Schwarzschild radius calculator
The event-horizon radius of a non-rotating black hole of mass \( M\): \( r_s = \dfrac{2GM}{c^2}\). Anything squeezed inside its own \( r_s\) becomes a black hole.
Idea 01 · Established · Schwarzschild, 1916
The Event Horizon
A boundary in spacetime you can cross, but never recross.
To leave the Earth you have to outrun its gravity, which means about eleven kilometres a second. Leave from a denser, heavier world and that escape speed is higher. Now keep squeezing the same mass into a smaller and smaller ball. At some point the speed you would need to escape reaches the speed of light. Push past that point and not even light is fast enough. That is a black hole.
A black hole isn't really an object. It is more like a boundary, a spherical surface called the event horizon. Outside it you can still, with enough effort, climb away. Cross it and you can't. The reason isn't that the walls are strong. It is that every direction which counts as "the future" now points inward. It is a one-way door. The more massive the hole, the bigger the door: a hole the mass of the Sun is just a few kilometres across, while the one at the centre of our galaxy is wider than the Sun itself.
The size of the horizon follows from a single, surprisingly simple formula, the Schwarzschild radius:
\[ r_s = \frac{2GM}{c^2}. \]
Put the Sun's mass in and you get about 3 km. Put the Earth's in and you get 9 mm. Anything compressed inside its own Schwarzschild radius becomes a black hole. Notice that the horizon depends only on the mass, not on what that mass is made of, nor on how it got there. The Schwarzschild radius calculator at the top of the page works out \(r_s\) for any mass you enter.
The horizon is a feature of spacetime, not a wall of material, so there is nothing locally special to feel as you pass. But it has a startling effect on what others see. As you fall toward it, a distant observer watching you sees your clock run slower and slower, and your light stretched redder and redder, until you appear to freeze on the horizon and fade to black. They never quite see you cross. From your own point of view, you sail straight through.
Outside a spherical mass, spacetime is the Schwarzschild solution of Einstein's equations,
\[ ds^2 = -\Big(1-\tfrac{r_s}{r}\Big)c^2\,dt^2 + \Big(1-\tfrac{r_s}{r}\Big)^{-1} dr^2 + r^2\,d\Omega^2, \]
with \(r_s = 2GM/c^2\). The apparent blow-up at \(r=r_s\) is only a coordinate singularity: switch to ingoing Eddington–Finkelstein or Kruskal coordinates and the geometry is perfectly regular there. What is real is that \(r=r_s\) is a null surface, a one-way membrane. The gravitational redshift factor \(\sqrt{1-r_s/r}\to 0\) as \(r\to r_s\), so signals from the horizon arrive infinitely redshifted; in Schwarzschild time the crossing takes forever, while the faller's proper time to reach it, and then the singularity at \(r=0\), is finite.
Inside, the light cones tip until every future-directed path has decreasing \(r\): the surface is a trapped surface, and the inward fall is now as unavoidable as the passage of time. Rotating holes (the Kerr metric) generalise this, splitting the horizon from an outer ergosphere where spacetime itself is dragged around.
Status: established. A prediction of general relativity since 1916, now seen directly: gravitational waves from merging horizons (LIGO, 2015) and the silhouette of a horizon imaged by the Event Horizon Telescope (M87*, 2019; Sgr A*, 2022).
A black hole as a funnel in curved space: matter streams down it and is swallowed at the dark throat, the event horizon. Drag to orbit. The bright loop is the horizon at \(r_s\). Outside it light can still climb away, but the closer in you start, the steeper the climb. Drag to orbit; it also turns on its own. The blue ring is the photon sphere, where light itself can orbit; at \(r_s\) the outgoing cone closes and nothing escapes. Drag to orbit.
Idea 02 · Established consequence of GR
Tides & the Singularity
Stretched head to toe, on a path that ends nowhere good.
Gravity gets stronger the closer you are. Fall feet-first toward a black hole and your feet are pulled harder than your head, and that difference grows enormously as you near the centre. You get stretched lengthwise and pinched at the sides, drawn out into a thin strand. Physicists, never the most solemn people, call this spaghettification.
And there is no stopping. Once you are inside, every path leads to the very centre, a point of infinitely crushing gravity called the singularity. This is where our best theory of gravity throws up its hands and stops making sense. That is usually read as a sign that something deeper, which we don't yet have, has to take over.
The stretching is a tidal effect, the same thing that raises ocean tides, only ferocious. The difference in pull across a small height grows as the inverse cube of distance, \(\sim 1/r^3\), so it climbs far faster than gravity's own \(1/r^2\).
That cube has a counter-intuitive consequence. For a supermassive black hole the horizon sits so far from the centre that the tide there is mild, and you could cross it without noticing. For a small stellar-mass hole the horizon is close in, and the tides would shred you long before you reached it. Bigger holes are, in this one sense, gentler.
Inside the horizon something stranger happens: the roles of space and time swap. Moving toward smaller \(r\) becomes as inevitable as moving toward tomorrow. The singularity is then not a place ahead of you in space but a moment ahead of you in time, one you cannot avoid any more than you can avoid next week.
Tides are geodesic deviation: neighbouring free-fallers separated by \(\xi^\mu\) accelerate apart as \(\tfrac{D^2\xi^\mu}{d\tau^2} = -R^\mu{}_{\nu\alpha\beta}\,u^\nu \xi^\alpha u^\beta\), governed by the Riemann tensor. The radial tidal component scales as \(\sim 2GM/r^3\) and diverges as \(r\to 0\).
That divergence is not a coordinate artefact. The curvature invariant, the Kretschmann scalar \(R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta} = 48\,G^2M^2/c^4 r^6\), blows up at \(r=0\), a genuine curvature singularity. The Penrose–Hawking theorems show such singularities are generic: once a trapped surface forms, geodesic incompleteness is unavoidable in classical GR, not a quirk of perfect symmetry.
Status: established, up to a point. Tidal physics and the inevitability of collapse are solid general relativity. The singularity itself is widely read as a breakdown of the theory: at the Planck curvature, quantum gravity must replace the classical prediction. What truly sits at the centre is unknown.
A ring of fallers drops toward the hole. Watch it draw out into a cigar, stretched the long way and pinched across. It falls on a loop. The arrows are the tidal pull, growing as \(1/r^3\): mild far out, violent close in. It falls on a loop. The stretch is geodesic deviation; the glow at the centre is the curvature singularity, where the invariants diverge. It falls on a loop.
Idea 03 · Predicted · Hawking, 1974 · not yet observed
Hawking Radiation
Even the darkest thing in the universe glows.
If nothing escapes a black hole, it should be perfectly black and last forever. In 1974 Stephen Hawking found that, once you allow quantum physics in, neither of those is quite true. A black hole glows, incredibly faintly, with a feeble warmth. So over almost unimaginable stretches of time it slowly leaks away and eventually evaporates.
The glow is far too dim to have ever been seen. But its very existence ties three great pillars of physics together, namely gravity, quantum mechanics and heat, and that is why physicists have spent half a century chasing what it means.
Hawking's result gives a black hole a real temperature:
\[ T = \frac{\hbar c^3}{8\pi G M k_B}. \]
The mass sits in the denominator, so the rule runs backwards from everyday objects: smaller black holes are hotter. A hole the mass of the Sun sits a fraction of a billionth of a degree above absolute zero, far colder than the cosmic microwave background, so today it absorbs more than it emits and actually grows. Only a tiny hole runs hot.
Because emission speeds up as the hole shrinks, evaporation is a runaway that ends in a final flash. The lifetime scales as the cube of the mass, \(\propto M^3\). For a solar-mass hole that works out to around \(10^{67}\) years, vastly longer than the present age of the universe.
Heuristically, the vacuum near the horizon is full of virtual particle–antiparticle pairs; tidal gravity can pull one across the horizon while its partner escapes to infinity as real radiation. The infalling partner carries negative energy relative to a distant observer, so the hole's mass decreases. In effect it pays for the radiation. The spectrum is very nearly thermal at the Hawking temperature, set by the horizon's surface gravity \(\kappa\) via \(T = \hbar\kappa/2\pi k_B c\).
This completes black-hole thermodynamics: the horizon carries an entropy proportional to its area, the Bekenstein–Hawking entropy
\[ S = \frac{k_B\,c^3 A}{4 G \hbar} = \frac{k_B\,A}{4\,\ell_P^2}, \]
and the laws of mechanics for horizons mirror the laws of thermodynamics exactly. It also poses the deepest puzzle in the subject: if the radiation is truly thermal, what happened to the information that fell in? The information paradox remains unresolved.
Status: predicted, not yet observed. The effect is far too faint to detect from any astrophysical black hole. But it is theoretically central, and laboratory analogues, such as sonic and optical horizons, have reported the corresponding thermal emission, which supports the mechanism.
The dark sphere is the hole; the faint motes streaming outward are its glow. Over a very long cycle it shrinks away. Drag to orbit. The gauge tracks its temperature, climbing as the hole shrinks, because \(T \propto 1/M\). Drag to orbit; watch it heat as it evaporates. Pairs split at the horizon: the blue quantum escapes, its red partner falls in carrying negative energy, so the hole loses mass. Drag to orbit.