Teleology and Local Undetectability

Two observers can occupy isometric spacetime neighborhoods and nevertheless disagree—correctly—about whether the central event lies on an event horizon, because their spacetimes differ outside those neighborhoods or later in time. Every local experiment then gives the same result, while the global escape classification differs. This is the operational content of event-horizon teleology and local undetectability; it is not a claim that future matter sends a force into the past.

Helpful background. Exact Solutions of Einstein’s Equations supplies collapse and stationary examples; Causality and Global Structure supplies causal boundaries; The Einstein Equation governs the later developments being compared; Hypersurface Normals and Tangential Projectors supplies the local surface geometry used by quasi-local alternatives.

Locally identical, globally different

Choose a small neighborhood $U$ in Minkowski spacetime and a later time well to its future. Construct two global developments that agree exactly on $U$. In the first, no matter arrives and every outgoing light ray from $U$ reaches future null infinity. In the second, a sufficiently massive spherical shell later collapses. Choose $U$ so that the event horizon selected by the second development passes through it before the shell arrives.

The metric, Riemann tensor, matter fields, and all of their derivatives agree on $U$. Freely falling laboratories therefore record the same tidal forces, clock comparisons, light propagation, and detector responses. Yet the central point belongs to $J^-(\mathscr I^+)$ in the first spacetime and lies on $\partial J^-(\mathscr I^+)$ in the second. No scalar built from a finite jet of the local fields can distinguish them.

This construction is stronger than the statement that Schwarzschild coordinates behave smoothly at $r=2GM$ after a coordinate change. It says that even the complete local geometry need not identify event-horizon membership. Booth emphasizes this limitation of the event-horizon definition in Booth 2005, §2.1, pp. 1075–1079.

What teleology means

The event horizon is defined using a final escape condition. In a collapse problem one can locate late generators near the settled horizon and integrate them backward. The boundary begins to expand before infalling stress-energy crosses it because the backward selection chooses precisely those null rays that fail to escape in the completed solution.

This resembles solving an ordinary differential equation with a final boundary condition. The selected solution at an earlier parameter value depends on the later condition, but no physical influence propagates against the differential equation’s causal characteristics. Likewise, changing the future spacetime changes which earlier null surface deserves the global label “event horizon”; it does not change the local field equations in the shared past neighborhood.

The distinction matters when discussing an “anticipatory” area increase. A nearby observer cannot use it to learn that a shell will fall in, because the observer has no local procedure for identifying the globally selected surface. Only an analyst who already knows the future development can say which local null congruence to monitor.

Measurements at a regular crossing

A small freely falling laboratory measures the metric in a local orthonormal frame, curvature through geodesic deviation, and matter fields through local couplings. At a regular event horizon of a sufficiently massive black hole, these readings can be finite and unremarkable. The equivalence principle does not erase curvature, but it does rule out interpreting the event horizon as a material membrane that must exert a singular local force.

Extended or accelerated experiments can reveal useful structure without locally proving horizon membership. An observer may measure redshift relative to distant clocks, the behavior of a chosen null congruence, or the inability of signals to return during a finite protocol. Those results depend on an extended region, an observer family, or a finite observation window. None establishes that escape is impossible for the entire future.

Similarly, detecting a marginal surface is not detecting the event horizon. The null expansion of a specified two-surface is quasi-local and can be calculated from data near that surface, but its value depends on the surface and, for apparent horizons, on the slicing. In stationary Schwarzschild the surfaces coincide; that example is a consistency check, not an identity of definitions.

A useful response analogy

For a stationary black hole perturbed weakly and then allowed to settle, the event-horizon expansion is often computed with a future regularity or settling condition. Schematically, a linear equation along a generator has the form

$$\frac{d\theta}{dv}-\kappa\theta=-\mathcal S(v),$$

where $v$ parametrizes the generator, $\kappa\gt0$ is the background surface gravity in a fixed normalization, and $\mathcal S$ contains matter and shear sources at the appropriate perturbative order. Imposing $\theta\to0$ after the disturbance gives

$$\theta(v)=\int_v^\infty e^{-\kappa(v'-v)}\mathcal S(v')\,dv'.$$

The future integral displays the teleological boundary condition. It should not be read as a retarded observable available to a local detector. A quasi-local dynamical horizon instead supports flux relations tied to the surface as it evolves, but answers a different geometric question. Ashtekar and Krishnan contrast the two roles in Ashtekar and Krishnan 2025, §§1–2.

Limits of the statement

Local undetectability is a statement about generic event-horizon membership, not a prohibition on all inference. In an exactly known stationary solution, local measurements plus strong prior knowledge of the global family can infer parameters and hence a horizon location. That inference relies on the prior model and boundary normalization. It is not a universal scalar horizon detector.

Nor are all horizons locally undetectable in the same sense. Killing horizons can be recognized when the relevant Killing field is known; marginal surfaces are diagnosed by null expansions; isolated horizons satisfy intrinsic conditions. Each test imports different data. The correct conclusion is not that horizons are unknowable, but that the event horizon’s defining criterion is irreducibly global.

Check the alleged detector

A proposed instrument measures the Kretschmann scalar and declares that it has crossed an event horizon when the scalar exceeds a fixed threshold. Give two reasons this cannot be a general event-horizon detector.

Solution

First, the curvature at a regular horizon is not fixed universally. For a large Schwarzschild black hole it can be arbitrarily small, and in the flat portion of a collapsing-shell event horizon it can vanish. Second, large curvature can occur outside a black-hole region or in a spacetime with no future event horizon. Curvature is local, while membership in $\partial J^-(\mathscr I^+)$ depends on global escape. The instrument may be a curvature-warning device, but its threshold does not implement the event- horizon definition.

Black Holes Beyond Asymptotic Flatness next asks what replaces $\mathscr I^+$ when the conformal boundary is cosmological, timelike, or finite.

References

• Ashtekar, Abhay, and Badri Krishnan. “Quasi-Local Black Hole Horizons: Recent Advances.” Living Reviews in Relativity 28 (2025): article 8. DOI.
• Booth, Ivan. “Black-Hole Boundaries.” Canadian Journal of Physics 83 (2005): 1073–1099. DOI.