Null Geodesic Path Integrals and Observational Signatures of Spacetime Curvature: From Shapiro Delay to Gravitational Lensing

Light travel time is not just a function of how far apart a source and observer are. Two sources at the same proper distance can send photons that arrive at noticeably different times, because their paths through the gravitational field differ. This paper derives the travel time difference Δt directly from the pseudo-Riemannian metric — no approximations — and recovers the weak-field and exact Schwarzschild results as special cases. The Shapiro delay (Shapiro 1964) and gravitational lensing time delays (Refsdal 1964) both fall out of the same equation. An asymmetry measure κ ∈ 0,1 is introduced that captures how different the two path integrals are, and pinpoints the symmetric case κ=0 exactly. The condition for Δt=0 is a metric isometry mapping one geodesic to the other; the case where the integrals cancel by coincidence without any such symmetry is also described. Extensions to the post-Newtonian hierarchy (2PN corrections), frame-dragging in the Kerr metric, and gravitational wave perturbations of the travel time are derived. Six primary measured systems — from 246 μs (Cassini solar conjunction) to 417 days (Q0957+561) — are tabulated and compared against the theory, alongside an extended catalogue of 14 additional gravitational lensing systems from the literature. Three full step-by-step worked examples are provided with complete numerical calculations. The open-source Python package NGO (Null Geodesic Observer, v0.1.0) implements the full framework: RK45 null geodesic integration, exact Schwarzschild travel time, the asymmetry measure, performance benchmarks, numerical error analysis, and a database of 20 known systems. The software is available at https://github.com/chiragrathiresearcher/null-geodesic-observer under the MIT license.