale and does not reverse orientation.

Equality of cone fields constrains local propagation directions. This page does not develop causal hierarchy, domains of dependence, conformal infinity, or global compactification. It also does not assert equality of curvature: derivatives of $\Omega$ enter the transformed Riemann and Ricci tensors. The conformal class determines neither a preferred clock nor a preferred volume measure.

A useful inventory is therefore:

• the sign of $g(v,v)$, orthogonality, time orientation, and the set of null directions are preserved;
• null geodesic images are preserved, but their tangent normalization and affine parameters are not;
• timelike and spacelike lengths, volume, the Levi-Civita connection, and curvature generally change; and
• the Hodge star is preserved only in the middle degree $p=n/2$.

This inventory also clarifies null hypersurfaces. If $df$ is null for $g^{-1}$, then it remains null for $\widetilde g^{-1}=\Omega^{-2}g^{-1}$, so the same level surface is null. Its raised generator changes:

$$(df)^{\widetilde\sharp} =\Omega^{-2}(df)^\sharp.$$

The generator’s line is preserved, but its normalization is not. Likewise, the screen metric on a chosen cross-section scales by $\Omega^2$ and its area form by the corresponding dimensional power.

The positivity of the conformal factor is essential to this smooth equivalence. Writing a rescaling as an arbitrary scalar multiple rather than a positive square can permit a sign change or a zero, altering the signature or destroying nondegeneracy. A conformal boundary intentionally uses such a singular representative to add ideal points; it is not another regular metric at the boundary itself.

Exercises

Let $\widetilde g=e^{2Ht}(-dt^2+dx^2)$ in two dimensions, with constant $H$. For the right-moving null line $x=t$, find an affine parameter $\widetilde\lambda$ for $\widetilde g$ in terms of $t$, choosing $\widetilde\lambda(0)=0$ and $d\widetilde\lambda/dt=1$ at $t=0$. State what happens when $H=0$.

Solution

The Minkowski parameter $t$ is affine and $\Omega=e^{Ht}$. Hence

$$\frac{d\widetilde\lambda}{dt}=e^{2Ht}.$$

For $H\ne0$,

$$\widetilde\lambda(t)=\frac{e^{2Ht}-1}{2H}.$$

When $H=0$, the limit is $\widetilde\lambda=t$, as expected because the metrics then coincide.

References

• Tong, David. General Relativity. University of Cambridge lecture notes, 2019, §4.4.1, pp. 168–170. Official course PDF.