Two clocks that disagree about the fall

Drop a clock into a black hole and watch it. The standard account says the clock, seen from far away, slows as it approaches the horizon, its light reddens, and it hangs at the surface for ever without ever quite crossing. The falling clock’s own account says it crossed in a few milliseconds and continued.

Both accounts appear in textbooks, and the disagreement between them is usually presented as a paradox to be marvelled at. It is not a paradox. It is two different quantities, and the marvelling has obscured the fact that the first account is also, as a description of what is seen, false.

The faller’s account, which is short #

Release a clock from rest at ten Schwarzschild radii and integrate its own elapsed time as it falls.

The answer for a fall from $r_0$ has a closed form, and it is the same cycloid that describes a Newtonian radial fall — which is a coincidence worth noticing, since almost nothing else about the two theories agrees this close in:

$$\tau = \sqrt{\frac{r_0^3}{2 r_s c^2}}\,\bigl(\eta + \sin\eta\bigr), \qquad r = \frac{r_0}{2}(1 + \cos\eta).$$

From ten Schwarzschild radii the crossing takes 49 in units of $r_s/c$, which for a ten-solar-mass hole is 4.8 milliseconds and for a four-million-solar-mass one is about ten minutes. Nothing happens at the crossing. The faller has no local means of detecting it, because the horizon is defined by where signals can eventually get to and that is not a local question.

The distant account, and what it is a statement about #

The coordinate time for the same fall diverges. That is a fact about the coordinate, and the coordinate is not nothing — it is the time kept by the family of observers who stay at fixed radius for ever, which is the natural family for anybody watching from outside.

The divergence has a clean origin. A static observer at $r$ has a clock running at $\sqrt{1 - r_s/r}$ of the distant rate, so the further in the emitter is, the more distant time passes per unit of its own.

At 1.05 Schwarzschild radii a static clock runs at 0.218 of the distant rate, and the curve’s approach to zero at the horizon is what makes the coordinate time diverge. The approach is by a square root — steep, but with no infinity in the geometry itself. What diverges is a ratio between two clocks, and a ratio is a relationship rather than a place.

But there is a second contribution, and forgetting it is the usual reason the distant account gets told wrongly. The light also has further to climb, and the climb takes longer as the emitter approaches the horizon: a photon emitted at $r$ takes a coordinate time that diverges logarithmically as $r \to r_s$. The received signal is therefore delayed twice over, and both delays go as the logarithm of the distance above the horizon.

Putting the two together, the emitter’s distance above the horizon falls off exponentially in received time, and the redshift with it. That is the hero figure. It is an exponential, and exponentials do not linger.

The image does not freeze; it goes out #

Take the exponential seriously and the picture everybody carries falls apart.

For a ten-solar-mass hole the e-folding time is about 0.2 milliseconds. After one millisecond the received frequency is down by a factor of $e^5 \approx 150$; after ten it is down by $e^{50}$, which is $5\times10^{21}$. Visible light emitted by the falling clock arrives as radio waves within a millisecond and as nothing measurable within ten.

The shape of the argument is borrowed from a process with no gravity in it. A decaying sample also never formally reaches zero, and it is nevertheless perfectly correct to say a sample is gone. An exponential’s tail is not a lingering; it is a statement about how quickly the thing has ceased. What makes the infall case feel different is that its exponential is written in a coordinate time, and coordinates do not carry the intuition that half-lives do.

There is also a hard limit that has nothing to do with intensity. The falling object emits a finite number of photons before crossing — proper time is finite and emission rate is finite, so the count is finite — and after the last one has been emitted there is nothing further to receive. The distant observer sees the object redden, dim, and stop. There is a genuine last photon, and after it the image is not faint, it is absent.

The claim that an infalling object “hangs at the horizon for ever” is therefore not an approximation that gets better with better instruments. It is a description of a mathematical limit that contains no photons, and stating it as a physical prediction is the same class of error as saying that a horizon is a place where physics breaks down.

The arithmetic of the last photon #

The finite-photon argument deserves numbers, because it is the one that closes the question and it is almost never made quantitatively.

Take a source emitting at a rate of $10^{15}$ photons per second — a milliwatt of visible light — falling from ten Schwarzschild radii into a ten-solar-mass hole. Its own clock records 4.8 milliseconds, so about $5\times10^{12}$ photons are emitted in total. That is the entire supply; there is no more.

Those photons arrive spread over an unbounded stretch of the receiver’s time, with the arrival rate falling by a factor of $e$ every 0.2 milliseconds. Integrating, half of them arrive in the first fraction of a millisecond, and the rate drops below one per second after about $0.2\ \mathrm{ms} \times \ln(10^{15}) \approx 7$ milliseconds. It drops below one per year within about ten milliseconds, and below one per age of anything within about twenty.

So the sequence of events for the distant observer is: a bright object, redward through the spectrum in a millisecond, a scatter of increasingly rare and increasingly red photons for a few more, and then nothing whatever — with the last photon of the whole supply arriving at a definite time. Nothing about that description contains a lingering image, and the arithmetic is entirely ordinary: a finite number of photons, spread over an exponential.

The same disagreement, with no gravity at all #

The strangeness here is not a property of gravity, and the cleanest way to see that is to reproduce the whole phenomenon in flat spacetime.

Take an observer accelerating uniformly for ever, and let a clock be released from the rocket at the moment the acceleration starts. The clock drifts inertially; the rocket keeps accelerating away. There is a plane behind the rocket, at a fixed distance in its own frame, from beyond which no signal can ever catch up — because a light ray chasing an ever-accelerating rocket can lose the race. That plane is a horizon, called a Rindler horizon, and it exists in empty space with no mass anywhere.

The released clock crosses it, and everything on this page repeats. In its own frame the crossing is an event of no significance whatever: it is drifting freely through empty space and nothing happens. In the rocket’s frame its signals redshift exponentially, its image fades and goes out, and it never quite arrives at the horizon.

The flat-spacetime version happens because an accelerating worldline is a curve that stays inside the light cones while continually changing which inertial frame it is momentarily at rest in. A light ray sent from far enough behind never catches it, and that failure defines a surface in the diagram — not a place in space, and not a property of anything material. Turn the engine off and the surface is gone.

The equivalence principle says this had to be so. A horizon is what an observer who refuses to fall freely creates, and a static observer outside a black hole is precisely an observer who is refusing, by continually accelerating away from free fall. What the distant view of an infall is really reporting is the cost of that refusal.

Why “when did it cross” has no answer #

The remaining discomfort is usually a hidden assumption: that there is a fact about what is happening to the falling clock now, at the distant observer’s now, and that the two accounts must be reconciled against it.

The assumption can be dismantled in flat spacetime before gravity is anywhere in sight. Two observers in relative motion slice spacetime into “now” differently, and neither slicing is preferred — so “what is happening there at this moment” is already a convention rather than a fact. Curved spacetime removes even the convention’s uniqueness: there is no natural way to extend a distant observer’s now across a horizon, and no way to choose between the unnatural ones.

In flat spacetime “now” is a choice of slicing, and different observers choose differently. Near a horizon the choice is worse than non-unique: the Schwarzschild slicing does not reach past the horizon at all, so the question “has it crossed yet, according to a distant observer” has no answer within that slicing rather than the answer “no”. Other perfectly good slicings — the ones an infalling observer uses — do reach past, and in those the crossing happens at a definite time.

That is the whole of the resolution. There is a fact about the crossing, and there is a fact about what the distant observer receives, and the thing that does not exist is a fact about which distant moment the crossing happened at.

What the faller sees looking outward #

The view in the other direction is worth its own section, because the popular account of it is as wrong as the popular account of the view inward, and wrong in the mirror-image way.

The story usually told is that an infalling observer sees the entire future history of the outside universe play out in fast-forward at the moment of crossing. It is false, and the reason is a race between two effects.

Light from outside is blueshifted on the way down, and the falling observer is also moving inward at a large fraction of $c$, which Doppler-shifts the incoming light back toward the red. The two very nearly cancel. Carrying the calculation through, an observer falling from rest far away and looking radially outward at the moment of crossing sees the outside universe with a redshift factor of about two — not an infinite blueshift, and not a fast-forward.

What the observer does see is the sky contracting. Aberration pulls the incoming light forward into a shrinking cone ahead, so the outside universe appears as a bright disc that gets smaller as the fall proceeds, with darkness around it. The disc does not vanish at the horizon: an infalling observer inside a large horizon still sees light arriving from outside, right up to the point where the calculation itself stops applying.

The total amount of outside history that arrives before the crossing is finite and modest — roughly the light-crossing time of the region fallen through. An observer falling into a stellar-mass hole sees a few milliseconds of the universe’s future, not all of it.

The star that was frozen #

The two accounts on this page were once two names, in two languages, and the split is a good record of which coordinate each community was thinking in.

Through the 1960s the standard Russian term for the object was zamyorzshaya zvezda — a frozen star. It is an exact description of the Schwarzschild-time picture: the collapse of a massive star, watched from outside, slows and asymptotically stops, and the surface hangs just above where the horizon will be, redder and dimmer for ever. Zel’dovich and Novikov used it, and it is what the mathematics of the distant view says.

Western usage at the same period was different and less committed — “collapsed star”, “Schwarzschild singularity”, and Wheeler’s earlier and unimprovable “gravitationally completely collapsed object”. The phrase that displaced them all appeared in the mid-1960s and was popularised by Wheeler in 1967, and it describes something else again: not what a clock reads, but what light does.

The two names are the two accounts. Frozen star is the distant static observer’s coordinate time, made into a noun. Black hole is the causal structure, made into a noun. Each is a correct description of the same object, and each conceals what the other reports.

What settled it was not a measurement but the spread of the understanding this essay is about. Once it was generally appreciated that the freezing is a property of a coordinate — that Finkelstein’s coordinates cross the surface smoothly and the faller notices nothing — “frozen star” stopped being a description of the object and became a description of one view of it. The Soviet authors adopted the new term within a few years, and by the mid-1970s the older one had gone.

It is unusual for a naming dispute to carry this much content, and the reason it does is that both names are accurate. Nothing had to be discovered to choose between them; what had to be noticed is that the two were answering different questions, which is the whole argument of this page arriving as a matter of vocabulary.

The membrane that is not there #

The distant view is awkward to compute with, because everything interesting happens at a surface where the coordinates fail. There is a reformulation that makes it easy, and its status is exactly the subject of this essay.

Replace the horizon, for the purposes of any external calculation, with a thin membrane sitting a little way outside it. Give the membrane ordinary physical properties: an electrical surface resistivity, a viscosity, a temperature, an entropy and a charge density. Then apply completely ordinary physics — Ohm’s law, Navier–Stokes, thermodynamics — and every prediction about the outside comes out right.

That is the membrane paradigm, developed by Damour and by Znajek in the late 1970s and set out at length by Thorne, Price and Macdonald. It is not an approximation in the usual sense: for anything an external observer can measure, the membrane and the horizon are interchangeable.

The properties are not arbitrary and one of them is delightful. The membrane’s electrical resistivity comes out at 377 ohms per square, which is the impedance of free space — the same $\sqrt{\mu_0/\varepsilon_0}$ that an antenna has to be matched to. Drop a charge in and it spreads over the membrane exactly as charge spreads over a conductor. Thread a rotating hole with a magnetic field and currents flow in the membrane, and the resulting circuit — a rotating conductor driving current through an external load — is the leading model for how a quasar launches a jet a million light-years long.

The paradigm’s value is that it converts an intractable problem into an electrical engineering one, and a great deal of high-energy astrophysics is computed this way.

And it is wrong for the faller. There is no membrane. An observer crossing the horizon finds no surface, no current, no viscosity and no temperature, and passes through a region where nothing is happening at all.

That is not a defect of the paradigm; it is a statement of its domain, and it is this essay’s thesis in another vocabulary. Two observers, two complete and correct descriptions, no experiment that compares them, and a contradiction only for somebody who insists that one of the two must be describing what is really there. The membrane is really there for everybody outside and really absent for everybody who crosses, and those two facts are compatible because nobody can occupy both positions.

What it costs #

Two of the quantities here cannot be measured by the observers who own them. The faller cannot detect the crossing; the distant observer cannot see it happen. The event is perfectly well defined and belongs to nobody’s experiment, which is a strange property for a physical event to have and is a direct consequence of the horizon being defined by the entire future of the spacetime rather than by anything local.

The horizon is teleological. Whether a given event is inside one depends on whether a signal from it ever escapes, which is a question about the infinite future. A horizon can therefore grow before the matter that will grow it arrives, and this is not a paradox but a consequence of the definition. Anybody expecting a horizon to be a locally identifiable object is expecting the wrong thing.

The two accounts cannot be compared by bringing the clocks together. That is what makes this different from the twin paradox, where the disagreement is settled by a reunion and a direct comparison of two readings. Here there is no reunion available, even in principle: one worldline ends at the singularity and the other never enters the region. The disagreement is therefore permanent in a way the twins’ is not, and any statement of the form “who is really right” has no experiment behind it — which is the honest reason the question dissolves rather than being answered.

The idealisation is a lone clock and nothing else. A real infall involves matter with pressure, magnetic fields and radiation, and none of that is in these figures. What is drawn is the geometry’s own contribution, which is the part that does not depend on what fell.

Where the model stops #

It is classical throughout. In a quantum treatment a horizon has a temperature and evaporates, and for the fall of a single object into a large hole those effects are utterly negligible — but they are what makes the question of whether information can be destroyed by this process a live one, and the answer is not known.

The fall from rest is a special case. Anything arriving from far away, or arriving with angular momentum, follows a different trajectory; the qualitative conclusions survive, and the numbers do not.

The exponential’s constant depends on the definition of the received time. Fitting the decay against a distant static observer’s clock gives about $2 r_s/c$; against other reasonable time coordinates the coefficient changes. The figure states which it used, which is the only defence available against a number that looks universal and is not.

Nothing here says what is inside. The trajectory has been integrated to the horizon and no further, and the interior — where the radial coordinate is a future direction — is the subject of the next rung.

What to keep from this #

Three sentences are worth carrying out of it. A coordinate diverging is not a physical event. An exponential does not linger, whatever the coordinate it is written in. And two observers computing different quantities are not in disagreement — they are answering different questions, and the appearance of a paradox comes from assuming that only one question was asked.

The ladder from here #

Later rungs on this anchor: the interior, and why arrival at the centre is a time rather than a place; the photon sphere, and the shadow an imaging array actually measures; the Kerr solution, where an ergosphere makes even standing still impossible outside the horizon; the Penrose process, which extracts energy from rotation; and the information problem, in which the classical statement that nothing comes back out meets the quantum requirement that nothing is ever lost.

The neighbouring ladders are simultaneity, whose flat-spacetime lesson is the one this essay leans on hardest, and gravitational time dilation, of which everything here is the limit as the factor goes to zero.