A black hole is invisible, but it's betrayed by light from superheated matter swirling nearby. That light is like a double agent: on one side of its orbit, it blueshifts as it races toward us; on the other, it redshifts as it recedes. But gravity itself (by warping spacetime) also repaints the rays, sapping their energy. To separate one effect from the other, scientists compare readings from two symmetric "informants." The difference in their colors, measured via spectral analysis, lets them use simple formulas to calculate the mass of a black hole and its distance — like weighing an invisible phantom by the tremor in its accomplice's voice.
The method can handle the system's overall motion: even if the hole and its entourage are hurtling through the galaxy, symmetry saves the day.
The color of light emitted near a black hole can shift so dramatically that ultraviolet radiation turns into infrared — purely due to gravity, without any motion involved.
In Interstellar, the bending of light around a black hole is shown — the very effect that new formulas use for 'weighing.'