Local light cones do not determine a spacetime’s global health

With signature −+++, a causal tangent is nonzero and has nonpositive squared norm. A time orientation chooses one continuous future cone. A future-directed causal curve keeps its tangent in that cone, piecewise smoothly if necessary. Timelike reachability and causal reachability are statements about the existence of entire curves between events, not merely a comparison of coordinate time labels. In curved spacetime there may be no preferred global coordinate time, and a locally valid timelike coordinate direction need not define a global ordering.

A closed timelike curve violates chronology even if every local metric neighborhood looks ordinary. Global hyperbolicity is a stronger causal condition involving an appropriate Cauchy evolution structure; local Lorentzian signature alone does not imply it. Horizons are also global or observer-region-dependent concepts. A Rindler horizon bounds an accelerated observer’s wedge in flat spacetime, while an event horizon is defined through reachability of an asymptotic region in a full spacetime. Neither concept is defined by declaring a divergent metric component to be physical.

A spacetime is geodesically complete if every maximally extended affine geodesic has parameter domain all real numbers, up to affine rescaling. Timelike proper time is affine; null curves need another affine parameter. An arbitrary finite segment is not evidence of incompleteness because it may simply be extended within the spacetime. The diagnostic concerns a maximal geodesic reaching a finite affine endpoint with no further event available in that manifold. Timelike, null, and spacelike completeness can also be considered separately.

An open subset of Minkowski spacetime can be incomplete merely because regular events were excluded. Such a subset can admit an extension. A curvature scalar diverging at a finite reachable endpoint can obstruct a smooth regular extension, but finite scalars do not prove completeness or exclude all singular behavior. In Schwarzschild geometry the invariant 48m²/r⁶ is finite at r=2m and diverges at r=0. Regular horizon-penetrating coordinates extend the first surface; they do not eliminate the latter curvature divergence. Statements about singularities must specify the spacetime and regularity class being considered.

A geodesic leaving a restricted wedge at finite affine parameter can continue in the containing Minkowski spacetime, so the wedge’s incompleteness is not a curvature singularity. Conversely a flat quotient with periodic timelike direction can be geodesically complete while containing closed timelike geodesics. Local curvature, global topology, causal restrictions, and completeness are independent pieces of geometric information. Singularity theorems combine precise global, focusing, and energy assumptions to infer incompleteness; they should not be paraphrased as a claim that every metric divergence proves infinite curvature.

The spacetime lab tests only the local null-cone aspect. Reset and set both controls to β=0,ct=0.6; with fixed x=0.6 predict transformed (x′,ct′)=(0.6,0.6) and interval readout zero. Reset and set β=0.6,ct=0.6, predicting (0.3,0.3), again null. The readout uses ct²−x², opposite our metric sign but also zero here. This one-event flat-space boost cannot establish global reachability, identify a spacetime quotient, or test geodesic completeness.