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Sources – Can you destroy a black hole?


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This video covers in a narrative format some of the most important results about black holes physics in the framework of classical general relativity (properties of event horizons, black hole uniqueness and “no-hair” theorems, non-conservation of baryon and lepton numbers, the properties of rotating and charged black holes, the Hawking-Penrose singularity theorems, the cosmic censorship conjecture and Hawking radiation). General relativity has a complicated mathematical structure, so in order to make these results accessible to everyone several simplifications have been made. This document provides context and material for further reading and explains many of the simplifications done in the video.



–Let’s create a tiny black hole, about the mass of our Moon, in the Kurzgesagt Labs and try to rip it apart. 


The size (radius) of a black hole is directly proportional to its mass. For a non-rotating black hole with no electric charge (a.k.a. “Schwarzschild black hole”), the relation is given by the formula:


R = 2GM/c2

Where R is the radius of the black hole, M is its mass, G is the gravitational Newton’s constant and c is the speed of light:

#Encyclopaedia Britannica: “Schwarzschild radius” (retrieved 2023)
https://www.britannica.com/science/Schwarzschild-radius
 Quote: “The Schwarzschild radius (Rg) of an object of mass M is given by the following formula, in which G is the universal gravitational constant and c is the speed of light: Rg = 2GM/c2”


Note that, in SI units, Newton’s constant is of the order of 10–11 and the inverse speed of light squared is of the order of 10–16. This implies that the factor G/c2 is extremely small, of the order of 10–27, meaning that, in SI units, the mass of a black hole will always be huge relative to its radius. For example, to get a black hole with a radius of the order of 1 km (103 m), we would need a mass of the order of 1030 kg, which is the order of magnitude of the mass of the Sun.


What would be the size of a black hole with the mass of the Moon? The Moon has a mass of 7.3·1022 kg:

#NASA (2021): “Moon Fact Sheet” (retrieved 2023)
https://nssdc.gsfc.nasa.gov/planetary/factsheet/moonfact.html 


So using the above formula we get that a black hole of that mass would have a radius of about 0.1 mm. This would be similar in the size to a speck of dust, barely visible with the naked eye.



–Black holes swallow whatever crosses their event horizon, matter and energy. And since E = mc2, all the energy that enters a black hole increases its mass. 


The defining characteristic of a black hole is the existence of an event horizon: a surface beyond which nothing, not even light, can escape:


#Max Planck Institute for Gravitational Physics: “Event horizon”, Einstein Online Dictionary (retrieved 2023)
https://www.einstein-online.info/en/explandict/event-horizon/ 

Quote: “Event horizon. In general relativity: A closed surface that is the boundary of a black hole. Whatever enters through this boundary from the outside can never again leave the inside.”


This means that neither matter nor energy can escape. Energy is always carried by some kind of particle, be it atoms or photons. But since not even light can escape, this means that absolutely nothing can leave the inside of a black hole. And via the well known relativistic equivalence between matter and energy (E = mc2), all energy that enters a BH will end up increasing its mass.



–The mass of a black hole is proportional to its size, so as we nuke our tiny black hole, it just gets bigger and more massive!


As explained above, the mass of a BH is proportional to its radius. We can now compute the size increase after throwing the world’s nuclear arsenal at our speck-sized BH.

The exact destructive power of the world’s nuclear arsenal is not known, but it has been estimated in about 3 billion tonnes of TNT. We explained this estimation in the source document of our video “What If We Detonated All Nuclear Bombs at Once?”

#Kurzgesagt (2019): “What If We Detonated All Nuclear Bombs at Once?”, Source Document.
https://sites.google.com/view/sourcesallthebombs/


One ton of TNT is a unit of measure equivalent to 4.184·109 joules. This means that 3 billion tonnes of TNT are


3·109 × 4.184·109 = 12.6·1018 joules

Via E = mc2, this corresponds to a mass of just 140 kg. Compared to the initial mass of our BH (7.3·1022 kg), this would mean a negligible increase of just 0.0000000000000000001 % (and the same increase in radius).


Of course, the energy of a nuclear bomb also comes from E = mc2, so we would basically get the same result if we throw the bombs undetonated, or if we threw the same amount of mass of any material, be it sugar or stones. Still, the computation is illustrative of how weird black holes are: The whole effect of a world’s nuclear arsenal launched onto an object not bigger than a speck of dust is to make it only slightly bigger.


–Matter and antimatter annihilate each other. What will happen if we throw a moon's mass of antimatter at it?


When a particle and the corresponding antiparticle collide, they vanish and all their mass is converted into energy (typically, in the form of high-energy photons) according once again to the formula E = mc2: 


#Feynman, Richard (1963): “Symmetry in Physical Laws – Antimatter”; Feynman Lectures on Physics, vol. I, chapter 52.
https://www.feynmanlectures.caltech.edu/I_52.html

Quote: “All the properties of these two particles [an electron and a positron] obey certain rules [...] but, more important than anything, the two of them, when they come together, can annihilate each other and liberate their entire mass in the form of energy, say γ-rays. [...] The most important feature, however, is that a proton and an antiproton coming together can annihilate each other. The reason we emphasize this is that people do not understand it when we say there is a neutron and also an antineutron, because they say, “A neutron is neutral, so how can it have the opposite charge?” The rule of the “anti” is not just that it has the opposite charge, it has a certain set of properties, the whole lot of which are opposite. The antineutron is distinguished from the neutron in this way: if we bring two neutrons together, they just stay as two neutrons, but if we bring a neutron and an antineutron together, they annihilate each other with a great explosion of energy being liberated, with various π-mesons, γ-rays, and whatnot”


Therefore, if we have a black hole made of matter, we may think that throwing an equal amount of antimatter would end up in the annihilation of the black hole.



–Unfortunately, when an object enters a black hole, the black hole will completely delete its past identity – including if it was made of matter or of antimatter.

All matter (and antimatter) in the universe comes in two types: baryons and leptons. Baryons are particles made of quarks, and leptons are particles like the electron or the neutrino. Each of these particles has a corresponding “baryon number” or “lepton number”: for particles (like the proton, the neutron or the electron) that number is +1, and for antiparticles (like the antiproton, the antineutron or the positron) that number is –1:

#Encyclopaedia Britannica: “Baryon” (retrieved 2023)
https://www.britannica.com/science/baryon
 Quote: “baryon, any member of one of two classes of hadrons (particles built from quarks and thus experiencing the strong nuclear force). Baryons are heavy subatomic particles that are made up of three quarks. Both protons and neutrons, as well as other particles, are baryons. (The other class of hadronic particle is built from a quark and an antiquark and is called a meson.) Baryons are characterized by a baryon number, B, of 1. Their antiparticles, called antibaryons, have a baryon number of −1. An atom containing, for example, one proton and one neutron (each with a baryon number of 1) has a baryon number of 2. In addition to their differences in composition, baryons and mesons can be distinguished from one another by spin: the three quarks that make up a baryon can only produce half-integer values, while meson spins always add up to integer values.”


An old result about black holes is that they have no baryon number at all:

#Bekenstein, J. D. (1972): “Nonexistence of Baryon Number for Static Black Holes”. Physical Review D, vol. 5, 1239.
https://journals.aps.org/prd/abstract/10.1103/PhysRevD.5.1239
 Quote: “Wheeler has conjectured that black holes should have no well-defined baryon number, and that as a result the law of conservation of baryons should be transcended in black-hole physics. We show here that a static black hole cannot have any exterior classical scalar or massive vector fields. We consider the modifications that would arise from a quantum-theoretical treatment, and we conclude that such a black hole cannot interact with the exterior world via virtual mesons such as the π and ρ. Because of this we find no way for external measurements to assign unambiguously a baryon number to such a black hole in agreement with Wheeler's prediction.”


This means that, once a black hole has swallowed a particle, the black hole will “delete” (or “forget”) if that particle was a baryon or an antibaryon. This has to be so because, if the black hole had any way to “remember” any information about the baryon number, we could use that information to assign a baryon number to the black hole itself. The same thing happens to the lepton number. What this means in practice is that black holes don’t distinguish between matter and antimatter, and that black holes themselves cannot be assigned any of these two categories – they are simply “black holes”, neither matter nor antimatter.


–Black holes just care about gravity, which only depends on the total mass-energy of an object. And the mass of a particle is the same as its corresponding antiparticle’s, so throwing an anti-moon has the same effect as throwing a moon. The black hole just gets more massive!


Particles and antiparticles have exactly the same mass:

#Encyclopaedia Britannica: “Antiparticle” (retrieved 2023)
https://www.britannica.com/science/antiparticle
 Quote: “antiparticle, subatomic particle having the same mass as one of the particles of ordinary matter but opposite electric charge and magnetic moment. Thus, the positron (positively charged electron) is the antiparticle of the negatively charged electron. The spinning antineutron, like the ordinary neutron, has a net electric charge of zero, but its magnetic polarity is opposite to that of a similarly spinning neutron.”


And since black holes “delete” their baryon or lepton number, the effect of throwing a particle (or a moon) is the same as that of throwing an antiparticle (or an anti-moon): increasing the mass of the black hole.


–This “deleting ability” of black holes is pretty interesting: It means that, despite their size and power, black holes are, in a way, similar to elementary particles. An elementary particle, like an electron, is an extremely simple object, fully specified by just three numbers: its mass, spin and charge. And amazingly, the same is true for black holes. They have a mass, they can rotate and carry an electric charge. Once a black hole forms, it doesn't matter if it came from a collapsed star, an anti-star or a banana: it will always be fully described by those three numbers, nothing else.


Black holes delete all properties of the objects they swallow except three: their mass, electric charge and angular momentum. This is because black holes themselves don’t have any other properties. Once these properties are known, the black hole is completely described. So just as two electrons are completely identical in all respects, two black holes with the same mass, charge and angular momentum are completely identical as well. Black holes are uniquely determined by these numbers:

#Max Planck Institute for Gravitational Physics: “Black hole uniqueness theorems”, Einstein Online Dictionary (retrieved 2023)
https://www.einstein-online.info/en/explandict/black-hole-uniqueness-theorems/ 

Quote: “Black hole uniqueness theorems. Theorems proved in the context of general relativity that answer the question: How many different kinds of black holes are there? If that question is restricted to stationary black holes (namely black holes that have settled down and do not change over time), then the answer is: Surprisingly few. Once you know a stationary black hole’s mass, angular momentum (roughly speaking, how fast it rotates) and electric charge, its properties are determined completely.”
 

This result is known as “no-hair theorem”, often paraphrased as “black holes have no hair” – the “hair” being all other possible features (shape, baryon number, whatever) that in principle could characterized a black hole but don’t.


#Max Planck Institute for Gravitational Physics: “How many different kinds of black holes are there?”, Einstein Online (retrieved 2023)
https://www.einstein-online.info/en/spotlight/bh_uniqueness/ 

Quote: “But as it turns out, in the long run, black holes are much more simple. They can be completely described in terms of a few parameters: the total energy, angular momentum (roughly, “how fast the black hole rotates”) and (of less interest to astrophysics), electric charge of the black hole. Once these parameters are chosen, the model universe – the way that spacetime geometry is distorted around the black hole – is determined completely. Sphere, cube or cigar – they all collapse to form a very simple kind of black hole that retains no traces of the original shape. This striking property of black holes has been popularized as “the no-hair theorem” by John Wheeler (with the term itself invented by an anonymous member of the audience of one of Wheeler’s lectures). The name alludes to the fact that only very few parameters are needed to describe those solutions – apart from the values of those parameters, black holes have no distinguishing characteristics (no “hair styles”).”



–A particle has the same mass as its corresponding antiparticle but opposite charges. 


As already mentioned above:


#Encyclopaedia Britannica: “Antiparticle” (retrieved 2023)
https://www.britannica.com/science/antiparticle
 Quote: “antiparticle, subatomic particle having the same mass as one of the particles of ordinary matter but opposite electric charge and magnetic moment. Thus, the positron (positively charged electron) is the antiparticle of the negatively charged electron. The spinning antineutron, like the ordinary neutron, has a net electric charge of zero, but its magnetic polarity is opposite to that of a similarly spinning neutron.”



–Since a black hole has mass and electric charge, its corresponding anti black hole should have the same mass and opposite electric charge. What if we make them collide? Sadly, the charge will just add up and cancel out. So after the collision, we’ll just get a new black hole twice as massive with no charge.


Even if black holes delete almost everything, there is a reason why they cannot delete mass, electric charge or angular momentum. This is because these features are “conserved quantities” in physics: regardless of what kind of physical process takes place, the total mass, total charge and total angular momentum of a system can never change: 


#Max Planck Institute for Gravitational Physics: “How many different kinds of black holes are there?”, Einstein Online (retrieved 2023)
https://www.einstein-online.info/en/spotlight/bh_uniqueness/ 

Quote: “So how do black holes become simple? How does a cigar-shaped object that collapses to form a black hole lose all its cigar-like properties? In general relativity, there is a natural mechanism for simplifying the complicated geometric structure of certain regions of space: The emission of gravitational waves can carry away complicated features, with the waves either escaping into deep space or being swallowed by the black hole. While a wide variety of features can be “radiated away” in this manner, there are exceptions. For instance, electric charge cannot be radiated away at all since neither the gravitational nor electromagnetic radiation carries such charge. More subtle restrictions apply to how much of the total mass or total angular momentum can be carried off (quantities for which there are so-called conservation laws).”


An antiparticle has the same mass of its corresponding particle but opposite charge. If we have two black holes of identical masses M and electric charges given by Q and –Q, the total mass of the whole system will be 2M and the total charge will be 0. Since these are conserved quantities, this means that after both black holes collide, we’ll get an object of mass 2M and zero electric charge. 


We could think that maybe that object doesn’t have to be a black hole. However, a black hole is the only possible result of a collision between black holes. The defining property of a black hole is the existence of an event horizon, which in turn is characterized by its total surface area. An old theorem in general relativity asserts that, in any process involving black holes, the total area of all event horizons can never decrease. If the area of the event horizon cannot decrease, then our final object will necessarily have an event horizon, meaning that it will necessarily be a black hole:

#Bardeen, J. M. (1973): “The four laws of black hole mechanics”. Communications in Mathematical Physics, vol. 31, 161.

https://link.springer.com/article/10.1007/BF01645742
 https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-31/issue-2/The-four-laws-of-black-hole-mechanics/cmp/1103858973.full (open-access PDF)
Quote: “The area A of the event horizon of each black hole does not decrease with time, i.e. 


δA>0.

If two black holes coalesce, the area of the final event horizon is greater than the sum of the areas of the initial horizons, i.e. 


A3 > A1 + A2 .” 


This “area theorem” has been recently confirmed experimentally by the detection of gravitational waves from black hole collisions:


#Chu, J. (2021): “Physicists observationally confirm Hawking’s black hole theorem for the first time”. MIT News

https://news.mit.edu/2021/hawkings-black-hole-theorem-confirm-0701 

Quote: “There are certain rules that even the most extreme objects in the universe must obey. A central law for black holes predicts that the area of their event horizons — the boundary beyond which nothing can ever escape — should never shrink. This law is Hawking’s area theorem, named after physicist Stephen Hawking, who derived the theorem in 1971. Fifty years later, physicists at MIT and elsewhere have now confirmed Hawking’s area theorem for the first time, using observations of gravitational waves. Their results appear today in Physical Review Letters. In the study, the researchers take a closer look at GW150914, the first gravitational wave signal detected by the Laser Interferometer Gravitational-wave Observatory (LIGO), in 2015. The signal was a product of two inspiraling black holes that generated a new black hole, along with a huge amount of energy that rippled across space-time as gravitational waves. If Hawking’s area theorem holds, then the horizon area of the new black hole should not be smaller than the total horizon area of its parent black holes. In the new study, the physicists reanalyzed the signal from GW150914 before and after the cosmic collision and found that indeed, the total event horizon area did not decrease after the merger — a result that they report with 95 percent confidence.”



–It is true that a black hole can carry spin and charge. But even for these crazy objects, there are limits. If the spin or the charge of a black hole becomes too large, something really weird will happen: the event horizon will dissolve.


As explained above, a general black hole is fully characterized by its mass M, its angular momentum J and its electric charge Q. However, the equations of general relativity imply that, in order to have an actual black hole (i.e. an event horizon), these parameters cannot take arbitrary values. In particular, the mass has to be always larger than a certain combination of the charge and the angular momentum. If this constraint does not hold, the resulting spacetime will describe a “naked singularity” (this term will be explained in detail below) with no event horizon. That is, the spacetime won’t be a black hole.

 In particular, if the total angular momentum J or the total charge Q of a black hole are such that the combination 


√ [(J/M)2 + Q2] > M ,


the event horizon will disappear:


#Jacobson, T. et al. (2010): “Might black holes reveal their inner secrets?”. FQXi essay competition "What is Ultimately Possible in Physics"

https://forums.fqxi.org/d/594-might-black-holes-reveal-their-inner-secrets-by-ted-jacobson-and-thomas-sotiriou 

Quote: “In general relativity black holes are fully characterized by three quantities: their mass M, spin angular momentum J, and electric charge Q. The spacetime in the vicinity of the black hole is described by the Kerr–Newman (K-N) metric — the measure of distances and times — which contains the three parameters M, J and Q. The K-N metric describes a black hole as long as the mass is sufficiently large compared to a combination of the charge and angular momentum, M2 ≥ a2+Q2 , where a = J/M. (We adopt units with Newton’s constant G and the speed of light c both set equal to unity. Displaying hidden factors of G and c, the quantities MG/c2 , a/c and Q√G/c2 all have the dimension of length.) The case where M2 = a2 + Q2 is called an extremal black hole, while for M2 < a2 + Q2 there is no event horizon and the K-N metric actually describes a naked singularity.”


For the special case of a black hole with no charge (Q = 0), the expression above means that, in order to have a proper black hole with event horizon, the angular momentum per unit mass (J/M) cannot be larger than the mass:


J/M ≤ M 


And in the special case of a black hole with no angular momentum (J = 0), the expression above means that, in order to have a proper black hole, the charge cannot be larger than the mass


Q ≤ M 


(As explained in the quote above, physical units can always be chosen so that J, Q and M can be directly compared. All expressions above use such units.)


But this also means that if we are able to violate any of these bounds, i.e. if somehow we manage to get a total angular momentum or a total charge such that


J/M > M


or


Q > M


then the event horizon will disappear.



–In a simplified way, we think of black holes as hiding a singularity inside – an infinitely compressed mass with such a strong gravity that absolutely nothing can escape from its surroundings, not even light. This is why a black hole looks like a “black sphere of nothingness”. The event horizon is the outer edge of this ultimate prison. Cross it, and you’ll never be able to come back.


All black holes hide a “singularity” inside the event horizon: a region of infinitesimal size where spacetime itself “breaks”, or ceases to exist:

#Max Planck Institute for Gravitational Physics: “Singularity”, Einstein Online Dictionary (retrieved 2023)
https://www.einstein-online.info/en/explandict/singularity/ 

Quote: “Singularity. Irregular boundary of spacetime in general relativity – region where spacetime simply comes to an end. Often, such boundaries are associated with spacetime curvature growing beyond all bounds and becoming infinitely large – so-called curvature singularities (notably Ricci singularities or Weyl singularities – but there are exceptions (for instance a conic singularities). According to general relativity, a singularity exists inside every black hole”


In a black hole, the event horizon is the boundary of the region of space surrounding the singularity beyond which not even light can escape:


#Max Planck Institute for Gravitational Physics: “Event horizon”, Einstein Online Dictionary (retrieved 2023)
https://www.einstein-online.info/en/explandict/event-horizon/ 

Quote: “Event horizon. In general relativity: A closed surface that is the boundary of a black hole. Whatever enters through this boundary from the outside can never again leave the inside.”



–But when a black hole rotates, it works a bit like a spinning washing machine. It is as if the rotation wants to repel nearby objects and push them out of the black hole – which doesn’t happen because its gravity is so strong. BUT – If the rotation gets too fast, this effect will win and the event horizon will disappear – nearby objects won’t be imprisoned forever anymore! The same thing happens with the electric charge. Make it too large, and our ironclad jail will break open.


As explained above, the event horizon marks the boundary of the region of “no escape” around the singularity. But if the angular momentum or the charge become large enough, we’ve also seen that the event horizon will disappear. And if the event horizon disappears, this means that the region of no escape will vanish, too. Objects near the singularity will be able to escape with (say) a powerful enough rocket.


In the case of a rotating black hole, we can understand this effect by using a loose analogy from classical physics: it is as if the centrifugal force associated with the rotation (the same force that pushes the clothes outwards in a spinning washing machine) would “help” objects to stay away from the singularity. And a similar effect happens with the electric charge. 



–If we managed to destroy the event horizon, the singularity would still be there. And objects would still naturally fall towards it. If you hit it, you would still die horribly and quickly. But the vicinity of the singularity won’t be an inescapable prison anymore! You could get as close as you want and come back.

It is important to notice that this does not mean that objects are repelled from the black hole: the gravitational field is still there, and objects will still tend to fall towards the singularity. But if the rotation or the electric charge are large enough, this “help” to stay away from the singularity will make the event horizon vanish and objects could escape with a powerful enough thrust (something impossible if an event horizon is present).



–All we have to do is to overcharge or over-spin the black hole. We could do this by throwing objects with a small mass and a lot of charge or angular momentum, so that the charge or spin increases faster than the mass. We have to overfeed the black hole until it reaches the point where it breaks open.


As explained above, if we have a rotating black hole with angular momentum J and somehow manage to increase its angular momentum to the point that


J/M > M ,


the event horizon will disappear. And the same thing will happen if we have a charged black hole with charge Q and somehow manage to make Q such that


Q > M .


In principle, we could achieve this by throwing objects with a small mass and a large angular momentum or a large charge. Imagine that we have a charged black hole with Q < M and that we throw a particle with q and mass m. Since both the mass and the charge are conserved quantities, after the particle has been swallowed by the black hole, the total charge and mass of the black hole will be:

New charge of the BH: Q + q

New mass of the BH: M + m


But if our particle has a lot of charge and a little mass, i.e. if it obeys q > m, then the charge will increase faster than the mass. So if we keep on repeating the process, throwing more and more particles with q > m) sooner or later we should reach the point where


Final charge of the BH > Final mass of the BH


and the event horizon will disappear. 


To be sure, there are many objects in nature whose charge q or angular momentum j is larger than its mass m (i.e. that satisfy the conditions q > m or j/m > m), from electrons to ordinary stars. This implies no contradiction with the bounds J/M ≤ M or Q ≤ M mentioned above, since the latter only apply to black holes, and neither electrons nor ordinary stars are black holes.

#Jacobson, T. et al. (2010): “Might black holes reveal their inner secrets?”. FQXi essay competition "What is Ultimately Possible in Physics"

https://forums.fqxi.org/d/594-might-black-holes-reveal-their-inner-secrets-by-ted-jacobson-and-thomas-sotiriou 

Quote: “We can already extract a useful piece of information from this last equation. Dividing both side by δE2 and observing that each term on the right hand side should by itself be smaller that the left hand side, we get 


δJ/δE2 > 2M/δE >> 1 


If δE were equal to the rest mass of the body (it can be much less if the body is deeply bound by the gravitational field of the black hole), and if δJ comes from spin (rather than orbital angular momentum), this would imply that the body has angular momentum far over the extremal ratio. In that case the body could not be a black hole. This does not mean that it would have to be a naked singularity itself, as there is no a priori upper limit to this ratio for bodies other than black holes. Stars for instance can easily have ratios much larger than 1.”


–However, whether you can actually do this has been a passionate argument among physicists. Think of a charged black hole. Equal charges repel each other, and the more of the same charges you squish together, the more they push back. So let’s say that we have a negatively charged black hole and we want to overfeed it with electrons, for example, whose charge is far larger than its mass. The electrons will feel an electrostatic repulsion. And the more electrons we throw, the larger the negative charge of the black hole will be and the stronger the repulsion. But once we reach the upper limit, the electrostatic repulsion will be so strong that it won't allow any more electrons to come in. At this point, the black hole will refuse to be overfed. With the spin it works similarly. Once the black hole reaches its upper limit, it won’t gobble more spin.


Long ago, physicists calculated what would happen if we were to do the kind of experiment explained above to overcharge or overspin a black hole. Experiments with actual black holes in the lab are not possible, but one can calculate what would happen in the framework of Einstein’s general relativity. Those calculations are complicated and one has to make several simplifying assumptions, but the first calculations of this kind showed that it is impossible to overcharge or overspin a black hole: once the black hole reaches the limit, the repulsive electrical force or the centrifugal force will prevent the black hole to gobble more particles:

#Wald, R. (1974): “Gedanken experiments to destroy a black hole”. Annals of Physics, vol. 82, 2

https://www.sciencedirect.com/science/article/abs/pii/0003491674901250
 Quote: “It is widely believed that the complete gravitational collapse of a body always results in a black hole (i.e., “naked singularities” can never be produced) and that all black holes eventually “settled down” to Kerr-Newman solutions. An important feature of the Kerr-Newman black holes is that they satisfy relation m2 ⩾ a2 + e2 where m is the mass of the black hole, e is its charge, a = Jm is its angular momentum per unit mass and geometrized units G = c = 1 are used. (For m2 < a2 + e2 the Kerr-Newman solutions describe naked singularities.) In this paper, we test the validity of the above conjectures on gravitational collapse by attempting to create a spacetime with m2 < a2 + e2 starting with a Kerr-Newman black hole with m2 = a2 + e2. Such a spacetime would either have to be a new black hole solution or a “naked singularity,” in violation of the above conjectures. In the first gedanken experiment we attempt to make the black hole capture a test particle having large charge and orbital angular momentum compared with energy. In the second gedanken experiment we attempt to drop into the black hole a spinning test body having large spin to mass ratio. In both cases we find that bodies which would cause violation of m2 ⩾ a2 + e2 will not be captured by the black hole, and, thus, we cannot obtain m2 <a2 + e2, although we can come arbitrarily close in the sense that m2 = a2 + e2 can be maintained in these processes.”



–That’s what the standard calculations say. But some scientists have discovered what looks like a loophole. If an instant before the black hole reaches the limit, we throw the right amount matter in just the right way, it looks like we could actually overfeed it. 


Years later, however, more complex calculations showed that it actually might be possible to overcharge a black hole if one throws the charged particles in a certain way. In particular, if the right amount of charge is thrown in the right way before the black hole reaches the limit:


#Hubeny, V. E. (1999): “Overcharging a black hole and cosmic censorship”. Physical Review D, vol. 59, 064013.

https://journals.aps.org/prd/abstract/10.1103/PhysRevD.59.064013 

https://arxiv.org/abs/gr-qc/9808043 (open-access version)

Quote: “The simplest way to examine the possibility of supersaturating the extremal bound of a black hole is to start at extremality and consider strictly test processes. Unfortunately, such processes cannot lead to destroying the horizon, since the particles which are allowed to fall in would at best maintain extremality ([5],[6],[9],[7]). Faced with these results, one is led to start with a near extremal black hole. One might then hope that such a set-up allows the black hole to “jump over” extremality, by capturing a particle with appropriate parameters. Perhaps somewhat surprisingly, we found that according to the test particle approximation, this can be indeed achieved. Namely, a broad range of configurations can be found in which, according to the test equation of motion, the particle falls into the singularity and the resulting object exceeds the extremal bound. This result, which seems to have been previously overlooked, is important in the context of these earlier attempts to violate cosmic censorship.”


A similar result was found later for rotating black holes:


#Jacobson, T. et al. (2009): “Overspinning a Black Hole with a Test Body”. Physical Review Letters, vol. 103, 141101

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.103.141101 

https://arxiv.org/abs/0907.4146 (open-access version)

Quote: “It has long been known that a maximally spinning black hole cannot be overspun by tossing in a test body. Here we show that if instead the black hole starts out with below maximal spin, then indeed overspinning can be achieved. We find that requirements on the size and internal structure of the test body can be met if the body carries in orbital but not spin angular momentum. Our analysis neglects radiative and self-force effects, which may prevent the overspinning.”



–Most scientists are skeptical, but let's give it a try anyway!


However –as it was acknowledged by the authors themselves– the calculations quoted above neglected the “backreaction”; that is, the possible changes in spacetime caused by the infalling particle. Such effects are generally thought to be negligible when the particle’s mass, charge and spin are much smaller than those of the black hole. However, a full computation should take them into account. And indeed, some more precise calculations seem to disprove the possibility of overcharging or overspinning a black hole:

#Sorce, J. (2017): “Gedanken experiments to destroy a black hole. II. Kerr-Newman black holes cannot be overcharged or overspun”. Physical Review D, vol. 96, 104014.

https://journals.aps.org/prd/abstract/10.1103/PhysRevD.96.104014 

https://arxiv.org/abs/1707.05862 (open-access version)

Quote: “We consider gedanken experiments to destroy an extremal or nearly extremal Kerr-Newman black hole by causing it to absorb matter with sufficient charge and/or angular momentum as compared with energy that it cannot remain a black hole. It was previously shown by one of us that such gedanken experiments cannot succeed for test particle matter entering an extremal Kerr-Newman black hole. We generalize this result here to arbitrary matter entering an extremal Kerr-Newman black hole, provided only that the nonelectromagnetic contribution to the stress-energy tensor of the matter satisfies the null energy condition. We then analyze the gedanken experiments proposed by Hubeny and others to overcharge and/or overspin an initially slightly nonextremal Kerr-Newman black hole. Analysis of such gedanken experiments requires that we calculate all effects on the final mass of the black hole that are second-order in the charge and angular momentum carried into the black hole, including all self-force effects. We obtain a general formula for the full second order correction to mass, δ2M, which allows us to prove that no gedanken experiments of the generalized Hubeny type can ever succeed in overcharging and/or overspinning a Kerr-Newman black hole, provided only that the nonelectromagnetic stress-energy tensor satisfies the null energy condition.”



–The event horizon of a black hole hides the singularity. So destroying the horizon would leave us with a “naked singularity”, one that is not hidden by an event horizon. And this poses a problem: it could mean the end of physics as we know it.


For reasons that will be explained in more detail below, it is thought that naked singularities cannot form in nature under any physically reasonable conditions. This is called the “cosmic censorship conjecture” (because, if true, it would mean that nature would “censor” the existence of “naked singularities”) and it is based on the severe pathologies that naked singularities would induce in the properties of space and time:


#Max Planck Institute for Gravitational Physics: “Cosmic censorship”, Einstein Online Dictionary (retrieved 2023)
https://www.einstein-online.info/en/explandict/cosmic-censorship/  

Quote: “Cosmic censorship. Possibly the most disturbing feature of Einstein’s general theory of relativity is the existence of singularities – most commonly, regions of spacetime in which density and curvature go to infinity. [...] The hypothesis of cosmic censorship states that, whenever a body collapses so completely as to result in the formation of a singularity, a black hole will be formed so that the singularity will be hidden behind the horizon, and thus completely unobservable for anyone outside the black hole.”



–There is a big dirty secret about black holes. Contrary to widespread belief, the singularity of a black hole is not really “at its center”. No. It is in the future of whatever crosses the horizon. Black holes warp the universe so drastically that, at the event horizon, space and time switch their roles. Once you cross it, falling towards the center means going towards the future. That’s why you cannot escape: Stopping your fall and turning back would be just as impossible as stopping time and traveling to the past. So the singularity is actually in your future, not “in front of you”. And just like you can’t see your own future, you won’t see the singularity until you hit it. But you also can’t hit something that is in your future, only sort of… experience it, like you will experience your next birthday when it happens. 


It is a widespread misconception that the singularity of a black hole is “at its center”, like if it was a point in space. Actually, in the simplest case of a neutral non-rotating black hole, the singularity is a moment in time, not a place in space. This is because, once an infalling object crosses the event horizon, space and time switch their roles, so the would-be “center” of the black hole is actually in the future of an infalling observer:

#Max Planck Institute for Gravitational Physics: “Changing places – space and time inside a black hole”, Einstein Online (retrieved 2023)
https://www.einstein-online.info/en/spotlight/changing_places/  

Quote: “Our analogy is useful in understanding one feature of a black hole singularity that is easy to get wrong. If you hear about a spherically symmetric black hole, bounded by its horizon and containing a central singularity, you are likely to picture a cross-section of the black hole which looks like this:

Here, the circle is meant to represent the horizon, and in the center of the black hole, there is a point – the singularity.


In our three-dimensional model, this picture can be obtained by looking at a plane that is orthogonal to the axis: The intersection of the plane with the horizon-cylinder is a circle, the intersection with the singularity-axis is a point. So is this a snapshot of a black hole, showing its interior structure?


Not quite. Only outside the cylinder does the intersection with a plane at constant height (“at constant time” as seen from the outside) correspond to a snapshot. Inside the cylinder, time and space have switched places. Inside, the intersection image doesn’t show a snapshot – it shows something much more weird: a caleidoscopic combination of many different times. After all, inside, time is not the axial, but the radial coordinate, and all the different distances from the “center” which you see in the sketch correspond to different moments in time. Instead of the spatial structure of the black hole, the sketch shows a strange mix of space and time!


Likewise, if you think about the unstoppable collapse of a body to form a black hole, you might think that the body ends up with all its matter concentrated in a single point of space – the singularity. But again, this picture of the spacepoint-singularity residing in the center of the black hole is simply wrong. Using our analogy, you can see why. The singularity is the whole of the axis – and the axis represents a space direction. Hence, the singularity is not a point in space – it is infinitely extended!”


The clearest way to see this is with the help of a “Penrose diagram”, an abstract representation of the spacetime around a black hole widely used by physicists. The details are complicated, but the only thing we need to know is that, in these diagrams, time is always the vertical axis, space (the radial coordinate) is always the horizontal axis, and light rays always travel at 45 degrees. For a neutral and not rotating black hole, the Penrose diagram looks like this:

#Hamilton, A.: “Penrose diagrams”. JILA – University of Colorado Boulder & NIST
https://jila.colorado.edu/~ajsh/insidebh/penrose.html 

Quote: “A Penrose diagram is a kind of spacetime diagram arranged to make clear the complete causal structure of any given geometry. They are an indispensable map for navigating inside a black hole. Roger Penrose, who invented this kind of diagram in the early 1950s, himself calls them conformal diagrams.

In a Penrose diagram:

The event horizon of the black hole is the diagonal line labeled “horizon”. Below that line we find the outside universe, and above that line we find the inside of the black hole. Once inside the black hole, it’s trivial to see why not even light can escape – because it moves at 45º. Any other objects move slower than light and therefore with an angle of more than 45º with the horizontal, so they also cannot exit the horizon and will always end in the singularity. The singularity is the extended wiggly line at the top. Since time runs along the vertical direction, such a horizontal line marks a moment in time: the moment where time itself ends. Time ends there because the diagram ends there: there’s nothing above the singularity. Also apparent in the diagram is the fact that, to escape the singularity, an infalling observer would have to stop time and travel to the past.

A last caveat: The Penrose diagram and the statements above about the properties of the singularity are only exact for the case of a black hole with no angular momentum and no electric charge. For rotating or charged black holes, the situation is more complicated and their corresponding Penrose diagrams are different. However, it is still true that the singularity lies in the future region of an infalling observer.



–Singularities that are in the future are not a problem because we can’t see them or interact with them. But a naked singularity would be in front of us, for all of us to see. What would we see? Well, the whole point is that it’s impossible to know. A singularity is a region of infinite gravity, and gravity is the bending of spacetime. At a singularity, the bending is so radical that the fabric of spacetime is literally broken. Space and time don’t exist anymore. This means that you cannot predict anything, since predicting means making a forecast about where and when something will happen. But “where” and “when” have lost their meaning!


According to the theory of general relativity, space, time and the theory of general relativity itself all break down at a singularity, making predictability impossible. Physicists think that this is an indication that Einstein theory stops being valid at very small distances and that the situation will be resolved once we have a quantum theory of gravity. However, until today no one has found such a theory.

#Max Planck Institute for Gravitational Physics: “Singularity”, Einstein Online Dictionary (retrieved 2023)
https://www.einstein-online.info/en/explandict/singularity/ 

Quote: “Irregular boundary of spacetime in general relativity – region where spacetime simply comes to an end. Often, such boundaries are associated with spacetime curvature growing beyond all bounds and becoming infinitely large – so-called curvature singularities (notably Ricci singularities or Weyl singularities – but there are exceptions (for instance a conic singularities). According to general relativity, a singularity exists inside every black hole, and the starting point of any universe described by a big bang model is a singularity as well. The occurrence of singularities is a failure of general relativity – and a strong indication that the theory is incomplete. Instead, one could describe the earliest universe and the interior of black holes using a theory of quantum gravity.


–So we have an unpredictable thing with infinite gravity, and therefore infinite energy. This means that anything could come out of it for no reason – from a pile of bananas, to lost socks or a solar system. Predictability, causality and physics as we know it would break down.


The potential consequences of naked singularities have been expressed in a similar way in an often quoted passage of the book Bangs, Crunches, Whimpers, and Shrieks – Singularities and Acausalities in Relativistic Spacetimes, by the philosopher of physics John Earman:

#Earman, J. (1995): Bangs, Crunches, Whimpers, and Shrieks – Singularities and Acausalities in Relativistic Spacetimes. Oxford University Press.
https://sites.pitt.edu/~jearman/Earman_1995BangsCrunches.pdf (PDF from Earman’s webpage)

–We think that singularities should exist in nature because we can prove that, under very general conditions, gravitational collapse leads to the formation of singularities.

 For years, physicists thought that the singularities predicted by general relativity might be a mathematical artifact of the theory, but that they don’t really form in any realistic physical process. But this changed in the 1960s, when physicists Roger Penrose and Stephen Hawking proved that, under fairly general physical conditions, singularities are an unavoidable consequence of gravitational collapse: 


#Penrose, R. (1965): “Gravitational Collapse and Space-Time Singularities”. Physical Review Letters, vol. 14, 57.

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.14.57 

Quote: “It will be shown that, after a certain critical condition has been fulfilled, deviations from spherical symmetry cannot prevent space-time singularities from arising. If, as seems justifiable, actual physical singularities in space-time are not to be permitted to occur, the conclusion would appear inescapable that inside such a collapsing object at least one of the following holds: (a) Negative local energy occurs. (b) Einstein's equations are violated. (c) The spacetime manifold is incomplete.  (d) The concept of time loses its meaning at very high curvatures —possibly because of quantum phenomena.”


#Hawking, S. W. et al. (1970): “The singularities of gravitational collapse and cosmology”. Proceedings of the Royal Society A, vol. 314, 1519.

https://royalsocietypublishing.org/doi/abs/10.1098/rspa.1970.0021 

Quote: “A new theorem on space-time singularities is presented which largely incorporates and generalizes the previously known results. The theorem implies that space-time singularities are to be expected if either the universe is spatially closed or there is an ‘object’ undergoing relativistic gravitational collapse (existence of a trapped surface) or there is a point p whose past null cone encounters sufficient matter that the divergence of the null rays through p changes sign somewhere to the past of p (i. e. There is a minimum apparent solid angle, as viewed from p for small objects of given size).”


Roger Penrose was awarded the 2020 Nobel Prize in Physics because of this discovery:

#Nobel Foundation (2020): “The Nobel Prize in Physics 2020”. Popular Information.
https://www.nobelprize.org/prizes/physics/2020/popular-information/ 

Quote: “Three Laureates share this year’s Nobel Prize in Physics for their discoveries about one of the most exotic phenomena in the universe, the black hole. Roger Penrose showed that black holes are a direct consequence of the general theory of relativity. [...] 


Not even Albert Einstein, the father of general relativity, thought that black holes could actually exist. However, ten years after Einstein’s death, the British theorist Roger Penrose demonstrated that black holes can form and described their properties. At their heart, black holes hide a singularity, a boundary at which all the known laws of nature break down. [..]


A trapped surface forces all rays to point towards a centre, regardless of whether the surface curves outwards or inwards. Using trapped surfaces, Penrose was able to prove that a black hole always hides a singularity, a boundary where time and space end. Its density is infinite and, as yet, there is no theory for how to approach this strangest phenomenon in physics.


Trapped surfaces became a central concept in the completion of Penrose’s proof of the singularity theorem. The topological methods he introduced are now invaluable in the study of our curved universe. [...]


Similarly, all matter can only cross a black hole’s event horizon in one direction. Time then replaces space and all possible paths point inwards, the flow of time carrying everything towards an inescapable end at the singularity”



–However, scientists think that nature forbids the formation of naked singularities. Something that enforces the creation of an event horizon around them, to prevent their insanity from infecting the rest of the universe.


This is the “cosmic censorship conjecture” mentioned above::


#Max Planck Institute for Gravitational Physics: “Cosmic censorship”, Einstein Online Dictionary (retrieved 2023)
https://www.einstein-online.info/en/explandict/cosmic-censorship/ 

Quote: “Cosmic censorship. Possibly the most disturbing feature of Einstein’s general theory of relativity is the existence of singularities – most commonly, regions of spacetime in which density and curvature go to infinity. It is quite likely that singularities are artefacts resulting from the fact that Einstein’s theory does not take quantum effects into account, and that they will be absent in a more complete theory of quantum gravity. Yet even if you leave aside quantum theory, and stay strictly within the framework of Einstein’s theory, it is likely that most singularities are, if not absent, then at least well-concealed. The hypothesis of cosmic censorship states that, whenever a body collapses so completely as to result in the formation of a singularity, a black hole will be formed so that the singularity will be hidden behind the horizon, and thus completely unobservable for anyone outside the black hole.”



–Without event horizons, physics may not make sense at all. So although black holes have been portrayed as the ultimate monsters of the universe, they may actually be the heroes that keep us safe from the madness of singularities.


These thoughts were expressed in a similar manner in another often-quoted sentence attributed to physicist Werner Israel: 


#Earman, J. (1995): Bangs, Crunches, Whimpers, and Shrieks – Singularities and Acausalities in Relativistic Spacetimes. Oxford University Press.
https://sites.pitt.edu/~jearman/Earman_1995BangsCrunches.pdf (PDF from Earman’s webpage)


–As far as we know, there’s just one safe method to destroy a black hole: Wait. All black holes emit tiny particles – a phenomenon called Hawking radiation. This process causes them to slowly lose mass until they eventually “evaporate”, leaving behind no horizon and no naked singularity.

The fact that quantum effects make black holes to radiate particles was discovered in 1975 by Stephen Hawking and is one of the most important and well-known results of black hole physics. This means that, without doing anything, any black hole will slowly lose mass until eventually disappearing:

#Hawking, S. W. (1975): “Particle creation by black holes”. Communications in Mathematical Physics volume 43,199.

https://link.springer.com/article/10.1007/BF02345020 

Quote: “In the classical theory black holes can only absorb and not emit particles. However it is shown that quantum mechanical effects cause black holes to create and emit particles as if they were hot bodies with temperature hκ/2πk ~ 10−6(𝑀⊙/𝑀) ºK where κ is the surface gravity of the black hole. This thermal emission leads to a slow decrease in the mass of the black hole and to its eventual disappearance: any primordial black hole of mass less than about 1015 g would have evaporated by now.”


–The time it takes for a black hole to completely evaporate depends on its mass. For our mini black hole the size of a speck of dust, it will be about 1044 years: 10 billion trillion trillion times the present age of the universe.


The time it takes for a black hole to completely evaporate grows with the cube of its mass. For most black holes, such a time is many orders of magnitude higher than the current age of the universe. In the case of our mini black hole with the mass of the Moon (7.3·1022 kg) and a radius of 0.1 mm, the evaporation time can be calculated to be of the order of 1044 years:


#Toth, Viktor T: “Hawking Radiation Calculator” (used 2023)
https://www.vttoth.com/CMS/hawking-radiation-calculator 

The age of the universe is about 14 billion years, i.e. of the order of 1010 years. Since 


1044 = 10·109·1012·1012·1010,

the time we’d have to wait for our mini black hole to evaporate would be about 10 billion trillion trillion times the present age of the universe.