The shadow
Without spin, the shadow is a circle about 2.6 times the radius of the event horizon. Light aimed anywhere inside that circle ends up in the hole.
Only 100% is the real value.
The Sun's surface is about 5,800 K.
Each button flies the camera to a good view of one feature.
Without spin, the shadow is a circle about 2.6 times the radius of the event horizon. Light aimed anywhere inside that circle ends up in the hole.
The disk is flat. The dome above the hole, and the arch below it, are the far side of the disk. Its light is bent around the hole and reaches you from above and below.
A thin bright line hugs the shadow. This light went around the hole one or more times, near the photon sphere (1.5 rs without spin), before it escaped toward the camera.
A spinning hole drags the space around it along. Light travelling with the spin can get closer before it is captured, so the shadow is squeezed flat on the side where the disk gas moves toward you. The innermost stable orbit also moves in, from 3 rs with no spin to about 0.6 rs near the maximum.
Without spin, the inner disk orbits at about half the speed of light. In True physics, gas moving toward you looks brighter and bluer, and gas moving away looks dimmer and redder. Gravity also reddens and dims light that climbs out of the hole. Film look turns both shifts off, as the film's effects team did, so the two sides of the disk match.
Pick a black hole and a height above its horizon. A clock there ticks slower than a clock far away.
This calculator ignores spin. For a clock hovering above a non-spinning black hole, rate = √(1 − rs/r). Getting the film's 7 years per hour this way puts you about 80 metres above the horizon of a 100 million Sun hole, and no rocket could hold that position. In the film, Miller's planet can orbit that close because Gargantua spins at almost its maximum rate, which allows a stable orbit near the horizon.
The page follows light rays through curved spacetime, which is also how the film's effects team made its images. It runs on your graphics card, many times per second.
Each pixel sends a ray out of the camera. The ray takes a few hundred small steps, and the spacetime around the hole bends its path at every step. For a spinning hole the steps use Boyer-Lindquist coordinates and two conserved numbers, L and Q.
r″ = R′(r) / 2
A ray can fall through the horizon, and the pixel stays black. It can cross the disk and pick up some glow. Or it can escape and sample the star field in the direction it ends up pointing.
The disk's heat output follows the Page-Thorne thin-disk model for the current spin. It is zero at the innermost stable orbit, peaks just outside it and falls with radius. Temperature is the fourth root of that flux, scaled to the peak set by the temperature slider, 4,500 K by default (real disks around giant black holes are far hotter and glow mostly in ultraviolet), and brightness goes as T⁴. Turbulent clumps ride on the gas at the orbital rate, so inner gas outruns outer gas and shears them into trailing spirals. Each clump is replaced after about an orbit. True physics multiplies the temperature by g and the brightness by g⁴.
T ∝ F^(1/4), I ∝ g⁴ T⁴