Proposition: Conformal ambiguity of causal data and clock lapse

Let $q$ be a time-oriented Lorentzian metric and $\tilde{q}={\Omega }^{2}q$, where $\Omega >0$ is smooth. The future causal curves, and hence their pointwise causal relation, are the same for both metrics.

For a fixed scalar $T$ with timelike gradient,

satisfies

Thus causal order alone does not determine this calibrated raw lapse or proper-time scale.

Proof. For every tangent $v$, $\tilde{q}(v,v)={\Omega }^{2}q(v,v)$. Its causal character and the chosen future cone are unchanged. The same curves are therefore future causal.

Since ${\tilde{q}}^{-1}={\Omega }^{-2}{q}^{-1}$, substitution gives (eq:conformal-lapse). For the proper-time statement it suffices to take $\Omega =\lambda >0$ constant: proper times and lengths are multiplied by $\lambda$, with causal order unchanged.