The horizon that nothing marks

The picture of a black hole most people carry is of a place where gravity is so strong that nothing survives arrival. Half of that is right and the half that is wrong is the interesting one: the horizon is where nothing leaves, and what happens to a body reaching it depends entirely on how large the hole is.

Why the tide falls as the square of the mass #

The tidal acceleration between two points separated by $\ell$ at distance $r$ from a mass $M$ is

$$a_{\text{tidal}} = \frac{2GM\ell}{r^3},$$

which is a difference of two inverse-square attractions and is the term free fall cannot remove. At a fixed distance a bigger mass gives a bigger tide, which is the expected direction. But a horizon is not at a fixed distance: it sits at $r_s = 2GM/c^2$, which is proportional to the mass. Substituting,

$$a_{\text{tidal}}(r_s) = \frac{2GM\ell}{(2GM/c^2)^3} = \frac{\ell c^6}{4G^2M^2}.$$

Two powers of the mass in the denominator. The bigger the hole, the gentler its horizon — not slightly, but as an inverse square, so a hole a thousand times more massive has a horizon a million times more benign.

That crossing is the whole content of the essay’s title. For a stellar-mass hole the destruction happens outside the horizon, in full view; for a galactic-centre hole it happens deep inside, unobserved, and the crossing itself is unremarkable.

The numbers, on a scale a body can be compared to #

The abstraction is worth converting into the units a human frame is specified in.

A tidal stretch is a difference of accelerations across a length, so the quantity to compare is a stretch per metre — the residue free fall cannot remove, measured at a horizon instead of in a lift. A person can survive a few tens of gravities of uniform acceleration briefly and a good deal less applied as a stretch; ten gravities per metre across a body is a generous limit and it is the one drawn in the first figure. On that scale:

A ten-solar-mass hole gives $1.9 \times 10^7$ gravities across 1.8 metres at its horizon, which is $10^7$ per metre — six orders of magnitude past lethal, and reached about three thousand kilometres out, well before arrival. A hole of $10^4$ solar masses gives roughly the threshold value exactly at the horizon. The hole at the centre of the Milky Way, at $4.3\times10^6$ solar masses, gives about $10^{-4}$ gravities per metre — comparable to standing up quickly. M87’s, at $6.5 \times 10^9$, gives $4\times10^{-11}$, which is a hundred times smaller than the tide the Moon raises across a person standing on the Earth.

The size of that last hole is worth stating too, because it explains the mildness. Its horizon is 190 astronomical units across, roughly five times the orbit of Neptune. Falling into it is falling into a region the size of a planetary system, and there is no reason for anything the size of a person to notice a boundary.

What is actually at the horizon #

Nothing, and the claim can be made precisely rather than asserted.

How large a freely falling laboratory may be before gravity becomes detectable inside it scales inversely with the tide — so a large hole permits a large laboratory. At a solar-mass horizon a metre-scale instrument would register the gradient immediately; at a supermassive one an entire laboratory could cross without anything registering at all. That is the quantitative form of “nothing marks the horizon”, and it says the answer depends on the mass rather than on the horizon.

What is left when the uniform part of gravity has been transformed away is convergence: two released bodies fall toward the same centre along different lines and drift together. That convergence is the tide, it is the only part of gravity a falling observer can detect, and it is finite and small at a large horizon. Everything else about the field has been removed by the choice of frame, which is what free fall does.

So the physical content of “the horizon is not a place” is that the one locally measurable gravitational quantity is small there for a large hole, and there is no other. Every other candidate is a coordinate artefact — the same distinction that decides which of two clocks is really running slow. The Schwarzschild radial coordinate misbehaves at $r_s$ and coordinates adapted to a falling observer do not; the metric written in the latter is perfectly regular, and every component is finite.

Two clocks, and why they disagree without contradicting #

The strongest source of confusion here is that the crossing takes a finite time and an infinite time, both correctly.

Neither curve is the true one. Proper time is what a wristwatch on the falling body records, and it says the crossing happens. An observer who never stops accelerating has a horizon of their own, built out of nothing but the acceleration. Coordinate time is a label a distant observer attaches to events, and it says the crossing never gets a label. Both are correct descriptions of the same worldline in different charts, and asking which is real is asking which coordinate system is real.

What the distant observer actually receives is a third thing again, and it is the one worth quoting because it settles the popular version.

The light a distant observer collects is emitted from successively deeper in the well, so the photons arrive later and redder and the received brightness falls exponentially — with a time constant of about $10^{-5}$ seconds for a solar-mass hole. So the image does not linger at the horizon in any practical sense: it fades to nothing in a fraction of a millisecond, and the “frozen star” of the older literature is a statement about coordinates rather than about anything visible.

So “the body appears frozen at the horizon” is true for about a millisecond and false thereafter, because there is nothing left to see. The infinity is in a coordinate; the observation is a fade.

Where the tide does its work, for a star rather than a person #

The same arithmetic applied to a body held together by its own gravity rather than by chemistry gives the tidal disruption radius, and it is worth doing because it is what is actually observed.

A star of mass $m$ and radius $R$ is torn apart where the tide across it exceeds its own surface gravity, which happens at

$$r_t \approx R\left(\frac{M}{m}\right)^{1/3}.$$

For the Sun near a $10^6$-solar-mass hole that is about $10^{11}$ metres, roughly the Earth’s orbit, and the horizon of such a hole is $3\times10^9$ metres — so the star is destroyed well outside, and about half its material is flung away while the rest falls back and lights up as it accretes. That is a tidal disruption event, and dozens have been recorded.

Raise the hole’s mass and the same competition of powers as before decides the outcome: the disruption radius grows as $M^{1/3}$ and the horizon as $M$, so above about $10^8$ solar masses for a Sun-like star the horizon overtakes the disruption radius and the star is swallowed whole, with no flare at all. The absence of tidal disruption events from the largest holes is a direct observational consequence of one power law being steeper than another — the same crossing as in the second figure, with a star in place of a person and a self-gravity in place of a tolerance.

What the horizon is a statement about #

If nothing local marks it, what is it? The definition is global: the horizon is the boundary of the region from which no future-directed path reaches infinity.

That is a claim about the entire future of the spacetime, not about any moment. It means, strictly, that whether a given event is inside a horizon cannot be determined by any measurement in its neighbourhood, and cannot be determined at all without knowing what the spacetime does afterwards — matter falling in later moves the horizon outward, so a horizon can grow through a region before the matter causing the growth arrives.

A horizon casts a shadow of angular radius $\sqrt{27}\,GM/c^2$ — larger than the horizon itself, because light passing nearby is bent inward — and that shadow is the mechanism by which a horizon is observed at all. Nothing is seen at the horizon; what is imaged is the boundary of the region from which light cannot reach the camera, and it sits well outside the surface it is named for.

What a falling clock actually measures #

The most useful way to see that nothing happens at the crossing is to write down what a falling observer measures and check that every entry is finite.

Take an observer released from rest far away and falling radially. In their own frame the metric can be written so that at the moment of crossing every component is of order one; the acceleration they feel is zero, because they are in free fall; the curvature they can measure is the tidal term already computed. Nothing in that list diverges, and for a large hole nothing in it is even large.

The crossing does have one signature, and it is not a local one. The observer’s future light cone, drawn in coordinates that cover both sides, tips over: before the crossing some outward-directed rays escape, after it none does. But an observer cannot measure the tipping of their own light cone by looking around them, because the cone is defined by where rays end up, and finding that out means waiting. Send a signal outward and wait for a reply, and the absence of a reply is evidence — but the waiting takes the rest of the observer’s finite proper time, which for a stellar-mass hole is microseconds and for a supermassive one is hours.

So the honest statement is that the crossing is detectable in principle and only retrospectively, by never hearing back. That is a strange kind of measurement, and it is the physical content of the horizon being defined globally: an event’s status depends on the future, and the only instrument that reads the future is patience.

Where the model stops #

Schwarzschild’s solution is for a non-rotating, uncharged hole in vacuum. Every real hole rotates, often near the maximum, and the rotating solution has a considerably more complicated interior with a second horizon inside the first. The tidal scaling with mass survives — it is dimensional — and the interior structure does not.

Classical general relativity, everywhere. Near the singularity the curvature reaches the scale where every model runs out at once, and nothing here describes what happens there. The claim is only that the horizon of a large hole is a region of small curvature, which is exactly why the trouble at the centre does not propagate outward to it.

A hole radiates at a temperature inversely proportional to its mass, so a large hole is colder than the background it sits in. That is the one quantum statement attaching to a horizon, and it points the same way as everything else here: the temperature is a property of the geometry rather than of any surface, and there is still nothing locally at the horizon for a falling observer to notice.

The falling body is a test particle. It is treated as having no mass of its own and no internal structure beyond a length. A body massive enough to disturb the geometry changes the problem, and one held together by forces strong enough to resist the tide crosses in a different state from one that is not.

The chart that covers both sides #

The essay has said several times that the trouble at the horizon is in the coordinates, and it is worth showing that a better chart exists rather than merely claiming one does.

The Schwarzschild coordinate $t$ is the time kept by a distant observer, and its defect is exactly the one the two-clock figure draws: a falling body’s worldline runs to $t = \infty$ without reaching the horizon, so the chart’s labels are exhausted before the physics is. The metric written in those coordinates has a component that blows up at $r_s$ as a result.

The fix is to relabel time by something that tracks the light rather than the distant observer. Define a new time coordinate that advances along an ingoing light ray — so that a radially infalling photon has a fixed value of it — and rewrite the metric. Every component is then finite at $r_s$, the falling worldline crosses at a finite coordinate value and continues inward, and the chart covers both sides of the horizon without interruption.

What the horizon looks like in that chart is instructive. Outgoing light rays, drawn as curves, are outward-sloping far from the hole; approaching $r_s$ they tilt over; exactly at $r_s$ an outgoing ray stands still, neither advancing nor falling; inside, every ray moves inward. The light cones tip, smoothly and continuously, and the horizon is simply the surface where the tipping has reached forty-five degrees. There is no discontinuity anywhere in the picture.

Eddington wrote down that coordinate in 1924, in a footnote to a book, and nobody noticed; Finkelstein found it again in 1958 and the significance was recognised. That is a thirty-four-year gap during which the field believed a horizon was a singularity, and the belief was a property of a chart that nobody had thought to change.

The general lesson has nothing to do with black holes. A quantity diverging in a calculation is evidence about the calculation until it is shown to be a divergence of something measurable — and the test is whether an invariant diverges, not whether a component does.

The horizon a computer can find #

The definition of an event horizon is global, which is elegant and useless to anybody trying to simulate one. A numerical relativity code integrates forward from a slice and does not know the future, so it cannot apply the definition at all.

What it uses instead is a different surface, defined on the slice it has.

Take a closed two-dimensional surface and shine light outward from every point of it. Ordinarily that outgoing wavefront grows: the area it sweeps increases. In a strong enough field it does not — the light still moves outward relative to the local geometry, and the geometry is falling inward faster, so the area of the outgoing front decreases. A surface with that property is called trapped, and the outermost surface for which the outgoing expansion is exactly zero is the apparent horizon.

That is a local condition given a slice: a computer can search for such a surface at each time step, and codes do, several times a second. It is how a simulated merger reports that a black hole has formed and how its mass and spin are read off.

The two horizons are not the same object, and the difference is instructive. The apparent horizon lies inside or on the event horizon, always, and they coincide when nothing is changing. During a merger they differ substantially: the event horizon, knowing the future, begins to form and grow before the two holes touch, reaching out to meet its counterpart in advance, while the apparent horizon appears only when the common trapped region does.

So the event horizon of a merger is found by running the simulation to completion and then integrating light rays backwards from the final quiescent hole — a computation that can only be done after the fact. The apparent horizon is found as the simulation runs.

Which is the practical form of this essay’s central claim. A horizon is not something an instrument in the region can detect, and the surface that can be detected locally is a different surface that happens to coincide with it when things have settled down.

Why “nothing special happens” is not the whole truth either #

The claim defended here is local, and it is worth marking where it stops being reassuring.

Nothing local happens at the crossing. What happens afterwards is not local and is not mild: every future path leads to smaller radius, the radial direction has become a direction in time, and the interior is a region in which the singularity is a moment rather than a place. The proper time remaining is bounded — at most $\pi GM/c^3$, which is a fraction of a millisecond for a stellar hole and about nine hours for M87’s — and no amount of thrust extends it. Accelerating, in fact, shortens it: the longest surviving path is free fall.

There is also a question this treatment cannot answer, and it is currently open. Quantum field theory in the presence of a horizon gives Hawking radiation, and reconciling that radiation with the unitarity of quantum mechanics has produced arguments that something must be present at the horizon after all — a firewall, in the version that provoked the most work. Those arguments are not about general relativity, which is unambiguous here, but about whether general relativity and quantum mechanics can both be right at a horizon. This essay’s claim is the classical one, and the classical one is not in doubt.

What the pictures cannot show #

None of these figures draws the interior. The Schwarzschild coordinates in which most of them are laid out do not cover it, and continuing them across the horizon requires a different chart — so the region past the crossing is not merely omitted but unrepresentable on these axes.

Nor can any of them show what the falling observer sees. The sky ahead and behind is distorted enormously, blueshifted in some directions and dimmed in others, and rendering it requires tracing null geodesics rather than plotting a quantity against a radius.

And the two-clock figure invites a question it cannot answer: what is happening “now” at the falling body, according to the distant observer. There is no answer, because simultaneity between separated events is a choice even in flat spacetime and is a more thoroughly arbitrary one here.

Where the ladder goes next #

The horizons ladder began with a surface that only lets things in and continued with two clocks that disagree about the fall. This rung asked what a crossing feels like and found the answer to be a question about mass. The rungs after it are the interior structure of a rotating hole, the shadow as an observable — which is an astronomer’s subject rather than this one’s — and the thermodynamic identity that gives a horizon an entropy proportional to its area.

The habit worth carrying away is the one the second figure supplies. Two quantities that both grow with a parameter can still cross, if they grow at different rates, and the crossing is often where the physics changes character. Here the horizon grows as $M$ and the danger radius as $M^{1/3}$, so the same object is a shredder at one scale and a doorway at another, with nothing changing but a number. When an intuition holds at one scale and fails at another, the useful question is which two competing quantities carry different powers, because the intuition was built where one of them happened to dominate.