The factor, against distance from the horizon

At its defaults it draws the factor, against distance from the horizon. √(1 − rs/r) against radius in units of the Schwarzschild radius, with its reciprocal beside it. The first is the rate of a clock held still there, as read by somebody far away, and is also the redshift of anything it emits; the second is how much a radial ruler is stretched. At 10 rs the clock runs at 0.949 of its distant rate, which is a large effect for a quantity most of physics is allowed to call one. Both curves are drawn to the horizon, where one reaches zero and the other has no value at all.

schwarzschild is one function in lib/figures/gravity.js — geometry, horizons, and the waves in it. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

√(1 − rs/r) against radius in units of the Schwarzschild radius, with its reciprocal beside it. The first is the rate of a clock held still there, as read by somebody far away, and is also the redshift of anything it emits; the second is how much a radial ruler is stretched. At 10 rs the clock runs at 0.949 of its distant rate, which is a large effect for a quantity most of physics is allowed to call one. Both curves are drawn to the horizon, where one reaches zero and the other has no value at all.

The effective potential for light outside a non-rotating mass, against distance in horizon radii. For a massive particle this function has a dip as well as a bump, and the dip is where stable orbits live; for light there is no dip. The single peak sits at 1.5000 horizon radii — one and a half, found by searching the drawn curve — and it is a maximum, so the circular light orbit there exists and is unstable: a ray on it leaves at the smallest disturbance, inward or outward. The three horizontal lines are the energies of rays with different impact parameters. One passes over the peak and is captured, one is turned back, and the critical one grazes it. The critical impact parameter is 2.5981 horizon radii, which is √27/2, and it is bigger than the photon sphere, which is bigger than the horizon — three different radii that are all sometimes called the size of a black hole.

The effective potential of the Schwarzschild geometry per unit mass, in units of c², at 4 angular momenta, against radius in Schwarzschild radii. Newton's version has a minimum — a stable circular orbit — at L̃²/GM for every angular momentum there is, however small, and it is the dashed curve at the same L̃, here with its minima at 10.13, 7.61, 6.00, 5.12 rs. General relativity adds one term, −GM L̃²/c²r³, and that minimum stops existing below a definite angular momentum. At L̃ = 4.50 GM/c the barrier and the well are still separate, at 1.83 and 8.29 rs. The two merge at L̃ = √12 GM/c, at 3 rs = 6GM/c² — the innermost stable circular orbit, where the curve drawn here has neither a maximum nor a minimum but a single inflection. Below that momentum — the curve at L̃ = 3.20 GM/c — there is no stationary point anywhere outside the horizon, so no circular orbit exists at all, at any angular momentum whatever. None of that is a property of matter; it is a statement about the geometry, and it fixes the energy of the innermost orbit at √(8/9) = 0.94281 of mc², so 5.719% of the rest mass has been radiated by anything that reached it. Every well drawn here was located on the emitted curve by golden section and checked against the closed form.

The Mach number of steady, spherical, isothermal flow round a point mass against distance from it, both on logarithmic axes, distance in units of the sonic radius, where the inflow reaches the speed of sound. Every curve is a contour of Bondi's conserved integral, which ties the Mach number to the distance. The faint curves are flows that never pass the speed of sound, or that exist only close in or only far out. Two curves cross at the saddle (1, 1): the solid one is accretion, starting from rest far away, reaching the speed of sound exactly at the sonic radius and falling supersonically within it; the dashed one is its time-reverse, a wind. Only the crossing flow joins a gas at rest to a hole that swallows it, and it fixes the rate: λ = exp(3/2)/4 = 1.120 in Ṁ = 4πλ(GM)²ρ∞/c³.

Bondi's transonic accretion flow for an isothermal gas, against distance in sonic radii on logarithmic axes: the inflow speed in units of the sound speed (solid), the free-fall speed from rest at infinity in the same units (dashed), and the density relative to the gas far away (dotted). Far out the gas barely moves and its density is the ambient one; at the Bondi radius, two sonic radii, it is 2.45 times ambient; at the sonic radius it moves at the speed of sound and is 4.48 times denser; well inside, the flow approaches free fall and the density rises as r^−3/2 — 80 times ambient at a tenth of a sonic radius. Pressure decides only how much gas is drawn in; deep inside, the gas falls as if pressure did not exist.

The dimensionless rate λ in Bondi's accretion rate Ṁ = 4πλ(GM)²ρ∞/c∞³, against the gas's adiabatic index γ, from Bondi's closed form λ = ¼[2/(5 − 3γ)]^((5 − 3γ)/2(γ − 1)). An isothermal gas, which can radiate away the heat of compression, gives λ = 1.120; diatomic air, γ = 7/5, 0.625; a monatomic gas that keeps its heat, 0.250. The stiffer gas resists being squeezed into the hole and is swallowed at under a quarter of the isothermal rate — but every λ is of order one, so the square of the mass and the cube of the sound speed set the rate to within a factor of five.

physicscheck asserts something about schwarzschild that could fail — it draws it and measures the result against a value reached some other way.

Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.