## Sunday, September 06, 2020

### Most laymen completely misunderstand what a black hole is

I am receiving lots of e-mail from young and armchair physicists. It's often a pleasant exchange. Sometimes the sender wants a theory to be confirmed. And among the layman senders, most of the theories or "theories" are seriously wrong. Many of the misconceptions keep on repeating all the time.

One of those invalid memes that I want to discuss – because I got several copies of that meme just in recent hours – is the idea that
The point of the black hole is the singularity. That is what makes a black hole a black hole and that's also where the mysteries of black holes hide.
This opinion is completely wrong. In this case, however, I have some empathy for those who think of black holes in this way. When I was 8, I (mis)understood black holes similarly – due to the influence of Newtonian prejudices combined with some popular explanations – despite the fact that some of the popular explanations, including those in the TV show "The Windows to the Universe Are Wide Open", went much deeper than to this misconception and I did watch them and read them.

Let me be more specific about the childish "singular" understanding of the black holes. People who are stuck in it assume totally Newtonian concepts everywhere, despite the fact that they superficially claim that they know something about general relativity. In Newtonian physics, the geometry of all spherically symmetric homogeneous bodies is always the same. You only distinguish them by the radius $$R$$ and the density $$\rho$$.

Celestial bodies may come in various sizes and the "size" is an extensive quantity that is assumed not to change things qualitatively: you may scale objects up, the Newtonian reasoning further assumes. So the "size" is the "quantitative" parameter while the "density" is the nicely localized "qualitative" parameter.

The air density is above 1 kg per cubic meter, water has 1,000 kg per cubic meter, iron has almost ten times as much. You may go to much more extreme materials such as those inside white dwarfs and neutrons stars. The density of a neutron star is huge. After all, it is a "big nucleus" and the nucleus is almost 100,000 times smaller than the atom. When you squeeze the nuclear (=atomic) mass to volumes whose linear dimensions are 5 almost orders of magnitude shorter than the atoms, the required volume decreases and the resultant density increases by almost 15 orders of magnitude (than e.g. water) – because $$V\sim R^3$$ – and that's indeed how dense neutron stars are.

In this sequence of material densities, the "black hole" is incorrectly understood to be the final step, bringing you to $$\rho=\infty$$, the infinite density. All the matter is concentrated to the point, the layman thinks, and the point is the "singularity". And because this infinite density is singular, the black hole is both "qualitatively different" from water as well as neutron stars (that are rather close in comparison) and also "mysterious".

But the novelty of black holes has nothing to do with the infinite density.

It is true that black holes are "qualitatively different" from all the previous "materials". But what makes them "qualitatively different" isn't the density and the relevant density of black holes isn't infinite, anyway. Instead, black holes aren't "materials" at all. They are objects in which the whole Newtonian way of thinking about celestial bodies completely breaks down. If you continue to use it, you are bound to produce gibberish.

And they aren't infinitely far from the neutron stars in the graph of radii $$R$$ and densities $$\rho$$. In fact, when a neutron star devours an extra dinner beyond a critical mass, it just turns into a black hole. The whole material of the star collapses, the homogeneity of the "material" becomes unsustainable, but the resulting object – the black hole – develops a new surface, the event horizon, which is as finite and spherical as the neutron star's surface used to be (just a bit smaller).

It is the event horizon, a spherical (in the simplest case) surface of a finite nonzero radius, that defines the black hole, not the singularity.

Is the event horizon very different from the surface of a neutron star? Yes, it is very different. The neutron star's surface is dim; but the black hole's event horizon is perfectly dark. It can't radiate at all. Also, the neutron star's surface is extremely hard, you don't want to hit it with your skull. On the other hand, the event horizon of a black hole is super soft. In fact, nothing is there at all. You may penetrate it completely smoothly to enter the black hole interior. Unlike the neutron star, the black hole (interior) may be a basically empty volume.

So which of the two surfaces, the neutron star's surface or the black hole's event horizon, is "more grave", a bigger separator between two regions? That is a great question (sadly, it is mine). The funny answer is that it depends on the "timescale" in which we observe the effect of the surface.

In the short term, the neutron star's surface is a big thing while the event horizon is non-existent. Your head hurts from the former but gets through the latter. In the long run, however, your head may recover from the neutron star collision. But it can't recover from falling beneath the event horizon.

The event horizon is what actually defines a black hole. It is the surface of no return. Once you fall beneath that surface, i.e. into the black hole interior, there is no way back. The way back is prohibited by the most universal laws of physics. In particular, it is banned by the condition that material objects never move faster than light because locally, the escape from a black hole is the same thing as a superluminal notion!

At some level, it is extraordinarily surprising and disappointing that almost no one gets this basic point. This point is by no means being hidden from the public. For example, the Wikipedia definition of a black hole starts with the sentence:
A black hole is a region of spacetime where gravity is so strong that nothing—no particles or even electromagnetic radiation such as light—can escape from it. [6]
That's right. It's a definition taken from Wald's 1984 book. A black hole is defined as a region (the black hole interior) from which nothing can escape. Whether there is also a "singularity" somewhere in the region is irrelevant for the black hole's being a black hole. General relativity allows you to prove that under some assumptions, there must be a singularity in this setup, somewhere. But a priori (and I would say even a posteriori), the singularity isn't really what matters at all (and whether it's singular at all depends on the very short-distance laws of particle physics which aren't "quite the same thing" as quantum gravity in general – quantum gravity involves characteristically quantum phenomena arising in situations with gravity that are manifested even at macroscopic distances).

OK, the event horizon is a characteristic concept that requires general relativity. How should you imagine it? In the Newtonian picture (and in the Schwarzschild-like coordinates), you are probably imagining the event horizon as the spherical surface $$R=a$$ that is stationary in time, just like the surface of a neutron star. But that is a totally wrong way to think about the event horizon if you want to describe the experiments that are done in the vicinity of that surface.

To study the local phenomena easily, you should better use very different spacetime coordinates for that vicinity of a point at the event horizon. And indeed, it's one of the repeating wisdoms of general relativity: you should choose many different coordinates and switch between them all the time. That's not something that you do in Newtonian physics where the Cartesian coordinates are really pretty much great at all times, you know. But this viewpoint assuming "one natural set of coordinates for all situations" is as wrong in general relativity as you can get.

So when the black hole is large and has characteristic dimensions $$a$$, e.g. if the event horizon is a sphere of area $$4\pi a^2$$, then the Riemann curvature tensor has components $|R_{\alpha\beta\gamma\delta}| \sim \frac{C}{a^2}.$ The curvature is inversely proportional to the squared distance scale (squared black hole radius, as measured from the area of the sphere). This scaling follows from the dimensional analysis; the Riemann tensor (which involves second derivatives of the dimensionless metric tensor) has the same units as $$1/a^2$$ and there is no other dimensionful parameter you could insert here (the "density of the singularity", even if it could be made meaningful, is infinite so the manifestly finite curvature cannot depend on it!). You may see that if the black hole radius is astrophysical in dimensions, the curvature goes to zero.

Because the curvature goes to zero, it becomes increasingly accurate to choose a local Lorentz frame, to describe the spacetime in a vicinity of the event horizon's point as a region in the Minkowski space. It simply has to be the case. So fine, there is some almost flat spacetime with the usual coordinates $$t,x,y,z$$ over there. What does the event horizon look like? The Newtonian thinking would lead you to think that at the North Pole of the event horizon, the event horizon would look like a $$z={\rm const}$$ surface in the spacetime that is just stationary in time $$t$$. But that's completely wrong.

Instead, the black hole event horizon is (in the local Lorentz coordinates) something like$z = ct.$ Cool. The spatial coordinate is proportional to time, the speed of light (in the vacuum) is the constant of proportionality. Locally, it looks like a surface in the Minkowski space that is moving by the speed of light (which is why it is a "null surface", a hypersurface whose 3D volume is neither spacelike nor timelike, it has a zero invariant volume). Maybe it's more pedagogic to describe the black hole event horizon as$R = ct.$ The surface of the black hole seems to grow by the speed of light in the local Lorentz coordinates. That's why you can't return outside once you are inside. To try to escape is equivalent to chasing a surface that is running away by the speed of light. You have no chance to win. The amazing ability of the spacetime curvature (a basic ingredient or consequence of general relativity) is that this sphere that looks like it is running away by the speed of light (which is highly time-dependent) looks time-independent in other, Schwarzschild (or similar) coordinates. Yes, the proper area of the event horizon (I mean its 2D section at one moment in this sentence) is constant in time. But the event horizon is still like something that "moves at the speed of light" in all locally pleasant coordinate systems.

The black hole may be imagined as a stationary object similar to stars if you are an observer who is outside the black hole and far enough from the black hole. But for the infalling observer, it is the interior bounded by a surface that runs at the speed of light (and there are no coordinates that allow you to describe the black hole interior as stationary in time; instead, $$R$$ and $$t$$ get interchanged so the black hole interior is translationally invariant in a spatial direction). These two pictures may look extremely incompatible within the Euclid-Newton framework but they are totally compatible with one another within the Riemann-Einstein curved framework. To understand why these two perspectives are not only compatible but "derivable from each other" within GR is one of the first tasks you need to solve once you really start to learn GR (and black holes) seriously.

Fine, once the observer falls inside, he can't get back. That is why I said that the event horizon is a "more far-reaching" surface "in the long run". There is no run that is long enough to escape from the black hole. While falling into the black hole may be painless or even undetectable by regular local apparatuses, it is one of the most irreversible decisions in your life that you can make (far more irreversible than a wedding and other acts). It matters in the long turn. The ban on the superluminal motion (and therefore the ban on the escape tricks from a black hole) is much more enforced by Nature than the ban on divorces.

Note that the irreversibility is directly contained in the geometry of the null surfaces such as $$z=ct$$. The point is that this surface is something in between two surfaces of different signature. It is somewhat close to the "spacelike" hypersurfaces such as $$t={\rm const}$$. Those surfaces clearly separate the spacetime into two parts, the past and the future. The fun is that the null surfaces such as the event horizon already do it, too. That's why you can always say whether the finite-volume black hole interior is "in the past" or "in the future". Basic thermodynamics involving spacetime implies that the black hole interior must always be "the future" (this statement is equivalent to "white holes can't exist"; the second law of thermodynamics directly restricts the allowed arrows of time in the spacetime geometry!). That's where your rest of the life may be; you can't ever turn the black hole interior into an episode of your past life because you can't escape from the interior.

We may continue to discuss the information loss paradox etc. But I must start by saying that it is completely silly and pretentious to discuss deep problems of quantum gravity (or borderline quantum gravity) before you understand basic classical general relativity such as the geometry of the event horizon etc.

The information loss puzzle deals with the fate of the information that is carried by the matter that crossed the event horizon i.e. fell into a black hole. For an external observer who wants to stay outside the event horizon at all times, the infalling matter and its information became permanently inaccessible. It is a marginally acceptable viewpoint to declare that "everything that happens inside black holes" is out of the realm of physics. Individual adventurous physicists may jump into a black hole, feel something, do an experiment or two, but they can't share their observations with a civilization that may develop science for a long time. That's why these observations are "unphysical" or "scientifically meaningless" from the viewpoint of the civilization that builds its knowledge outside all black holes.

This is just one viewpoint. Of course, an individual physicist still feels something which is why many physicists want to understand the proper theory of phenomena inside the black hole. At any rate, the mystery already begins when you cross the event horizon. You don't need to be crushed at the very center, the final singularity. The final singularity at the end of your life isn't too physically interesting, anyway. It's clear that in the Planckian vicinity of the singularity, all conceivable gadgets break down (literally). Most of them break down much earlier. That's why the phenomena near the singularity may really be declared unphysical. They just can't be precision-measured. Any "problem" arising due to some diverging numbers near the singularity is physically non-existent because no one can ever see the "divergence" and physics is an industry producing predictions of measurements.

But if you assume that some Newtonian notions are OK everywhere away from the singularity, even in the black hole interior, I assure you that your theory will imply that the information is really lost and we know that it is wrong. To solve the information loss puzzle, you must surely understand that some sort of nonlocal behavior occurs already near the event horizon. Because that surface is macroscopic in size, it is clear that the nonlocality has to act on macroscopic (and not just Planckian, ultramicroscopic) distance scales.

In recent decades, we have figured out that the information is certainly preserved. "Why the nonlocal communication" is allowed, to what extent, and how to describe it quantitatively, remains a bit misunderstood question largely because our descriptions that give us clear answers to the biggest question (the information is preserved, indeed) are not manifestly local in the bulk (like the CFT in AdS/CFT is obscuring the bulk locality). Because these stringy descriptions of ours don't seem "manifestly local" to start with, we can't easily quantify "how much nonlocality" they imply and where the nonlocality comes from.

But pretty much everything that used to look "singular" about black holes has completely disappeared in our modern understanding of black holes. The singularity at the event horizon is just a coordinate singularity, an artifact of coordinates. The singularity at the black hole center may be "more real" but you can't really measure its properties by apparatuses that survive for a suffficient amount of time. The black hole itself doesn't qualitatively differ from other "kinds of matter"; a black hole microstate may be understood as a species of elementary particles that are much heavier than the Planck mass. There is a quasi-continuous (gradual yet quantized) transition between the light elementary particle species and macroscopic black holes.

The main message is "Please don't send me would-be deep e-mails about black holes if you still believe that the singularity is what is important in a black hole." Everyone who believes in this misconception is a 100% layman who hasn't started to understand general relativity (let alone quantum gravity) at all.

And that's the memo.