The glow of eternal black holes

We compute the optical appearance of a maximally extended (eternal) Schwarzschild black hole surrounded by a geometrically thin accretion disk. Unlike astrophysical black holes formed by gravitational collapse, the eternal solution contains a white hole (WH) region connected to a past singularity. Following Markov's hypothesis of a limiting density of matter, we replace the past singularity with a spacelike "surface of last scattering" at $r=r_{\rm m}$ and assume that it emits black-body radiation with temperature $T$. Past-directed rays that cross the past horizon terminate on this surface, resulting in a bright spot at the center of the shadow of an eternal black hole. The radial profile of the spot is set by the gravitational frequency shift, with a blueshifted center when the spacelike surface is sufficiently close to the singularity. We use ray tracing to generate an image of the disk and the spot, and we derive a closed-form interferometric signature of the spot. Its visibility is a pure exponential in the baseline length, in contrast to the power-law envelopes of the disk and the photon ring. We constrain the product $r_{\rm m} T$ from the observed 230 GHz fluxes of M87* and Sgr A*. We also consider a Planck-star scenario in which the value of $r_{\rm m}$ is determined on dimensional grounds, and show that it is out of reach of any current or planned facilities. Nevertheless, the darkness of the observed shadows remains a test of whether these black holes are eternal.