Null Geodesic Integration

Backward ray tracing starts each image pixel with a local null momentum and integrates it through a specified spacetime. The central numerical claim is not that a plotted curve “looks bent,” but that it remains null, preserves every available constant of motion, crosses charts correctly, and approaches an analytic trajectory under tolerance and resolution refinement. These controls separate geometric lensing from integrator artifacts.

Required background. Affine and Variational Geodesics supplies the geodesic equation, affine parametrization, and constants generated by spacetime symmetries.

Helpful background. Formulations of General Relativity supplies numerical spacetime representations; Differential Equations for Relativity supplies adaptive integration and event location; Black-Hole Astrophysics supplies the source-model context; and Computational Relativity supplies the shared verification framework.

Hamiltonian rays and their residuals

Assume geometric optics in vacuum, or a medium whose refractive correction to the ray trajectory is negligible. Write $p^\mu=dx^\mu/d\lambda$ for a future-directed null tangent and $p_\mu=g_{\mu\nu}p^\nu$. Its constant normalization is arbitrary for the trajectory. To use the cyclic frequency $\nu$ of the transfer pages, normalize it by $-p\cdot u_{\mathrm{obs}} =\nu_{\mathrm{obs}}$; with $\hbar=1$ the physical photon four-momentum is $P^\mu=2\pi p^\mu$. Evolve the covector with the Hamiltonian

$$\mathcal H(x,p)=\frac12 g^{\mu\nu}(x)p_\mu p_\nu=0.$$

Hamilton’s equations are

$$\frac{dx^\mu}{d\lambda}=g^{\mu\nu}p_\nu, \qquad \frac{dp_\mu}{d\lambda} =-\frac12\partial_\mu g^{\alpha\beta}p_\alpha p_\beta.$$

The affine scale is arbitrary; the camera frequency fixes a convenient normalization. This first-order form avoids explicitly constructing every Christoffel symbol, while a second-order coordinate implementation provides a useful independent route. Vincent and collaborators give both practical geodesic equations and Kerr tests in Vincent et al. 2011, §§2.1.1–2.1.2, Eqs. (1)–(8), and Appendix A.

Monitor a dimensionless null residual, for example $|\mathcal H|$ divided by a positive scale formed from the separate metric-momentum terms. The denominator must not vanish through cancellation. In a stationary axisymmetric spacetime, also track $E=-p_t$ and $L_z=p_\phi$; in Kerr, track the Carter constant $Q$. These are stronger diagnostics than coordinate position because they test different contractions of the numerical state.

A capture threshold that can be calculated by hand

In the equatorial plane of Schwarzschild, let $m=GM$, $E=-p_t$, and $L=p_\phi$. The null condition gives

$$\left(\frac{dr}{d\lambda}\right)^2 +L^2\frac{1-2m/r}{r^2}=E^2.$$

The effective potential is largest where $d[(1-2m/r)/r^2]/dr=0$, namely $r=3m$. At that maximum its value is $L^2/(27m^2)$, so a ray arriving from infinity has critical impact parameter

$$b_{\mathrm{crit}}=\frac{|L|}{E}=3\sqrt3\,m.$$

For $b\lt b_{\mathrm{crit}}$ it falls through the horizon; for $b\gt b_{\mathrm{crit}}$ it turns and escapes. The exactly critical ray approaches the unstable photon orbit asymptotically. This is a stronger capture test than weak deflection because it challenges both event classification and long integration near an unstable orbit.

Near the threshold, small initial-data or integration errors can switch the outcome. Report a converged interval bracketing the critical pixel rather than expecting a discontinuous binary label to converge uniformly. Conservation of $E$ and $L$ alone is insufficient: their Hamilton equations have exactly zero right-hand sides in these coordinates even if the radial motion is poorly integrated.

Integration, charts, and stopping events

Use adaptive local-error control, but do not equate the requested tolerance with achieved global accuracy. Repeat with tolerances $\epsilon$, $\epsilon/4$, and $\epsilon/16$, and check step halving over representative rays. Integrate a ray backward to its source or termination surface, reverse its momentum, and return to the camera. Reversibility error exposes accumulated phase and event-location error, although dissipative interpolation or an intentionally nonreversible integrator can alter the expected order.

Coordinates introduce failure modes unrelated to the physical ray. Horizon-penetrating charts avoid a coordinate singularity at a black-hole horizon; multiple charts may be needed near axes or numerical-grid boundaries. At each chart transition, transform both position and covector momentum and re-evaluate $\mathcal H$. Termination rules must distinguish capture, escape, entry into an emitting region, and departure from the available numerical metric domain. A ray stopped because interpolation data ended is not a captured ray.

For time-dependent numerical spacetimes, metric interpolation adds spatial and temporal error. Refine the ray tolerance beyond the point where interpolation dominates, then refine the metric data separately. High-order geodesic integration cannot recover small-scale geometry absent from the supplied snapshots.

A versioned benchmark ladder

The benchmark dataset is synthetic rather than observational. Pin an exact release or repository revision of an analytic ray suite, its configuration checksum, the metric formulas, and the geodesic software revision. As of the evidence cutoff 2026-08-18, no telescope likelihood or measured photon dataset is used on this page.

Start in flat spacetime, where rays are straight and reversal is exact up to numerical error. Continue to weak Schwarzschild deflection with impact parameter $b\gg GM$,

$$\Delta\phi=\frac{4GM}{b} +O\!\left(\frac{G^2M^2}{b^2}\right).$$

This tests the sign and leading normalization but fails near the photon region, where the asymptotic series is invalid. Next use exact Kerr first integrals over a grid of spin, inclination, and impact parameters. Demand convergence of trajectory differences together with $E$, $L_z$, $Q$, and the null residual. Gold and collaborators describe a code-comparison geodesic test in Gold et al. 2020, §§3.1 and 5.1, Fig. 1.

Calibration, selections, and uncertainty

Calibration fixes camera radius, tetrad normalization, frequency scale, image handedness, and the map from pixel centers to initial momenta. Selections fix spin, inclination, field of view, pixel mask, ray termination threshold, and which rays are excluded near chart or data boundaries. Declare these before inspecting image differences.

Use trajectory norms at common affine or physical events, conserved-quantity residuals, endpoint errors, and pixel-classification disagreements as comparison metrics. These are deterministic numerical residuals, not likelihoods; there are no priors or posterior claims here. Report sensitivity to coordinates, integrator, tolerance, metric interpolation, camera radius, and termination rules as separate systematic variations.

A clean reproduction path begins with the versioned metric and camera configuration, regenerates the initial null states, runs the tolerance and metric-resolution sweep, and emits trajectories, constants, residuals, event labels, and checksums. This specification does not claim that the calculation has been executed. Reject the result if null or symmetry residuals plateau above the target, reversibility fails at the expected order, event labels change without convergence, or coordinate choices alter invariant endpoints beyond uncertainty.

Check your understanding

A Kerr ray conserves $E$ and $L_z$ to $10^{-12}$ but the Carter constant drifts by $10^{-4}$ and the final pixel classification changes under step halving. Is conservation of the first two quantities sufficient?

Solution

No. Stationarity and axisymmetry can preserve $E$ and $L_z$ even when the meridional motion is inaccurate. The $Q$ drift and unstable event label show that the finite claim is unresolved. Tighten tolerance, inspect chart transitions and metric derivatives, compare with the separated Kerr first integrals, and require the classification and all residuals to converge together.

Continue with Observer Cameras and Local Tetrads.

References

• Gold, Roman, et al. 2020. “Verification of Radiative Transfer Schemes for the Event Horizon Telescope.” The Astrophysical Journal 897, 148. DOI.
• Vincent, Frédéric H., Thibaut Paumard, Éric Gourgoulhon, and Guy Perrin. 2011. “GYOTO: A New General Relativistic Ray-Tracing Code.” Classical and Quantum Gravity 28, 225011. arXiv.