Illustrated Physics
physics, drawn
Generator

The factor, against distance from the horizon

One function in the gravity library, called 26 times across 6 essays. Below: what it draws at its defaults, what it draws at every branch an essay asks for, whether the site's own gate puts a claim to it, and everywhere it is called.

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.

At its defaults

Drawn even though every essay passes options, because a default nothing exercises is a trap for the next essay to call this with none — which has happened here twice.

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.

√(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.

A potential with a maximum and no minimum at all

The options are the ones The circle light cannot leave passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.

A potential with a maximum and no minimum 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 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 term that abolishes the inner orbits

The options are the ones The circle light cannot leave passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.

The term that abolishes the inner orbits. 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 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 bigger the hole, the gentler the horizon

The options are the ones The horizon that nothing marks passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.

The bigger the hole, the gentler the horizon. The difference in gravitational pull between the two ends of a 1.8 metre body, at the moment it crosses the horizon, against the mass of the hole. Both axes are logarithmic and the line is straight, of slope -2.00: the tidal acceleration at a horizon falls as the square of the mass, because the tide goes as the mass over the cube of the radius and the radius itself is proportional to the mass. At the low end — a hole of a few solar masses — the stretch is 1.9e+7 times Earth's gravity across a person, which pulls them apart long before they arrive. At the high end it is 1.9e-11, which is nothing at all: crossing the horizon of a large enough hole is locally unremarkable, and an observer would notice no boundary being passed. The two meet at 1.0e+4 solar masses, above which the horizon can be crossed intact. That is the sharpest available statement of what a horizon is and is not. It is not a surface, nothing is there, and nothing local marks it — it is the place from which no future path leads out, which is a statement about the whole of the future rather than about anything present.

The difference in gravitational pull between the two ends of a 1.8 metre body, at the moment it crosses the horizon, against the mass of the hole. Both axes are logarithmic and the line is straight, of slope -2.00: the tidal acceleration at a horizon falls as the square of the mass, because the tide goes as the mass over the cube of the radius and the radius itself is proportional to the mass. At the low end — a hole of a few solar masses — the stretch is 1.9e+7 times Earth's gravity across a person, which pulls them apart long before they arrive. At the high end it is 1.9e-11, which is nothing at all: crossing the horizon of a large enough hole is locally unremarkable, and an observer would notice no boundary being passed. The two meet at 1.0e+4 solar masses, above which the horizon can be crossed intact. That is the sharpest available statement of what a horizon is and is not. It is not a surface, nothing is there, and nothing local marks it — it is the place from which no future path leads out, which is a statement about the whole of the future rather than about anything present.

Two radii, and where they cross

The options are the ones The horizon that nothing marks passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.

Two radii, and where they cross. Two lengths against the mass of a black hole, both logarithmic. One is the horizon radius, which is proportional to the mass. The other is the radius at which the tidal stretch across a 1.8 metre body reaches 10 g per metre, and it grows only as the cube root of the mass, because the tide depends on mass over radius cubed. Two different powers of the same variable have to cross, and they do: below the crossing the lethal radius is outside the horizon, so a falling body is destroyed before it arrives; above it the lethal radius is inside, so the body crosses the horizon whole and is destroyed later, out of sight. The crossing is at 1.0e+4 solar masses. The two failures look identical from outside and are entirely different from within, and the whole difference is that one power law is steeper than the other.

Two lengths against the mass of a black hole, both logarithmic. One is the horizon radius, which is proportional to the mass. The other is the radius at which the tidal stretch across a 1.8 metre body reaches 10 g per metre, and it grows only as the cube root of the mass, because the tide depends on mass over radius cubed. Two different powers of the same variable have to cross, and they do: below the crossing the lethal radius is outside the horizon, so a falling body is destroyed before it arrives; above it the lethal radius is inside, so the body crosses the horizon whole and is destroyed later, out of sight. The crossing is at 1.0e+4 solar masses. The two failures look identical from outside and are entirely different from within, and the whole difference is that one power law is steeper than the other.

The factor, against distance from the horizon

The options are the ones The horizon that nothing marks passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.

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.

√(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.

What checks it

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.

Where it is called

Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.

Astrophysics

The circle light cannot leave

A black hole has three radii and they are all sometimes called its size. The horizon is where nothing can come back from; the photon sphere, half again as far out, is where light can circle and cannot keep circling; and the shadow a distant observer sees is larger than either, because the ray that just escapes was bent on its way out.

 Astrophysics

The horizon that nothing marks

A falling body is torn apart by the difference in gravity between its ends. At the horizon of a black hole that difference goes as the inverse square of the mass, so a large enough hole can be entered intact — and nothing local happens at the crossing to say it has occurred.

 Astrophysics

The orbit that cannot be made smaller

Newtonian gravity allows a stable circular path at every radius, however tight, and there is no innermost one. General relativity adds a single term to the expression that says so, and below 6GM/c² no stable circular path exists at any angular momentum whatever. Anything arriving there has radiated 5.72% of its rest mass, which is eight times what hydrogen fusion converts.

 Astrophysics

The orbit that has to shrink

Two masses in orbit radiate gravitational waves and lose energy, so the orbit tightens, so they go faster and radiate harder. The runaway takes 10²³ years for the Earth and the Sun and eight minutes for the last thousand kilometres of a black-hole pair — and the same one-line formula gives both.

 Astrophysics

The surface that only lets things in

An eighteenth-century calculation asking where the escape speed reaches the speed of light gives exactly the right radius, by reasoning that is wrong in every step. What is actually there is not a surface in space at all, and nothing local happens when it is crossed.

 Astrophysics

Two clocks that disagree about the fall

A clock falling into a horizon crosses it in a few milliseconds by its own reckoning and never crosses it at all by a distant one. Both accounts are right, and the thing everybody remembers about the second — that the image hangs there for ever — is wrong.

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