Imagine you’re caught in a maze and the only thing you have to guide you is a device that measures the fractional distance between the entrance and exit. At one point, you note that you are 6/7 of the way to the finish line. Then a while later, after several more dead-ends, you see that you’re 5/6 of the way there. You wonder whether that means you’re now closer to the finish line than you were at 6/7, or further from it.
There are a few ways to answer the question.
If you happen to remember an abstract rule — that when two fractions have the same difference between numerator and denominator, the one with larger numbers is overall larger — you’ll know right away that 6/7 was closer to the end than 5/6.
But in the stress of being trapped in a maze, would you trust your recollection of an abstract rule?
If not, you could work it out from scratch: 5/6 is 1/6 away from the endpoint, and 6/7 is 1/7 away from the endpoint, and since 1/7 is smaller than 1/6, you must have been closer when you were only 1/7 from the finish, when your device showed 6/7.
That works — but it’s a multi-step chain of reasoning to follow while you’re running around lost and trapped. You don’t want to rely on that either. There’s a third way, which is both simpler and more definitive, and therefore more reliable to work with in stressful or uncertain situations than the other approaches. It has two moves.
First, notice that the sequence 5/6, 6/7, 7/8, 8/9, … must be moving in some consistent direction as the numbers grow. The overall values are either getting steadily larger, or steadily smaller — they can’t be doing both, and they can’t be jumping around. That much follows from the structure of the sequence alone, without any calculation.
Second, find out which direction they increase by checking an extreme case. Compare 1/2 with 99/100. The first is only halfway to 1; the second is 99% of the way there. So the entire sequence must be increasing. Therefore 6/7 is larger than 5/6, and you were closer to the finish line then.
The whole thing takes one glance to recognise the trend and half a thought to check the extremes. You might even spare another half thought to confirm your proof, noting that if you add more 9s to the top and more 0s to the bottom of 99/100, you can get arbitrarily close to 1.
And notice what this achieves: you don’t need to know about and trust your memory of an abstract rule; and you don’t need the multi-step calculation to prove the rule. You only need to recognize that the sequence has to be monotonic — moving consistently in one direction; a trend — and then you need a single extreme case to prove which direction has larger values.
This two-step method — identify the consistent structure, then test it at an extreme to fix the direction — is one of the most powerful tools in mathematics. It works wherever you can recognize that something must vary smoothly or consistently across a range of cases, because then a single extreme case settles the whole range and you don’t need to fuss about minutiae. It’s a way to see, without computation, that things you didn’t know must nevertheless be true.
The following discussion uses tools like this — analogy; recognizing consistent structure; testing trends at extremes to make small differences obvious — to resolve longstanding confusion about a particular geometric feature of the spacetimes we use to describe physical reality: the metric singularity.
These show up in one of the most consequential places physics has ever asked us to think about: the surface of a collapsing star, in its infinite future limit — the event horizon.
The confusion is about what kind of thing an event horizon actually is. The standard picture treats it as a surface — a sphere of a certain radius, sitting out there in the universe right now — that you could fly toward and cross as the galaxies and whatnot in the outside world go on existing. A surface that harbors a massive singularity at its center. A surface that might radiate.
The picture this article will build, geometrically and from the ground up, reading trends in the geometry and testing them at their extremes — clarifying the misinterpretation that led to the standard picture — is different.
What you’ll see is that the event horizon actually isn’t a place in space at all. It’s an event — in fact, a whole set of events that all occur instantaneously at a specific location. And this event horizon does not occur at just any instant, either: from the perspective of every point in the universe outside the collapsing star — to everything that stays in the outside world and never falls in — this singular event happens the very moment after the end of time; — it’s a boundary at time’s infinite limit.
That single shift has consequences. Taken to the extreme, it means that no matter how long you wait before diving in toward a black hole, when you arrive at your horizon event so does a flash of light you sent towards the black hole arbitrarily long before. You could shine some photons at a black hole, sit around for a trillion years drinking daiquiris, then dive in. You’d only ever be moving more slowly, at every point in space, than those photons you sent a trillion years ahead of you did when they were there; yet you still must arrive at the event horizon together with them, having smoothly caught right up to them the whole way along.
In fact, you’d also arrive at the same moment the original collapsing star itself finally shrinks to the horizon radius — its horizon event occurring coincidentally with yours, with the horizon event of the photons you sent ahead of you, and with everything else that ever falls in throughout the entire history of the universe.
This pure geometric result, which eluded us for more than a century — remaining remarkably well hidden as it sat in plain sight — has significant consequences. It means the densities that have worried physicists since the early days of relativity — the ones that lead to the standard claim that general relativity “breaks down” at a black hole’s central singularity — never actually form anywhere in our universe at any finite time. Instead, mass stays finite in size, every particle that falls in asymptotically approaching its own horizon event. Densities stay bounded. The pathological interior is permanently deferred to sometime after a moment that never quite arrives.
Unless one dives in. Then the universe does come to an end, its time becoming infinite at the moment of one’s horizon event. This is just like a tangent function when the angle gets to π/2: for positive values, the tangent function is defined at every angle arbitrarily close to π/2, but it blows up and is undefined exactly at π/2. On the range of angles [0, π/2), the tangent function spans the interval [0, ∞). These intervals have to be written as open on the right because the tangent function does not exist at π/2. That angle exists just fine, so the angular range itself could just as well be [0, π/2]. But the tangent function blows right up and doesn’t have a value there since infinity is not a number.
Similarly, the whole outside universe’s time blows up, approaching infinite values, as the things that fall into a black hole come to their own individual horizon events. Those horizon events all occur at once, in a precise, geometrically forced sense. And that singular instant technically has no corresponding value in the outside universe’s time, since all times in the external universe blow up to infinity when that instant is finally reached.
Just as π/2 occurs smoothly in a sequence of angles, but the tangent function does not exist at π/2, when things fall into a black hole they smoothly reach horizon events that all occur in an instant when there’s technically no more universe; when it’s been left in the past. The event horizon is the very next instant that occurs after the universe, after its time has blown right up; an instant in one sense, that occurs one beat after the universe itself has run out of instants.
These are strong claims, and I won’t ask you to take them on faith. The argument is geometric, and once the geometry is in place it’s the kind of thing you’ll be able to see for yourself — the way you can see, once you’ve compared 1/2 with 99/100, that fractions like 6/7 and 5/6 must follow the rule they follow.
What we’ll need to get there is a working understanding of how light moves through spacetime, what it means for two events to be separated in space and time, and what an image really is — what you’re actually seeing when you open your eyes and look at the world. From those pieces, the geometry of the event horizon falls out almost on its own, and a small theorem about it — trivially provable, once you see what’s going on — finishes the job.
The article begins, then, with the speed of light.
The speed of light in a vacuum
The speed of light in the vacuum of outer space is 299792.458 kilometers per second. As far as we know, nothing moves faster than this. And according to Einstein’s special and general theories of relativity, light must move at this speed regardless of how fast we move relative to it.
If you shine a flashlight towards the Moon at the same time as launching a rocket towards it, the photons in that flash of light will move away from you at 299792.458 kilometers per second AND they’ll move away from the rocket at 299792.458 kilometers per second — even as the rocket itself blasts away from Earth at 11 km/s.
That’s the single weirdest thing about Einstein’s relativity, in a nutshell. The photons in that flash of light do not recede from the rocket at 299781 km/s (i.e. 299792 km/s — 11 km/s). They move away from the Earth AND the rocket both at exactly 299792.458 km/s.
Einstein’s light postulate is remarkably unintuitive. And it is the most basic principle of relativity theory — one that’s at the heart of every prediction we’ve ever confirmed, from E = mc2, to gravitational waves, to the continued functioning of GPS.
In the language of pure mathematics, Einstein’s light postulate translates to a rather unintuitive geometric structure: a non-Euclidean one.
The familiar Euclidean geometry — first laid out by Euclid in his Elements around 300 BCE — is the geometry of the world we walk around in. Within Euclidean geometry, distances combine according to the Pythagorean theorem: if you walk some distance ∆x1 in one direction, then pivot 90 degrees and walk ∆x2 in a perpendicular direction, the overall distance between your start and end points is given by (∆s)2 = (∆x1)2 + (∆x2)2. Walk three meters, pivot, walk four meters, and you end up exactly five meters from where you started, because 32 + 42 = 52. The hypotenuse of a right triangle is always longer than either of its legs, in any orientation, in any region of Euclidean space.
But spacetime doesn’t follow this rule. Spacetime — the map of every “here-and-now” point that ever occurs, every location across space, at every instant throughout eternity — is a non-Euclidean geometry. The coffee shop down the street, at 2:58 pm yesterday, is one event in spacetime — one point in four-dimensional spacetime; the exact time and location the asteroid hit the Earth that wiped out the dinosaurs is another event — another point. A single photon’s location at the moment it leaves your flashlight is an event, and that same photon’s location half a second later, 150 thousand kilometers towards the Moon, is another event. Between any two events there is some spatial separation and some temporal separation — and there is also a third quantity, the spacetime separation — which is what the geometry of relativity is built around.
Here’s the rule. In the Lorentzian geometry of relativistic spacetime, the spacetime separation between two events is given by the difference, not the sum, of their squared spatial and temporal separations:
(∆s)2 = (∆x)2 – (∆t)2.
That single sign-flip — the minus where Euclidean geometry has a plus — is the defining feature of Lorentzian geometry. The way Lorentzian geometries measure distance is fundamentally different from how they’re measured in Euclidean space, and the consequences are immediate and strange.
The most important consequence is that the spacetime separation between two events can actually be zero, even when their spatial and temporal separations are individually nonzero — provided those two contributions are exactly equal. The hypotenuse of a triangle drawn in Lorentzian space has zero length whenever the legs are equal in size — whenever the triangle is isosceles.
And that is precisely what light does. Along light’s path through spacetime, the spatial distance covered always, exactly equals the temporal distance taken to cover it. The two contributions in the difference cancel in the Lorentzian version of the Pythagorean theorem. The spacetime separation between any two events on a photon’s path is identically zero.
So Einstein’s weird postulate — that light moves at exactly the same speed from every perspective — translates, in formal mathematics, to an equally weird geometric law: light moves equal distances through space and through time, so the spacetime separation between any two events on its path is identically zero. It’s an isosceles right triangle, whose hypotenuse has zero length by definition.
Algebraically, we can define structures like this in Lorentzian geometry without difficulty. But when we try to draw them — on graphs in our Euclidean surroundings, on the pages of articles like this one — it’s easy to forget that the lines we draw, which the geometry has defined to have zero length, actually do have zero length, rather than whatever length they appear with on the page. In such cases it’s worth reminding ourselves that whatever Euclidean distance such a line appears to span on a graph — say a few centimeters — the geometry that line is depicting still assigns it a length of “a few centimeters multiplied by zero”. The graph is mapped onto a Euclidean surface; what it’s actually depicting is Lorentzian, and we don’t have Lorentzian paper. The rules don’t match, and the eye will follow the wrong ruleset unless reminded otherwise.
This is the formal expression of Einstein’s light postulate, as it is translated from physics to geometry. By construction, relativity theory describes light as travelling along these null paths, on which every event is at zero spacetime separation from every other, in every reference frame, by basic identity — in fact, by the fundamental defining aspect of the geometry. By the very aspect that formally encodes the invariance of light’s speed in a vacuum, in the basic mathematical architecture of relativity.
This is the single fact the rest of the article rests on. Everything that follows is geometry built on top of it — on top of the fundamental defining principle of relativity, Einstein’s light-postulate in its pure mathematical form: in Lorentzian geometry, the length of an isosceles right triangle’s hypotenuse is identically zero. And such triangles describe the paths of photons in Lorentzian spacetime, using triangles with equal spatial and temporal lengths, so their spacetime lengths are always zero.
And just look at what this fact already gives us. The photon emitted from your flashlight, and the same photon when it’s halfway to the Moon, are distinct events. They’re also causally ordered — the first event happens, and the later one happens after it. From any reference frame, you’ll measure both spatial separation and temporal separation between them. Different frames will measure different amounts. But every frame will agree, by construction, that the spacetime separation between them is zero. The line they lie on, in the geometry of relativity, has no length.
A line of distinct, causally ordered events, with no geometric length between them. That’s the object the rest of this article is going to put to work.
It starts with understanding what an image is.
What an image is
You see, we don’t ever see any thing-in-itself: we see an image of a thing. We directly interact only with photons that have travelled across space and time from distant objects to eventually reach our eyes — not with the things themselves. What we see is evidence of those things’ existence, which comes to us with information about their physical properties — colour, brightness, etc. And since light travels at a finite speed, the further anything is from you the further back in time that thing was when it scattered the photons you’re seeing now. You only ever have access to information about distant things in their pasts, never in their present state.
Step out into the street: the house closest to you presents an image composed of photons that show you what the house looked like a moment ago, when those photons started toward you. And, at the same time, the next closest house is similar, only its image comes from a moment further back in time. You see the next house along as it was a bit further into the past, and so on. So when you look out at the world you’re not seeing “the world, now”, but rather the world as it was at earlier and earlier moments the further away the photons originated from.
A picture of a person standing in front of a mountainous backdrop is not an image of the person and the mountain range at the same time, but an image showing what the mountain looked like before the moment of the person’s life that’s recorded in the photograph. And both of those moments occurred before the moment when the picture was actually taken, when the photons arrived at the camera.
What you’re looking at, in any given image, is a particular kind of geometric object. All the photons that reach your eye at any single instant originated on a surface that extends outward from you in every direction, infinitely far in space and, by the same amount all along, infinitely far back in time. It’s the surface defined by every spacetime event that could have sent a photon to reach you exactly now. We call this surface your past light cone, and in the mathematical language of relativity it carves out a null hypersurface — a three-dimensional slice of four-dimensional spacetime, with the property that the spacetime separation between every event on the slice is by definition zero.
Like every path light travels, your past light cone is built entirely of null lines — and so it inherits the defining property of those lines. Every event on it is at zero spacetime separation from every other event on it, and from you, by construction. The entire universe of distinct, causally ordered events you can currently see — at every moment — lies, geometrically, at zero distance from you, and all of them from each other.
Notice what kind of object this makes the past light cone. It is a surface of infinite extent in space and time, on which infinitely many distinct events lie in a definite causal order — the photon from the near house was more recent in time than the photon from the distant star, by thousands of years — and yet it is also a surface on which every measure of spacetime separation is identically zero. The light cone is infinite as a set of events. It is null as a spacetime structure. The difference between those two things is the entire content of what makes Lorentzian geometry strange.
And here is something more to notice about this object that’s going to matter later. Your past light cone is yours. It belongs to you, here, now. As you move forward in time, the apex of your past light cone moves with you — the cone continuously sweeps through spacetime, taking in new events whose separation from the apex (your present position) is null, while older events grow further into the past. Every observer at every event has their own past light cone, and the cones are all different from each other. The past light cone is an observer-dependent, instantaneously-defined null hypersurface that follows each of us through our lives — the structure in which all the world we ever experience continuously appears.
You have lived your entire life on the apex of one. Every image you have ever seen has been a snapshot of the events lying on it.
That is what an image is.
The shrinking Andromeda thought experiment
The geometry of an image — of the null hypersurface that is your past light cone — is strange enough on its own. Now we’re going to bend it, and watch what happens when the spatial extent of a section of it shrinks asymptotically to zero.
We start with the image of Andromeda, our partner in the Local Group of galaxies. Andromeda is roughly a hundred thousand light-years across and edge-on, so the image of it we see at any moment is composed of photons from the near side, originating at events that occurred a hundred thousand years before the events from the far side. The image gets deeper into the past the further out in space you’re looking.
This is just like every other image you see, only more visceral, because the object whose extent spans a hundred thousand light-years and a hundred thousand years makes the temporal-depth-of-an-image fact unmistakable. The image you see is not “Andromeda, at some moment,” but “Andromeda, as it was across a hundred thousand light-years of lookback depth” — across a region of a null hypersurface, in fact!
Now imagine that somehow — maybe magically — Andromeda starts to shrink in radial depth from our perspective. The near side and the far side begin to come together. And they come together in exactly such a way that they reach an exact point radially, infinitely far into the future.
This kind of behaviour is called asymptotic, and it’s very common in nature. If you set a cup of coffee on the table, it’ll be roughly room temperature in an hour. But technically, in an enclosed setting where the room is held at a constant temperature forever, it’ll actually take an infinite amount of time for the coffee to come all the way to room temperature.
Similarly, in the present thought experiment, we’re imagining that Andromeda asymptotically shrinks to a point along its radial depth, infinitely far into the future.
The moment Andromeda finally shrinks to a point — when the near side and the far side coincide — is not something anyone in the outside universe could ever see. The asymptotic approach takes an infinite amount of time, and there’s no finite moment in our future from which we could observe what happens in the infinite limit. Andromeda just goes on shrinking, in our evolving image of it, forever.
But it does happen, in this thought experiment, in a geometrically precise way. So let’s think about what that final image would look like, even though no one in the outside universe is around to see it.
Throughout the shrinking, Andromeda’s image has depth in both space and time — that’s what every image has, and you can trace it on the page: photons from the far side originated at events further from you and further in the past; photons from the near side, closer in both. As Andromeda shrinks, that depth shrinks with it. The emission events at the near side and the far side get closer together — closer in space, because the near side and the far side themselves are getting closer; and closer in time, because there’s less and less radial distance for the photons to traverse between them.
In the limit — in the asymptotically final image that ever gets emitted, which no one who is spatially separate can ever observe — the depth has shrunk to nothing. The events at which the last photons leave the near side, the far side, and every point between the near and far sides — they all occur at the exact same place, at exactly the same time.
But look what hasn’t changed. When Andromeda was a hundred thousand light-years deep, there were infinitely many distinct points across the radial depth of that galaxy, and the events we see in any image span a hundred thousand years of lookback depth in time and space, occurring in a definite order: near side, then a little further out, then a little further still, all the way to the far side — an event that occurred one hundred thousand light-years further away and one hundred thousand years earlier than the first along the line. As the galaxy shrinks, the spatial and temporal separation between those points goes to zero, but the points themselves do not vanish. None of the points are removed from the set as the distance shrinks. None of them merge with each other. They are still all there, in the same order, with the same identity. The set is preserved. Only the geometry laid over it collapses.
This is exactly the move the past light cone made — infinite as a set of events, zero as a geometric structure — but applied here in a different and stranger way. The past light cone is a snapshot: a single instant of an infinite set of events at zero spacetime separation. What we have at the asymptotic limit of the shrinking Andromeda experiment is the same kind of object, but built up by a process — a sequence of images, which are already null hypersurfaces, whose spatial and temporal extent, and therefore geometric extent in every sense, shrinks asymptotically to zero while the topological set remains infinite. The final image — that final null hypersurface that never intersects anyone’s past light cone — is a metric singularity.
The word “metric” in this term simply means it’s about the geometry’s measure of things. And “singular” means what it ordinarily does in English: that distinct things come together there into one, with no measure between them, neither spatial nor temporal. The events along the asymptotic limit are still distinct from one another — they remain individual events — individual points of the underlying manifold itself — in a definite causal order. What’s become singular is the geometry between them, the part that measures distance and duration. That geometric measure has gone to zero. The set of events is unchanged; the measurements laid over it have collapsed. The structure is singular in its metric; it’s got no holes or missing points in its fundamental substrate — so everything can get there smoothly; and when they get there, “there” is actually a well defined thing.
Such as the confluence of events that occur everywhere across Andromeda’s radial extent, when those points all reach the same distance from us, at the final moment when they do that.
And here is the small theorem about metric singularities I promised in the intro. Take any two events that lie at zero spatial separation from each other and zero spacetime separation from each other. Then their temporal separation is zero too.
The proof writes itself, given the difference-structure of the Lorentzian version of the Pythagorean theorem. The spacetime separation is built as the difference between the squared spatial and temporal lengths of a right isosceles triangle in Lorentzian geometry. If the spacetime length between the two events is null, and the spatial length is also null because they occur at the same invariant place, the temporal separation is forced to zero as well. There is nothing left for it to be.
You can still draw the two events spread out along a timeline, with their causal order intact — the picture on the page will show them one after another. But that line on the page has no geometric length.
The events may be spaced across three centimeters of paper, but if those three centimeters lie along a line that is null by identity it means the distance between any two points along it is three centimeters multiplied by zero. And anything multiplied by zero is zero. The causal order along the metric singularity is real; the temporal extent is not.
Two out of three is enough. The third must follow.
It’s an incredibly simple geometric structure: a thing of zero extent, spatially and temporally, that’s chock full of distinct, individual events that are causally ordered. The events all occur together, at once, one right after the other.
And the really neat thing about these structures is that they actually occur in nature!
The magic flashlight and the greedy astronaut
Consider the collapse of a massive star’s core during a supernova. There’s a remarkable theorem — proved by a young Norwegian mathematician named Jørg Tofte Jebsen in 1921, shortly before he died, and independently by the American physicist George Birkhoff in 1923 — which says that for a non-rotating star whose core simply collapses radially, the geometry of spacetime events outside the collapsing matter is uniquely determined. It’s the geometry Karl Schwarzschild had written down in 1916, just months after Einstein published the field equations of general relativity. It doesn’t depend on the details of what’s happening inside the star, and it doesn’t depend on time. The Schwarzschild geometry is what you get outside any spherically collapsing mass — full stop.
And the Schwarzschild geometry, it turns out, contains a metric singularity of exactly the kind the Andromeda thought experiment built up to.
The metric singularity sits at a finite radius — the Schwarzschild radius, rh — which is a function of the collapsing star’s mass. It’s a null hypersurface that occurs exactly at rh, with zero spatial extent. It has zero spacetime extent by definition, zero spatial extent because every horizon event that ever happens does so exactly at rh, and therefore by the weird little Pythagorean theorem of isosceles right triangles in Lorentzian spaces, it also has zero temporal extent.
In the spacetime diagram in the left panel of the figure, you can see the metric singularity right at the graph’s left-edge. All the red lines on the diagram are outgoing null lines — the paths of outgoing photons in the Schwarzschild geometry — and the outgoing null line at the very left edge points perfectly vertically. Unlike every other null line in the geometry, this one has zero radial extent — zero spatial extent — so, being a null line it also must have no temporal extent by the Pythagorean theorem of Lorentzian isosceles right triangles.
And while the graph in the figure appears to show vertical length at the Schwarzschild radius, algebra also proves that that apparent length is an illusion. In the coordinate chart displayed in the graph, along vertical lines of constant r, the geometry requires every incremental length of “funny time” to be multiplied by a factor (1 – rh/r). Therefore, right at rh, the geometry of this graph says, “Every increment along the ‘funny time’ direction must be multiplied by zero.”
That’s two different ways of seeing that rh must be a metric singularity, in which any way you slice the geometry all the events at rh occur at once — spatially, temporally, and spatiotemporally. Every point that lies along the vertical line at rh occurs at once in every spatial and temporal sense, by two versions of the same simple theorem: by law and algebraic proof.
But in this piece, right from the start, we’ve been careful not to rely on abstract rules or algebraic proofs, since it’s more instructive to reason geometrically so we can understand things like “no temporal extent along a line.” Since it’s far better to truly understand why something must be than to just know that it must be. And it’s instructive, and therefore worth taking the time to do so even in the simplest cases, where the rule is as simple as “when one leg of an isosceles right triangle is zero, so is the other by definition”, or when the algebra is as simple as “anything multiplied by zero is zero”.
By the epistemological method we’ve followed so far, what we want to do is understand the causal nature of this metric singularity by identifying monotonic trends, considering extreme examples, and seeing clearly what they force. That’s our best and most foolproof approach to understanding things that might naively appear to be otherwise, even when we know they can’t be.
We need that here because there is a small temptation. When you look at the diagram, there’s a temptation to interpret horizontal slices as lines of constant “time”, and to read the graph as showing that the collapsing star reaches rh first, then the ingoing beam, then the astronaut, etc. And, causally speaking, that is indeed what happens — only it’s also true that all of those horizon events occur with no spatial or temporal extent between them. But there is a temptation to look at the graph and see all those red outgoing null lines and read the picture as showing: as the collapsing star shrinks to smaller and smaller r as funny time passes, while it collapses it shines light outwards along those red null lines, which take more and more funny time to make their way out in r; then, when the star finally reaches the size defined by rh, the “outgoing” photon it emits just “hangs there at rh forever, never able to progress to any larger r than that. Just stuck there until something else reaches rh and detects it.”
The problem with this explanation — which is what we’ve told ourselves happens for six decades now — is that the temporal extent at rh is always zero: the line only shows the causal order of horizon events, not actual temporal extent since the geometry says we have to multiply the length of the line there by zero to work out how much time passes there. So we know this is true, by the basic rules of the geometry.
But there’s really a much better way to see this. Rather than simply seeing that the way we’ve been looking at this for the past sixty years is wrong, it’s really so much better to get a clear picture of what actually must be instead, and why. Because the geometry actually unambiguously forces an explanation that’s fundamentally different from “light just hangs there indefinitely”; that the nature of the metric singularity really has to be something else that’s quite different from that. Then, we’ll also understand how that metric singularity actually must occur in a causal sense as well, in relation to every other event on the page.
To see this, let’s take a closer look at the magic flashlight beam and the greedy astronaut that are also depicted in the graph above (I’m dropping the exact same graph below for convenience). The scene is like this. Some future civilization on Earth sees a star collapse from some finite distance away. The Earth stays a fixed distance away from the collapsing star forever. Sometime after the collapse starts, after the star has gotten small enough that no one on Earth could shine a beam of light towards the collapsing star and have those photons reach the collapsing star’s surface before its horizon event occurs (at least, in a causal sense), the Earthlings flash a beam of magic photons at the place where the star collapsed.
Now, these magic photons that these future Earthlings flash at this collapsed star have a special property: intermittently, the whole way in, a few photons are reflected backwards, so they come back out to Earth. You can see two of the reflected photons on the graph: the first two thick red lines from the right that make it all the way back to Earth before the greedy astronaut jumps in and intercepts the magic photons en route; those are representative of all the photons that make it all the way back to Earth before the greedy astronaut jumps in later in the story.
So in the time before that, the Earthlings are expecting that what they will see happen is: the beam of magic photons will appear to travel inwards, asymptotically approaching a finite radius, rh. They’ve seen the graph of the null lines in the Schwarzschild geometry before, so they’re expecting to see what’s left of the magic beam of photons arrive — having travelled along those red lines — forever; appearing to come from radii that asymptotically approach rh. The beam of magic photons will, because they reflect some photons back forever, appear to recede forever towards that finite distance, from rh. Like the graph of a tangent function as the angle approaches π/2. The Earthlings’ time goes to infinity just like the tangent function goes to infinity, as the magic light beam’s lookback distance approaches rh just like the angle approaches π/2. The image of the star will forever appear to recede asymptotically so the flash of magic photons appears to reach rh exactly in the infinite future.
That’s what they’re expecting to see in the image when the magic photons arrive back at Earth. This image of the magic photons making their way asymptotically towards rh.
But it turns out that after a long while — like, let’s say the star is actually observed to collapse sometime in our near future, then the beam of magic photons is sent towards the collapsing star in a million years, then let’s say another trillion years pass while these future Earthlings watch their magic photons make their way towards the collapsing star, just sipping on daiquiris the whole time as they watch this magical relativity experiment in action (because that’s really what it is)… —
OK, trying this again: — so they’re sipping away on daiquiris and then a trillion years later there’s this greedy astronaut who decides he just has to have all those magic photons for himself, so he takes off in a spaceship after the image that everyone on Earth had been enjoying for a trillion years and he intercepts the magic photons all on his way down to rh. And he heads after the photons and he can see the image of them as he goes after it, and the Earthlings never see another magic photon because he’s gobbling them all up with his eyes. And as time goes on he sees that the image is coming from closer and closer! The spatial (and, therefore, temporal) distances travelled by the magic photons along the null triangle — right, isosceles — are getting smaller and smaller — regardless of how long those lines appear to be when projected onto Euclidean space. The greedy astronaut sees that he is clearly gaining on the outgoing magic photons, as the reflection continues to originate from closer and closer, having travelled along those null lines the whole way down. The radial separation between the reflection events and the observation events get smaller and smaller until they coincide spatially (and, therefore, temporally). So he sees himself gaining on this magic beam of light — that the most recent reflected-back photons came from closer and closer to him the whole way down. And eventually — at rh, actually — he sees that he finally caught up to the magic beam of light — that the last photon he observed exactly as he reached the beam and the photon had to travel zero distance to get to him because he’d finally caught up to the source that had been reflecting them back the whole time. That’s what’s shown in the graph above.
But if on the other hand the greedy astronaut turns out to be an inquisitive astronaut, who just wanted to touch every one of the photons for a single beat — for as long as there is between each successive observation, and then he sends the one he held for a beat back along its way to Earth — if that were instead to happen — then the people on Earth would instead observe one paused beat in their evolving image of the magic beam as it makes its way towards rh, never quite getting there. And what they’ll see now, if the inquisitive astronaut puts his fingerprint on the magic photons he handles for that beat, is that he too appears to approach rh forever. Both of them appear to be asymptotically approaching rh, as if they’ll both arrive there together after infinite time has passed.
And from the inquisitive astronaut’s perspective he’s going to get to rh — which is a finite distance in space and time into his future — in a finite amount of time as he travels towards it at a finite speed. So he will arrive at rh — which he arrives at just as he touches the magic beam, catching up to it on its trip in — in finite time.
And from the perspective of the outside universe that trip will appear to asymptotically slow down. They’ll appear to freeze as they approach rh, as their proper passage of time gets chunked into smaller and smaller increments as he approaches both the magic beam and rh.
And it turns out that this picture is representative of every event that occurs across the entire metric singularity — every horizon event that ever happens as anything ever reaches the horizon — they all occur coincidentally as each one of those things reaches rh: — an event composed of all the horizon events that ever occur; an event of events — an event horizon.
(We had the name the entire time. It’s a horizon of events, which are each themselves horizon events. Another name for a horizon of events is literally an event horizon).
And every one of them — every horizon event that can ever happen — happens without spatial or temporal separation from any other. They form the metrically singular event horizon that anyone in the outside universe is going to see all infalling things approaching asymptotically into the future. At infinite cosmic time — just a single beat after the largest finite time that happens — that metric singularity finally happens, and everything that ever falls towards rh finally gets there.
And since it’s a metric singularity the infalling things actually do arrive there one after the other. The star’s surface reaches rh first, then the magic beam, then the greedy/inquisitive astronaut, and everything else. They’re all distinct events, and they happen in order. But they happen at the same place and time: the event horizon.
And that is a black hole.
cosmiCave.org 
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