= Solution
Since the real <conformal factor> $\Lambda$ is nowhere zero, $\Lambda^2>0$ at every point. For any <tangent vector>,
$$
\widetilde g(V,V)=\Lambda^2g(V,V).
$$
The sign is unchanged. With <metric signature> $(-,+,+,+)$, negative, zero and positive values therefore describe the same <timelike vectors>, <null vectors> and <spacelike vectors> for both <metric tensors>. In particular the two <null cones> coincide.
Choose the same <time orientation> for the two <Lorentzian metrics>. A future-directed <causal curve> or <timelike curve> for one has the same character for the other. Hence the <causal future>, <causal past> and <chronological future> of every point agree:
$$
\boxed{
J^\pm_{\widetilde g}(p)=J^\pm_g(p),\qquad
I^\pm_{\widetilde g}(p)=I^\pm_g(p).
}
$$
This is the preservation of <causal structure> by a smooth, nonvanishing <conformal transformation>. It does not assert preservation of <proper time> or of <affine parameter> lengths.
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