Schwarzschild Radius Calculator

What Is the Schwarzschild Radius Calculator?

How to use the Schwarzschild radius calculator

1. Choose the unknown. The Solve for menu opens on Horizon radius, and the other choice is Mass. There are only two, and that is correct: c is a constant, so the two fix each other and there is no third unknown.
2. Type the mass. It takes solar masses or kg and opens on 10 solar masses. A solar mass is 1.988416e30 kg, so the two options are the same quantity written two ways.
3. Type the distance. It is in horizon radii and opens on 3, the innermost stable circular orbit. It is an input in both modes, so it never leaves the form — and it never touches the headline.
4. Type the body length. It takes m or cm and opens on 1.80 m. It is the length the tidal stretch is measured across, and only the last three readings notice it.
5. Read the headline. Four significant figures, with trailing zeros dropped, in the unit that box was showing — switching to scientific notation outside the range 0.0001 to 10,000,000. Set the unit while the quantity is still an input and then point Solve for back at it.
6. Read the photon sphere and the ISCO first. They are 1.5 and 3 times the horizon radius, and neither moves when you change the distance box. That is the first claim of the page in two readings.
7. Read the clock rate. 0.8165 at the innermost stable circular orbit, 0.5774 on the photon sphere, 0.0000 at the horizon. It depends on the distance alone, so it reads the same for a one-solar-mass hole and a ten-billion-solar-mass one.
8. Read the two tidal stretches. The first is the stretch at your distance; the second is the stretch at the horizon, which falls as one over the mass squared — so a bigger hole is gentler at its own horizon.
9. Open Show working. The steps give GM, 2GM, c^2 and the quotient on four separate lines, then the proportionality as a check, then every reading with its arithmetic and your figures in every unit the boxes offer.

This page exists for the one step that turns a mass into a horizon, and then says what happens to clocks and bodies near it. The black hole physics guide is the prose half of it, with the mechanism, the worked problems and the misconceptions set out in full. The constant in the denominator has its own story too: the speed of light is exact by the SI definition of the metre, which is why the c^2 line of the working contributes no uncertainty at all.

Three mistakes account for most wrong answers here. The first is expecting the distance box to move the headline: it cannot, because x does not appear in R_s = 2GM/c^2 at all. The second is reading the clock rate as something an infalling observer would see — it is the rate of a static clock held at that radius, and nothing on this page models anything falling in.

The third is confusing this effect with the one a moving clock shows. A clock that is merely moving also runs slow, but for a different reason and by a different formula: that is velocity time dilation, and the time dilation calculator is the tool for it. The rate printed here comes from where the clock is, not from how fast it is going.

Worked example: change one thing at a time

The table starts at the defaults, moves the unknown through both quantities, then loads each preset in turn, walks every unit menu, and finally reaches the entries the calculator declines. Rows 1 and 2 are one set of boxes solved two ways; row 3 is the same thing once the horizon carries eight figures instead of four. Every answer and reading cell was read out of the running widget rather than worked out by hand, so where a cell and the tool ever part company, believe the tool.

Rows 1 to 3 are the rounding trap, caught in the act. The boxes open on 29.53 km and 10 solar masses, and those two cannot both be exactly true. Ten solar masses give 29.532501 km, so the four-figure 29.53 km in the box is 0.008468 per cent below it and row 2 inverts to 9.999 solar masses.

Row 2 shows the knock-on in the readings too: the ISCO there reads 8.859e+4 m rather than 8.860e+4 m. Nothing is wrong — a radius typed to four figures simply does not pin a mass to four figures. Row 3 removes it: any preset writes the horizon to eight significant figures, and the mass direction then returns a clean 10 solar masses with every reading matching the forward direction.

Rows 4 to 6 move the distance and nothing else. The headline does not budge from 29.53 km in any of them, and neither does the photon sphere or the ISCO — which is the first claim of the page in three rows. What moves is the clock rate: 0.5774 on the photon sphere, 0.0000 at the horizon, 0.9129 at twice the ISCO.

Row 5 is the one to look at twice. At the horizon your distance is the horizon radius, so the two tidal readings become the same number.

Rows 7 to 11 are the mass scan, and the third claim. One solar mass gives 2.953 km and ten gives 29.53 km, exactly ten times. Follow the second-to-last column down those rows — 1.891e+9 g, 2.102e+8 g, 0.9996 g, 1.023e-4 g, 1.891e-11 g — and at each hole's own horizon the stretch falls as one over the mass squared.

Rows 12 and 13 run the inverse direction. A 1.000 km horizon needs 0.3386 solar masses, which is below the sourced three-solar-mass floor. A horizon the size of the Sun, 7.0e8 m, needs 237000 solar masses — and because that input carried only two significant figures, the honest reading is about 2.370e+5.

Rows 14 and 15 are the one comparison that reverses, and they are the most important two rows here. Both put a body at the same 2.953e+10 m from the centre: the ten-solar-mass hole at x 1000000 and the 4.3-million one at x 2.3255814. At that fixed distance in metres the stretch is 1.891e-11 g and 8.133e-6 g, so the bigger hole is now 4.300e+5 times stronger — exactly the mass ratio.

Row 14’s clock cell wants one caveat before you read it. At x 1000000 the rate rounds to 1.000 at four significant figures, but the exact value is sqrt(1 - 1/1000000) = 0.9999995. That clock is still running slow, just not by enough to survive four figures — and no finite distance ever gets the rate to 1.

Compare rows 5 and 10 instead and the big hole is the gentler one, because there each is read at its own horizon. Both statements are true, and they are about different comparisons.

Rows 16 to 19 change nothing but the units and the body. 1.988416e31 kg is the same 10 solar masses and 180 cm the same 1.80 m, so both return the identical 29.53 km with identical readings. Row 18 shows the other half of the rule: the answer comes back in whichever unit the target box was showing, so the horizon reads 29530 m there.

Row 19 shrinks the body to one centimetre. Only the two tidal readings move, and they move by the factor of 180 the linearity demands.

Rows 20 to 25 sit outside the domain and are declined there rather than answered. A zero mass gives a zero horizon, so the tide divides by zero, and a negative mass gives a negative radius. A distance of 0.9 is inside the horizon, where the clock-rate expression has a negative number under its square root and no static observer exists to carry a clock.

A zero body length makes the stretch vanish, and a zero or negative horizon cannot be inverted to a mass. In every case the widget says so in words instead of printing an infinity or a NaN.

Formula and symbol reference

One relation does the first half of the work. The Schwarzschild solution of general relativity describes the geometry outside a non-rotating, uncharged mass, and the coefficient 1 - R_s/r in it vanishes at R_s = 2GM/c^2. That radius is the event horizon, and rearranged the other way it reads M = R_s c^2 / 2G — the two positions of the Solve for menu.

A Newtonian calculation reaches the same algebraic expression, and it is worth knowing why that is a coincidence rather than a derivation. Set the escape velocity of a mass equal to the speed of light and rearrange for the radius, and 2GM/c^2 drops out — but light does not slow down and fall back the way a stone does, and the honest account of a horizon needs general relativity. The escape velocity calculator is the tool for that quantity in its own right; this page takes R_s as the relativistic result and goes no further down that road.

The second half is what happens near the horizon. Two radii follow from the same geometry: the photon sphere at 1.5 R_s = 3GM/c^2, where a light ray can circle the hole on an unstable orbit, and the innermost stable circular orbit at 3 R_s = 6GM/c^2, inside which no circular orbit survives. Both are quoted here as radii of the geometry and neither is worked out from an orbital period.

Two readings then describe where you are standing. The clock rate is sqrt(1 - R_s/r), which in horizon radii is sqrt(1 - 1/x) and so depends on x alone. The tidal stretch is 2GMh/r^3 across a body of length h, and at the horizon that same quantity can be written h c^6 / (4 (GM)^2), which is where the one-over-mass-squared behaviour becomes visible.

The physics: why the clock, not the radius, is the signature

The horizon radius is just a length, and lengths scale. Ten times the mass gives ten times the radius exactly — 2.953 km for one solar mass, 29.53 km for ten — so a radius on its own tells you nothing about what makes the surface special. What is special is what happens to time there.

A static clock at a radius r ticks sqrt(1 - R_s/r) times for every tick of one far away, and at r = R_s that is exactly zero. Load the at the horizon itself preset and then the ten-billion-solar-mass giant: the headline changes by a factor of a billion while the clock rate reads 0.0000 in both. That mass-independence is what makes a horizon a place in the geometry rather than a size.

The rate never reaches 1 either, however far out you go. It is still below 1 at ten thousand horizon radii, which is why a clock at a finite distance is always running slow and the honest phrasing is “far enough away for the difference to be negligible”. The limit c that sets the scale of all of this is set up in special relativity, and the horizon is where the geometry takes that limit seriously.

The tidal stretch is the third claim, and it is the one that surprises people. At its own horizon a bigger hole is gentler: across a 1.80 m body the stretch is 1.891e+9 g at the horizon of a one-solar-mass hole, 1.891e+7 g at ten solar masses, 0.9996 g at 4.350e+4 and 1.023e-4 g at about 4.3 million. Those four come from the if the Sun became a black hole, ten-solar-mass stellar black hole, horizon tide of about 1 g and Sagittarius A* buttons in that order, which is not the order they sit in the grid — or read rows 7 to 11 of the table above, which already run in order of mass.

Where the Schwarzschild radius calculator breaks down

The arithmetic here is exact to the last digit a double can hold. What goes wrong is physical: a hole that spins, a question about the inside, a comparison made at the wrong distance, and a mass the formula will happily accept for an object that has never been seen.

Real black holes spin, and these are the non-rotating radii

Every figure here is the Schwarzschild case: non-rotating and uncharged. Spin changes the horizon, the photon sphere and the innermost stable circular orbit, and it is not a small correction — the fetched source gives M87*'s rotational parameter as 0.90 plus or minus 0.05, which is close to the maximum a black hole can have. Treat these radii as an idealisation, and do not apply them to a named object except as one.

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

This is the one place the page could ship something false, so the qualifier appears every time the claim does. At a fixed multiple of the horizon the stretch falls as one over the mass squared. At a fixed distance in metres it is simply proportional to the mass, so the bigger hole is the stronger one there by exactly the mass ratio — a factor of 4.300e+5 between a ten-solar-mass hole and one of about 4.3 million. Both statements are true, and they are about different comparisons.

The clock rate is for a static observer, and nothing here falls in

The reading is what a clock held at a fixed radius does, measured against one infinitely far away. It is not what an infalling clock reads, it is not a Doppler shift, and it is not a prediction about anybody's experience of crossing a horizon. No figure on this page describes infall, and the distance box stops at 1 partly because below it there is no static observer for the reading to belong to.

Nothing here describes the inside, and nothing here is quantum

The three relations this page evaluates all describe the geometry outside the horizon. Hawking radiation, evaporation times, black-hole entropy, the ergosphere, frame dragging, the singularity and the interior are all outside the model, and no figure for any of them appears anywhere on this page. The same goes for accretion discs, jets, lensing geometry and gravitational waves: they are real physics and they are not this calculator's.

The formula is arithmetic; what exists is a separate question

Point Solve for at the mass, type a 1.000 km horizon, and the answer is 0.3386 solar masses. That is correct arithmetic for an object below the sourced three-solar-mass observational floor — the limit above which a compact object can only be a black hole. The page quotes no formation mechanism for a sub-floor object and does not call one observed.

Two input figures do not make a four-figure answer

Type one solar radius, 7.0e8 m, into the horizon box and the mass comes back as 237000 solar masses. The input carried two significant figures, so the honest reading is about 2.370e+5 solar masses, and the extra digits the headline prints are a property of the arithmetic rather than of the answer. The same applies to the mass of Sagittarius A*, which is why its preset name says about 4.3 million.

The linear tidal form needs no caveat at human scale, but it has one

2GMh/r^3 is the stretch linearised in h. Measured against the exact difference of fields it is off by 1.859e-7 per cent at h = 1.80 m — five orders below the four figures this page prints — so for any real body the form is exact as far as the page can show. It fails only when h becomes comparable with the horizon itself, where the error reaches 17.71 per cent at h = 18000 m, and nothing on this page uses a body that long.

A refusal is an answer, not a fault

A zero or negative mass, a zero or negative horizon radius in the inverse direction, a distance below 1 and a zero or negative body length are all declined in words rather than answered with an infinity or a NaN. The distance case is the one worth understanding: x = 1 is not an error and prints a clock rate of 0.0000, while x = 0.9 is inside the horizon and has no static observer at all.

The calculator measures nothing — you supply the numbers

Two quantities and two auxiliaries go in, one relation is rearranged, and the answer describes the idealised hole you described. The preset names are shorthand for the four figures behind them; the only one that names a real object quotes its mass exactly as the source states it, about 4.3 million solar masses, and presents its horizon as a consequence of that stated mass.

Where the Schwarzschild radius calculator is actually used

Turning a mass into a size you can picture

This is the forward question and the reason the page opens where it does. Ten solar masses give 29.53 km: a city-sized radius holding ten Suns. One solar mass gives 2.953 km, and the proportionality means you never need the formula twice — multiply the solar-mass figure by the number of solar masses and you have it.

Going the other way, from a horizon to a mass

Point Solve for at the mass and the question inverts: a size you can imagine, and the mass it would take. A 1.000 km horizon needs 0.3386 solar masses; a horizon the size of the Sun, 7.0e8 m, needs about 2.370e+5. This is the direction in which a rounded radius costs you most, because the mass is strictly proportional to it.

Working out how much a clock loses at a given orbit

Set the distance to 3 and the clock rate reads 0.8165: a clock at the innermost stable circular orbit ticks about four fifths as fast as one far away. Set it to 1.5, the photon sphere, and it falls to 0.5774. Because the reading depends on the distance in horizon radii alone, both figures hold at those two distances for a non-rotating hole of any mass — and only for a non-rotating one, because spin moves the two radii off 3 and 1.5 in the first place.

Deciding which holes a body could survive approaching

Take the presets in order of mass rather than in the order they sit in the grid, and one reading runs 1.891e+9 g, 2.102e+8 g, 1.891e+7 g, 0.9996 g, 1.023e-4 g, 1.891e-11 g — six values from eight buttons, because three of them share the ten-solar-mass hole. The crossing point is the interesting one: around 4.349e+4 solar masses the horizon stretch across a 1.80 m body is roughly your own weight, and the preset's 4.350e+4 gives 0.9996 g rather than exactly 1. Past that mass the horizon reading for the same body comes out below 1 g, which is a figure this tool prints and not a verdict on anybody crossing.

Separating the gravitational clock effect from the velocity one

Two different effects slow a clock, and they are easy to run together. The rate printed here comes from where the clock is; the other comes from how fast it is moving, and the Lorentz factor calculator is the tool for that one. They have different formulae, different causes and different inputs, and a real satellite clock carries both at once.

Checking where GM for your object comes from

The mass box wants solar masses, and in practice what is measured for a real system is the product GM, from an orbit. The gravitational force calculator is where that Newtonian force law lives, and this page takes GM as an input rather than deriving it — the innermost stable circular orbit is quoted as a radius here, never worked out from a period.

Where to go next

For what a black hole is, how the horizon, the clock and the tide fit together, the worked problems and the misconceptions, read the black hole physics guide, which this tool is the arithmetic half of. The special relativity guide is where the constant speed of light is set up, and the speed of light guide is the constant itself.

Two more posts pick up what this page leaves out. The time dilation guide is the velocity kind of clock slowing, a different effect with a different cause, and the escape velocity guide is the Newtonian calculation that lands on the same algebra for the wrong reason.

Four tools sit next to this one. The time dilation calculator handles the velocity effect; the escape velocity calculator handles the speed needed to leave a mass behind; the gravitational force calculator is the Newtonian force law; and the Lorentz factor calculator is how far relativity stretches time and length at a given speed.

The full physics lab library and calculator index are open too.