Black Hole Simulator

What Is the Black Hole Simulator?

How to use the black hole physics simulator

1. Read the eight readings before you touch anything. The four cards are already answering for a ten-solar-mass hole with you standing at three horizon radii: Horizon radius 29.53 km, Clock rate here 0.8165, Tidal stretch here 7.005e+5 g and Stretch at the horizon 1.891e+7 g. The compact row under them reads 44.30 km, 88.60 km, 88.60 km and 10.00 Msun = 1.988e+31 kg.
2. Notice that the mass is two sliders, not one. Mass (x 10^n solar masses) carries the mantissa and Mass exponent n carries the power of ten, so the chips read 1.00 and 10^1 at load. That split is what lets a named mass be reached exactly: 4.35 with 10^4, or 4.30 with 10^6, rather than whatever a logarithmic slider happened to land on.
3. Drag Mass exponent n alone and watch the first card and the drawing together. 10^0 gives 2.953 km, 10^1 gives 29.53 km, 10^2 gives 295.3 km, 10^3 gives 2953 km. Each whole step is a factor of ten in the radius and nothing else in the relation changes.
4. Now drag Distance (x the horizon radius) and watch the first card refuse. Take it from 1.00 to 20.00: Horizon radius reads 29.53 km at every stop, and so do Photon sphere and the innermost stable circular orbit. The grid reading that moves is Your distance, from 29.53 to 590.7 km.
5. Read Clock rate here while you do it, because this is the card the lab exists for. 0.0000 at 1.00 R_s, 0.5774 at 1.50, 0.7071 at 2.00, 0.8165 at 3.00, 0.9747 at 20.00. Change the mass by ten decades and those five strings do not change at all.
6. Compare the two tidal cards, which answer to different things. Tidal stretch here falls as you move out — 1.891e+7 to 2364 g across the distance slider. Stretch at the horizon ignores the distance entirely and stays on 1.891e+7 g, because it is always evaluated at the horizon rather than where you are.
7. Read the drawing in two halves, and read the scale bar first. The upper scene is the hole face-on with its two rings and your marker on it; the lower one is the whole clock-rate curve with your distance marked by a filled dot. The scene auto-fits, so the disc never looks much bigger or smaller — the short bar underneath prints the kilometres that make the picture honest.
8. Press Pause orbit when you would rather read than watch. The dot is already circling when the page loads, so the button reads Pause orbit rather than asking to be started, and pressing it changes no reading at all. Reset puts the three sliders back to 1.00, 10^1 and 3.00 R_s and leaves the dot circling.

Dragging is the right way to find out which readings a control is allowed to move, and a poor way to hit an exact figure — the mantissa steps in hundredths, the exponent in whole powers of ten and the distance in hundredths of a horizon radius. When you already have a mass, or a radius you want the mass behind, the Schwarzschild radius calculator takes typed input and solves the relation in either direction. It has no drawing and no sliders; this panel has no unit menus and no inverse. The written guide to black hole physics is where the subject itself is set out at length.

Worked example: change one thing at a time

Nothing in the three tables below was worked out here. Each row is a setting the lab was put into and then read back from, cell by cell, so every string in them is the panel’s own. Each table moves one slider down its column and pins the others, which is what makes the column that changes diagnostic.

One: the mass, with the distance pinned at three horizon radii

The distance slider stays on 3.00 R_s in all five rows. Only the two mass sliders move, a factor of a thousand from top to bottom.

Read the three radius columns down the page first. Every one of them scales by exactly the same factor as the mass: ten times the mass gives ten times the horizon radius, ten times the photon sphere and ten times the innermost stable circular orbit. Rows one and three are the two buttons worth pressing one after the other — 2.953 becoming 29.53 km is the whole of the first claim, in one click.

Now read the clock column, which does not move at all. 0.8165 in all five rows, from one solar mass to a thousand. The distance slider has not been touched, and the clock rate answers to that slider and to nothing else — which is why the curve the lab draws needs one axis and carries no mass anywhere in it.

The last column is the one that surprises people. 1.891e+9, 2.102e+8, 1.891e+7, 1.891e+5, 1891 g: the horizon tide falls as the mass rises, and it falls as the square. Ten times the mass divides it by a hundred. The qualifier that makes this true is at its own horizon, and the third table is where it is worth reading slowly.

Two: the distance, with the mass left alone at ten solar masses

Now the two mass sliders are pinned at 1.00 and 10^1, and only the distance moves, from the floor of its slider to its ceiling.

Two whole columns are constant, and that is the point of the table. 29.53 km and 44.30 km down all eight rows, with 1.891e+7 g beside them. Three of these eight rows are the preset buttons A 10-solar-mass stellar black hole, On the photon sphere and At the horizon itself, so you can press the claim instead of reading it.

The clock column is where the cliff is. It climbs gently over most of the slider — 0.8165, 0.9129, 0.9487, 0.9747 — and then collapses in the last hundredth before the horizon: 0.09950 at 1.01 R_s and 0.0000 at 1.00. That is the shape the drawn curve shows, and it is why the lab plots the rate on a plain linear axis from zero to one.

The rate never reaches 1.000 anywhere on this slider. Twenty horizon radii out it is still 0.9747, so a clock at any finite distance is running slow and the honest phrasing is always “far enough away for the difference to be negligible”. A clock that is merely moving also runs slow, for a different reason and by a different formula, and that is the kind of time dilation a moving clock shows rather than this one.

Three: the tide ladder, with the distance pinned at the horizon

The distance slider sits on 1.00 R_s in all seven rows, so every reading is taken at each hole’s own horizon. Only the mass moves, over ten decades.

Read the last column top to bottom and the claim is unmistakable. 1.891e+9 g down to 1.891e-11 g as the mass climbs from one solar mass to ten billion: twenty decades of tide across ten decades of mass, because the fall is as the square. The fourth row is where it crosses something you can feel — 0.9996 g, which is why that preset is named for a tide of about 1 g and not exactly one.

The second column goes the other way, and that is the whole trick. The horizon radius rises with the mass while the tide at it falls, so the bigger hole is gentler at its own horizon — and only there. Line two of these holes up at the same distance in metres instead and the comparison reverses, because at a fixed radius the stretch is simply proportional to the mass.

The clock column is constant down every row, and that is the sharper claim. 0.0000 at one solar mass and 0.0000 at ten billion, a factor of ten billion apart in radius. The horizon is a place in the geometry rather than a size, and the clock is what says so. The speed of light in the denominator of the horizon relation is exact by the definition of the metre, so none of these radii carries any uncertainty from it.

Formula and symbol reference

The lab prints its own relations under the drawing, in the fixed definition line, in plain ASCII so that no radical sign and no subscript glyph is ever needed. Written out the same way, with sqrt spelled and every subscript an underscore:

R_s = 2GM/c^2 is the horizon radius; x = r / R_s is the distance slider, so clock rate = sqrt(1 - R_s/r) = sqrt(1 - 1/x) for a static observer; and stretch = 2GMh/r^3 across the fixed h = 1.80 m body, which at the horizon becomes h c^6 / (4 (GM)^2). That last form is where the one-over-mass-squared behaviour of the fourth card comes from. The photon sphere sits at 1.5 R_s and the orbit limit at 3 R_s.

Those two rings are quoted here as radii and are never worked out from a period. This panel has no orbital period in it at all: the Kepler third law calculator is where a period comes from a semi-major axis and a mass, and nothing on it transfers to a ring this close to a horizon. A Newtonian calculation does reach the same algebra for the horizon radius, for the wrong reason, and that route belongs to the escape velocity guide rather than here.

Twelve rows follow and only three of them are something you drag. The rest are cards, cells of the compact row, or quantities the panel names without giving you a control for. For the sliders the right-hand column gives the end stops and the step; for everything else it gives the strings the lab printed and the span it printed them over.

If you read one row twice, make it the clock rate row. It is the only reading on the panel whose span is the same two strings for every mass the sliders can reach, and that is what separates this lab from a radius calculator.

The physics: the three claims the panel lets you test

The first is that the horizon belongs to the mass and to nothing else. Two sliders set it, the third cannot touch it, and you can try to break that: hold the mass still, sweep the distance from 1.00 to 20.00, and the first card’s sub-line stays byte-identical at R_s = 2GM/c^2 - set by the mass alone. The drawn disc does not change size either, although the scale bar under it does.

The second is the one the lab exists for, and it needs the curve as well as the cards. A static clock at your distance ticks sqrt(1 - 1/x) times for every far-away tick, and at the horizon that is exactly 0.0000. Sample it at three masses ten decades apart and the string does not change: the cliff in the drawn curve is in the same place for a one-solar-mass hole and a ten-billion-solar-mass one.

That mass-independence is why the curve caption says what it says rather than naming a mass. A radius is a length and lengths scale; a clock stopping is not a length. Any account of a black hole that defines it purely by how big it is has described the easy half, and this panel puts the hard half on a card next to it.

The third claim is the awkward one, and it comes with four words it is false without. At each hole’s own horizon the tide across the fixed 1.80 m body falls as one over the mass squared, so a bigger hole is gentler there. At a fixed distance in metres the bigger hole is the stronger one instead, by exactly the mass ratio, and the lab’s own definition line under the drawing says both halves of that in one breath.

The constant that ties the three claims together is the speed of light, which appears squared in the horizon relation and to the sixth power in the horizon tide. Special relativity is where it is set up as a limit rather than as a constant in a denominator, and the horizon is where the geometry takes that limit seriously.

Where the black hole simulator’s readings break down

Every slider position this lab can reach is answered, and the arithmetic is not where it can let you down. The drawing is, and so is the model behind it. Nine limits sit under the readings on this page; four are choices the panel makes on purpose, and five are bounds on the physics rather than on the panel.

Real black holes spin, and every radius here is the non-rotating one

The whole lab is the Schwarzschild case: non-rotating, uncharged, isolated, with nothing falling in. Spin changes the horizon, the photon sphere and the innermost stable circular orbit, and it is not a small correction. The Wikipedia black hole article gives the rotational parameter of M87* as 0.90 plus or minus 0.05, which is close to the maximum a black hole can have. Treat every figure on the panel as an idealisation, and do not apply it to a named object except as one.

“A bigger hole is gentler” is only true at its own horizon

This is the one place this lane could ship something false, so the qualifier rides with the claim everywhere it appears: on the fourth card’s sub-line, in the sentence strip, in the definition line and in every paragraph above. At a fixed multiple of the horizon the tide falls as one over the mass squared. At a fixed distance in metres it is simply proportional to the mass, so there the bigger hole is the stronger one by exactly the mass ratio. Both statements are true, about different comparisons.

The clock belongs to a static observer, and nothing in this lab falls in

The second card answers for a clock held at that radius, against one infinitely far away. Nothing in this lab falls, so no reading on it predicts what a clock would do on the way through a horizon, and nothing here is a shift caused by motion towards or away from you. That is also half the reason the distance slider stops at 1.00: below it nothing could hold a radius for a card to answer about.

The scene auto-fits, so the drawn disc is never a measurement

The horizon radius runs from 2.953 km to 2.953e+10 km across the presets alone, and no fixed scale is legible over ten decades. The scene therefore scales itself until whatever it draws furthest out uses about seventy per cent of the room it has, which means the disc looks about the same size whether the first card says 2.953 km or 2.950e+11 km. The short bar beneath it prints the kilometres it stands for, and that printed value is the only honest way to read a distance off the picture.

The ring names are dropped whole on a narrow screen, never half-drawn

The upper scene cannot label its rings in place, because the orbiting dot and its dotted path sweep through every angle and at three of the eight presets that path lies exactly on a named ring. The names therefore live in a reserved column beside the scene, and below roughly 290 pixels of scene width that column plus a legible drawing will not fit, so it is dropped whole rather than clipped. Nothing is lost: the caption underneath still prints the horizon radius and the bar’s own value, and the compact row names both rings with their figures a little further down.

The orbiting dot claims no rate, and there is nothing to read off it

No string anywhere in the lab states a speed, a period or a number of orbits, and that is deliberate rather than an omission: a real orbital period at a given radius is a calculation this lane does not make. The dot marks where you are standing and keeps the scene alive. At 1.00 R_s it is drawn stationary on the horizon, because there is no circular orbit there at all, and one hundredth of a radius further out it circles again.

The scene stops naming a case as soon as you touch a slider

The title above the drawing reads a preset’s name only when all three sliders match one of its eight stored value sets. Nudge any one of them off — even by a single hundredth of a horizon radius — and it prints A non-rotating black hole instead. That is a refusal rather than a fault: every card carries on answering exactly as before, and the band is declining to claim a case the sliders are not on.

The interior is outside the model, and so is everything quantum

Each of the three relations behind the eight readings is a statement about the geometry outside the horizon, and no slider can leave it. There is no control, card or drawn mark anywhere in the lab for an evaporation time, a temperature, an entropy, an ergosphere, a dragged frame, a singularity, an accretion disc, a jet, a bent light path or a gravitational wave, and no figure for any of them is printed or implied. Even the photon sphere is a bare ring here, with no light path drawn on it.

The relation is arithmetic; what exists is a separate question

The mass sliders start at one solar mass, which is below the sourced three-solar-mass observational floor — the limit above which a compact object can only be a black hole. The preset at 8.860 km is named for that floor for exactly this reason. The lab will happily draw a lighter one, and it claims no formation mechanism for it and does not call it an observed object.

Where black hole physics is actually used

Watching one claim at a time, which is what the groupings are for

This is the forward use of the panel and the reason the presets are in the order they are. Press buttons one and three and only the mass has changed, so the 29.53 against 2.953 km is the proportionality and nothing else. Press three, four and five and only the distance has changed, so the 0.8165, 0.5774, 0.0000 is the clock and nothing else. Two claims, five clicks, nothing to take on trust.

Showing that the cliff sits in the same place for every mass

The drawn curve is the quickest answer to “how close is close?” that this subject has. Because the horizontal axis is in horizon radii, the shape is the same for every mass: 0.5774 on the photon sphere and 0.8165 at the orbit limit hold for a one-solar-mass non-rotating hole and for a ten-billion-solar-mass one alike. Those two numbers are worth carrying around, and the panel is where you can check that they really do not move.

Finding the mass at which a horizon stops being a shredder

Walk the tide ladder table and watch one column: 1.891e+7, 1891, 0.9996, 0.1891, 1.023e-4 g. The crossing is the interesting part — somewhere around the fourth rung the tide across a 1.80 m body at the horizon is roughly your own weight, and the preset’s 0.9996 g is as close as the mantissa slider lands. Above that mass the tide across that 1.80 m body at the horizon is under 1 g.

Reading a radius off a drawing without being fooled by it

This is a transferable habit rather than a fact about black holes. Any picture that has to span ten decades must auto-fit, and an auto-fitted picture with no stated scale tells you only the ratios of what is in it. The bar under this scene prints its own length in kilometres at every setting, which is the thing that turns the drawing back into a measurement — and it is why the caption beneath the scene always carries that figure.

Separating what the mass sets from what the distance sets

Split across two controls, the readings sort themselves into two groups you can name. The mass alone fixes the horizon radius, the photon sphere, the orbit limit and the tide at the horizon. The distance alone fixes the clock rate. Only your distance as a length and the tide where you are standing need both. That is a cleaner division than the formulae look like they give, and it is the thing dragging one slider at a time shows you.

Keeping the gravitational clock effect apart from the velocity one

Two different effects slow a clock and they are easy to run together. What this panel prints comes from where the clock is; the other comes from how fast it is going, and the Lorentz factor calculator is the tool for that one. One is a fact about position and the other a fact about motion, they take different inputs, and a clock in a real orbit is subject to both at once. Nothing on this panel is evidence about the moving kind.

Where to go next

A panel can show you which readings a control is allowed to move, and that is the end of what a panel is for. Black hole physics is where the subject itself is written out: what a black hole is, how the horizon, the clock and the tide fit together, the worked problems and the misconceptions. The calculator that takes a typed mass or a typed radius and solves either direction heads the related tools below.

Two posts carry the effects this panel deliberately keeps out. The time dilation guide belongs to clocks that are moving rather than clocks that are deep in a field, and the escape velocity guide owns the calculation whose algebra arrives at the horizon radius without the geometry that justifies it.

For the constant underneath all of it, the special relativity guide sets up the speed of light as a limit and the speed of light guide is the constant itself. Further out, the law of gravitation guide and the universal gravitation lab are the Newtonian field this geometry replaces close in, while the Kepler laws guide and the orbital velocity guide are where orbits are the subject rather than two quoted radii.

Beyond those there is the simulation library, an index of calculators, and the blog for everything written out at length.