Chapter 39: The EHT Re-Read: The Ring Is the Donut

In April 2019, the Event Horizon Telescope (EHT) collaboration EHT2019 released the first image of a black hole shadow: the supermassive object M87* at the center of the galaxy Messier 87. The image showed a bright ring of emission surrounding a dark central region—the shadow—with a diameter of ∼ 42 μas, consistent with the Schwarzschild prediction for a 6.5 × 10^9 M_☉ black hole at a distance of 16.8 Mpc. In 2022, the EHT released EHTSgrA2022 a second shadow image: Sgr A*, the 4.3 × 10^6 M_☉ object at the center of the Milky Way.

Both images were interpreted within the framework of general relativity, where the shadow is cast by a spherical (S²) event horizon. In Category τ, the interpretation is fundamentally different. The shadow is a toroidal (T²) readout. The bright ring is not the photon sphere of a Schwarzschild metric; it is the equatorial emission from the accretion funnel

wrapped around the donut, not the sphere.

This chapter develops the Shadow Shape Theorem , which predicts specific differences between T² and S² shadows: inner-shadow ellipticity, photon subring spacing, and polarization winding patterns. These are falsifiable predictions: next-generation EHT observations (with space baselines) can distinguish the two topologies.