Relativistic Image Formation

Relativistic image formation maps an emitting plasma to specific intensities on an observer’s screen. Null geodesics determine which events connect to each pixel; frequency shifts, aberration, emission, and absorption determine the brightness accumulated along each ray. The resulting annulus, asymmetry, and central depression are therefore joint spacetime–plasma predictions. Even a perfect ray integrator cannot make a reconstructed dark region a direct horizon image without an emissivity, opacity, sampling, and calibration model.

Helpful background. Black-Hole Geometry and Mechanics supplies null geodesics and observer tetrads; Relativistic Matter and Compact Objects supplies emissivity, absorption, and electron thermodynamics; Black-Hole Astrophysics supplies accretion-flow context; and Computational Relativity supplies geodesic integration and covariant-transfer validation.

Frequencies and invariant radiation variables

Let $k^\mu=dx^\mu/d\lambda$ be a future-directed null tangent normalized as the photon four-momentum divided by Planck’s constant $h$, and let $u^\mu$ be an observer or emitter four-velocity. This normalization makes $-k\cdot u$ the ordinary frequency $\nu$ in cycles per unit time; using the photon momentum itself would give energy $h\nu$, or angular frequency $2\pi\nu$ when $\hbar=1$. Then

$$\nu=-k_\mu u^\mu>0, \qquad g\equiv\frac{\nu_{\rm obs}}{\nu_{\rm em}} =\frac{-k\cdot u_{\rm obs}}{-k\cdot u_{\rm em}}.$$

The redshift factor $g$ contains gravitational shift and special-relativistic Doppler shift in a coordinate-independent ratio. It depends on the emitting velocity and ray, not merely on radius. Liouville’s theorem motivates the invariant intensity, emissivity, and absorptivity combinations

$$\mathcal I=\frac{I_\nu}{\nu^3}, \qquad \mathcal J=\frac{j_\nu}{\nu^2}, \qquad \mathcal A=\nu\alpha_\nu.$$

These invariants prevent an easy error: inserting one factor of $g$ for photon energy while forgetting arrival rate and solid-angle transformations. Younsi and collaborators formulate the covariant transfer problem and its invariant coefficients in Younsi et al. 2012, § 2 and § 2.1, equations (9) and (13)–(18).

Deriving the formal solution

In the emitter’s local orthonormal frame, transfer over positive proper path length $d\ell_{\rm prop}$ toward the observer is

$$dI_{\nu_{\rm em}}= \left(j_{\nu_{\rm em}}-\alpha_{\nu_{\rm em}}I_{\nu_{\rm em}}\right) d\ell_{\rm prop}, \qquad d\tau_{\nu}=\alpha_{\nu_{\rm em}}d\ell_{\rm prop}.$$

For a future-directed affine parameter, $d\ell_{\rm prop}=(-u_{\rm em}\cdot k)d\lambda$. A code may trace rays backward from the camera, but a positive absorption coefficient must attenuate radiation. The accumulated optical depth increases along propagation; the remaining optical depth from an emission point to the observer decreases as that point approaches the observer. Combining the local equation with $I_\nu/\nu^3$ invariance gives, for no incident background,

$$I_{\nu_{\rm obs}}= \int_{\rm ray}g^3j_{\nu_{\rm em}} e^{-\tau_\nu(\ell)}d\ell_{\rm prop}, \qquad \nu_{\rm em}=\frac{\nu_{\rm obs}}{g}.$$

The optical depth in the exponential is measured from the emission event to the observer. Thus every coefficient is evaluated at the local emitted frequency. Integrating the screen intensity gives flux density,

$$F_{\nu_{\rm obs}}=\int I_{\nu_{\rm obs}}(\alpha,\beta)\,d\Omega.$$

Two checks fix normalization and sign. In flat spacetime with comoving source and observer, $g=1$ and the ordinary transfer equation returns. For a homogeneous cell with constant source function $S_\nu=j_\nu/\alpha_\nu$, the solution approaches $I_\nu=S_\nu$ as $\tau\to\infty$, whereas its optically thin limit is $I_\nu\simeq j_\nu\ell$. A negative optical depth or an optically thick intensity that grows without approaching the source function reveals a ray-orientation error.

How geometry and plasma share the image

Lensing changes path multiplicity and dwell time near unstable null motion. The $g^3$ factor makes approaching material brighter and receding material dimmer, while gravitational redshift suppresses radiation emitted deep in the potential. Absorption can move the photosphere outward. The synchrotron coefficients depend on magnetic-field strength and direction, electron density, electron distribution, and temperature; a GRMHD snapshot does not determine these coefficients until an electron prescription is supplied.

The EHT M87 physical-model study makes this dependence concrete. Its paper-defined library contains 43 ideal-GRMHD simulations in SANE and MAD magnetic-flux states and about 60,000 ray-traced images. BHAC, H-AMR, iharm, and KORAL supplied simulations; ipole, RAPTOR, and BHOSS supplied cross-checked transfer calculations. The grid varied spin, inclination, snapshot, and $R_{\rm high}=1,10,20,40,80,160$, a parameter controlling the ion-to-electron temperature ratio in gas-dominated regions. Images were made at 230 GHz with a $160\,\mu\mathrm{as}$ field of view and $1\,\mu\mathrm{as}$ pixels, then rescaled in angular size, position angle, and flux for visibility comparison Event Horizon Telescope Collaboration 2019, §§ 3.1–3.3 and Eq. (8).

Changing $R_{\rm high}$ moves emission between disk and funnel wall even when spacetime and fluid snapshot are fixed. An alternate electron distribution, radiative cooling, tilted flow, nonthermal tail, or slow-light treatment can change the image again. The robust inference is therefore a conditional comparison across a declared library, not a metric-only photograph.

Named visibility release and inference boundary

The observational anchor is EHT data product 2019-D01-01, “First M87 EHT Results: Calibrated Data,” DOI 10.25739/g85n-f134. It contains calibrated Stokes-$I$ visibility products for the low and high bands on 2017 April 5, 6, 10, and 11, with the release flags and station metadata Event Horizon Telescope Collaboration 2019, data product 2019-D01-01. Paper III describes correlation, fringe fitting, a priori amplitude calibration, network calibration, and validation in Event Horizon Telescope Collaboration 2019, §§ 4–7. This page performs no new image-library fit.

For a reproducible comparison, the dataset version and every selected day, band, station, scan, flag, signal-to-noise rule, coherent averaging interval, total-flux constraint, and visibility product must be recorded. Calibration must preserve gain, phase, clock, network-calibration, polarization-combination, angular-scale, flux-scale, and scattering conventions. Although interstellar scattering toward M87 is small at 230 GHz, applying or omitting a kernel remains a declared operation.

The likelihood belongs in complex-visibility and closure-quantity space after the synthetic movie has been time averaged, Fourier sampled at the observed coordinates, flagged, and corrupted with thermal and gain errors. Mass-to-distance ratio, flux, orientation, station gains, and variability are nuisance parameters. Priors and posterior summaries must be stated. Thermal noise and finite sampling set statistical uncertainty; sparse coverage, calibration, fast-light approximation, electron thermodynamics, numerical resolution, physical-library incompleteness, and source evolution are distinct systematics and sources of model dependence.

Results and limitations through 2026-08-18

Invariant transfer and the $g^3$ formal solution are durable. The detailed 230-GHz emitting region and electron distribution remain model dependent. A faithful software reproduction should first pass flat-spacetime, optically thin, and constant-source-function tests. It should then ray trace named GRMHD snapshots with pinned code commits, camera, frequency, field of view, pixel scale, and electron model; compare at least two $R_{\rm high}$ values and an alternate emissivity prescription; and finally pass the images through the released array sampling and fit them in data space.

This is an honest reconstruction recipe, not a claimed rerun of the EHT simulation library. The paper-defined library is not replaced by an undocumented “similar” snapshot. A future result should publish snapshot identifiers, transfer-code version, build environment, random seeds or deterministic sampling, and the exact visibility-selection file needed to regenerate its comparison.

Common pitfalls

Using $g$ instead of $g^3$. One redshift factor accounts only for photon energy. Invariant intensity supplies the full cubic transformation before local frequency dependence is evaluated.

Ray tracing a fluid variable as brightness. Density is not emissivity. Electron thermodynamics, magnetic field, frequency, absorption, and path length must enter radiative transfer.

Solved diagnostic: an emitting cell

A homogeneous cell has constant local source function $S_\nu=10$ in arbitrary intensity units, optical depth $\tau=2$, redshift factor $g=0.8$, and no incident background. Find the observed intensity. Compare it with the optically thin approximation and explain the check.

Solution

The exact constant-cell solution is

$$I_{\nu_{\rm obs}}=g^3S_\nu(1-e^{-\tau}) =(0.8)^3(10)(1-e^{-2})=4.43.$$

The optically thin approximation would replace $1-e^{-\tau}$ by $\tau$, giving

$$I_{\nu_{\rm obs}}^{\rm thin}=g^3S_\nu\tau=10.24.$$

It fails because $\tau=2$ is not small and exceeds the source-function limit $g^3S_\nu=5.12$. The exact result lies below that limit and approaches it as optical depth grows. This simultaneously checks the redshift normalization, absorption sign, and large-optical-depth behavior.

References

• Event Horizon Telescope Collaboration (2019). “First M87 EHT Results: Calibrated Data.” Data product 2019-D01-01. DOI.
• Event Horizon Telescope Collaboration (2019). “First M87 Event Horizon Telescope Results. III. Data Processing and Calibration.” The Astrophysical Journal Letters 875, L3. DOI; Open PDF.
• Event Horizon Telescope Collaboration (2019). “First M87 Event Horizon Telescope Results. V. Physical Origin of the Asymmetric Ring.” The Astrophysical Journal Letters 875, L5. DOI; Open PDF.
• Younsi, Z., Wu, K., and Fuerst, S. V. (2012). “General Relativistic Radiative Transfer: Formulation and Emission from Structured Tori around Black Holes.” Astronomy & Astrophysics 545, A13. DOI; Open PDF.