  The Event Horizon Telescope recently produced the first images of a black
hole. These images were synthesized by measuring the coherent correlation
function of the complex electric field measured at telescopes located across
the Earth. This correlation function corresponds to the Fourier transform of
the image under the assumption that the source emits spatially incoherent
radiation. However, black holes differ from standard astrophysical objects: in
the absence of absorption and scattering, an observer sees a series of
increasingly demagnified echos of each emitting location. These echos
correspond to rays that orbit the black hole one or more times before reaching
the observer. This multi-path propagation introduces spatial and temporal
correlations into the electric field that encode properties of the black hole,
irrespective of intrinsic variability. We explore the coherent temporal
autocorrelation function measured at a single telescope. Specifically, we study
the simplified toy problem of scalar field correlation functions $\langle
\Psi(t) \Psi(0) \rangle$ sourced by fluctuating matter located near a
Schwarzschild black hole. We find that the correlation function is peaked at
times equal to integer multiples of the photon orbit period; the corresponding
power spectral density vanishes like $\lambda/r_{\rm g}$ where $r_{\rm g} = G M
/ c^{2}$ is the gravitational radius of the black hole and $\lambda$ is the
wavelength of radiation observed. For supermassive black holes observed at
millimeter wavelengths, the power in echos is suppressed relative to direct
emission by $\sim 10^{-13} \lambda_{\rm mm}/M_{6}$, where $\lambda_{\rm mm} =
\lambda/(1\,{\rm mm})$ and $M_6 = M/(10^6 M_\odot)$. Consequently, detecting
multi-path propagation near a black hole using the coherent electric field
autocorrelation is infeasible with current technology.
