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 ⟨Ψ(t)Ψ(0)⟩ 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 λ/r _g where r _g = GM/c ² is the gravitational radius of the black hole and λ 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 ∼10⁻¹³ λ _mm/M ₆, where λ _mm = λ/(1mm) and M ₆ = M/(10⁶ M _⊙). Consequently, detecting multi-path propagation near a black hole using the coherent electric field autocorrelation is infeasible with current technology.

事件视界望远镜最近拍摄了第一张黑洞图像。这些图像是通过测量在位于地球上的望远镜测量的复杂电场的相干相关函数来合成的。在假设光源发射空间非相干辐射的情况下，该相关函数对应于图像的傅立叶变换。然而，黑洞与标准的天体物理物体不同：在没有吸收和散射的情况下，观察者会看到每个发射位置的一系列逐渐缩小的回声。这些回波对应于在到达观察者之前绕黑洞运行一次或多次的射线。这种多径传播将空间和时间相关性引入电场，编码黑洞的特性，不考虑内在的可变性。我们探索了在单个望远镜上测量的相干时间自相关函数。具体来说，我们研究了标量场相关函数 ⟨Ψ(t )Ψ(0)⟩ 来源于位于史瓦西黑洞附近的波动物质。我们发现相关函数在等于光子轨道周期整数倍的时间达到峰值；相应的功率谱密度像λ / r _g一样消失，其中r _g = GM / c ²是黑洞的引力半径，λ是观察到的辐射波长。对于在毫米波长观察到的超大质量黑洞，回波的功率相对于直接发射被抑制了 ~10 ^-13 λ _mm / M ₆，其中λ _mm= λ /(1mm) 和M ₆ = M /(10 ⁶ M _⊙ )。因此，使用相干电场自相关检测黑洞附近的多径传播在当前技术下是不可行的。