Black holes are a paradox.

They are paradoxical because they simultaneously  must exist but can’t, and so they break physics as we know it.

But many physicists will tell you that the  best way to fix broken physics is with string.

String theory, in fact.

And in the black holes of string  theory - fuzzballs - are perhaps   even weirder than the regular type Einstein’s general theory of relativity tells us that if the density of matter is sufficiently ridiculous, ultimate gravitational collapse is inevitable.

The matter should contract to a single point of infinite density - the singularity - which is surrounded by a surface from which even light can’t escape - the event horizon.

And we see these black holes - or at least their incontrovertible evidence - in many places out there in the universe.

But black holes are also impossible.

At the central singularity, the known laws of physics break down - general relativity comes into irreconcilable conflict with quantum mechanics.

There’s our first paradox.

And at the event horizon another paradox arises.

On the one hand, general relativity insists that no information from the matter that fell into a black hole should be observable on the surface - the so-called no-hair theorem, which says that the only properties that we can observe from outside a black hole are its mass, electric charge, and angular momentum.

That’s not very many bits of information.

On the other hand, if we calculate the amount  of information that goes into a black hole to build it up, we get a prodigiously large number - 10^10^77 bits for a black hole the mass of our Sun.

Black holes should have an enormous number of  so-called microstates - hidden configurations - and this translates to an enormous entropy.

The formula for black hole entropy was discovered  by Jakob Bekenstein, who, incidentally, also inspired the colourful phrasing that black holes “have no hairs” to describe the absence of observable microstates in a black  hole, which, on the surface, seems to contradict his black hole entropy formula.

Here’s an analogy.

We can describe the behavior of the air in this room with very few numbers: its bulk motion, its temperature, pressure, etc.

But these few numbers emerge from the motion  of every air molecule — and to describe that we’d need the positions and momenta of 10^27ish particles.

So the air has the very high entropy of lots of information that is hidden under very few macroscopic properties.

But in the case of air, the hidden  information still exists and is,   in principle, still measurable.

Not so in the black holes of general relativity.

The no-hair theorem says that there’s no information beyond charge, mass and spin that’s observable above the event horizon.

The information contributing to black hole entropy could be beneath the horizon, but if we add one more fact we run into serious trouble.

Black holes evaporate by emitting Hawking radiation.

We talked about this previously, but the TL;DW  is that Hawking radiation should be completely random, and so leaks away the black hole’s mass without any of the information that went into building the black hole.

When the black hole eventually vanishes, so does its enormous information content.

And that violates a very deep and fundamental law  - the law of conservation of quantum information.

This threat of erasure of quantum  information in a hairless black hole is the black hole information paradox.

So yeah, black holes seem to be paradoxes.

And yet they definitely exist.

But you know what doesn’t exist?

Actual paradoxes.

When we see an apparent paradox in physics,  it’s really a clue pointing to a gap in our understanding.

Solving the paradox can lead us to new knowledge.

While we’ve covered these black hole paradoxes  before, and have even hinted at possible solutions - we’ve never actually solved them.

That’s what we’ll try to do today.

Ultimately, the black hole paradoxes stem from the disagreement between quantum mechanics and general relativity.

The impossibility of the central singularity is the most obvious place where we need a theory of quantum gravity.

But this is also the problem at the event horizon.

A theory of quantum gravity would  have really helped Stephen Hawking in deriving his eponymous radiation.

But such a theory didn’t exist back then in the 70s and still doesn’t.

Instead Hawking used a hack - something  called the semi-classical approximation.

He was able to describe the black hole using  pure general relativity but then analyze its effect on the surrounding quantum fields, which only worked if the gravity at the horizon was relatively weak, in which case quantum gravity effects shouldn’t play a part.

But that assumption may be wrong.

If gravity starts to get quantum even above the event horizon, then it may be possible to actually encode information on that surface  in a way that saves that information from destruction as the black hole evaporates.

And it turns out that our most advanced theory  of quantum gravity can do this quite neatly.

Black hole paradoxes may  be solved by string theory.

In string theory, black holes are not hairless at all - in fact all of those strings make them positively fuzzy.

And so we come to fuzzballs.

Well, very nearly.

First a spot of string theory.

We’ve covered this horrendously complex topic in detail before, so I’m only going to give you the crudest overview.

In string theory, all elementary particles are oscillations in 1-dimensional strands that themselves are embedded within complex, many-dimensional spaces.

String theory immediately solves the problem  of the black hole singularity, because instead of collapsing all of a black hole’s mass into a single point, it gets distributed around the ring structure of these strings.

Lose that pesky infinite density and we can start making sense of physics again.

It turns out that though, that string theory can make sense of the black hole event horizon also.

The first step towards this was in 1996, when  Andrew Strominger and Cumrun Vafa created a black hole using string theory  - in theory-space, not actually.

Now I’ll come back to the some of details later - for now just the result.

These guys found a way to count the microstates on  the horizon- the number of possible configurations of stringy structures - the 1-D strings themselves  and higher-dimensional structures called D-branes.

And the number they found exactly agreed with  the Bekenstein formula for black hole entropy.

It was hard to believe that this was a mere coincidence — it seemed they’d stumbled on the mechanism by which information  could be encoded on the event horizon.

The infinitesimal strings and branes of string  theory might be the analog of the molecules that store the entropy of our room full of air.

I should add that Strominger and Vafa did this for the somewhat unrealistic case of a black hole horizon with 4 spatial dimensions,  but it was a decisive step and it gave us a compelling reason to think that  string theory might explain where the microstates of black holes live.

But in order to solve the information paradox,  we still have to get that information out of the black hole as it evaporates.

The breakthrough came quickly.

A year after the Strominger-Vafa result, Samir  Mathur at Ohio State dug into that model to see if it could reproduce properties of black holes beyond the entropy.

And it did, almost too well.

For instance, the profile of the radiation emitted by the strings precisely matched that of traditional Hawking radiation.

This was another stunning match to theory, and also a way for stringy black holes to leak out their information.

It was looking pretty good, but I realize that I haven’t really told you what these stringy black holes look like, or how they form, or how structure can actually be supported on the event horizon.

For that we need to finally get to the idea of the fuzzball.

First, a thing that’s weird about even regular black holes.

If you were to dial up the strength of gravity, black holes would get bigger.

That’s different to pretty much everything else in the universe, which are crunched down to smaller size if you increase the power of gravity.

This property of black holes is actually quite  hard to reproduce in theories of quantum gravity.

But while he was exploring stringy black holes,  Samir Mathur found that the strings that formed the black hole would increase in size as the strength of gravity increased.

In fact, if you have a bunch of strings dense enough to form a black hole in general relativity, it wouldn’t actually collapse.

Rather it would grow to produce an agglomerate  of strings with the same radius as a classical black hole.

If this is right, then black holes don’t have an empty event horizon at all, but rather a real surface that looks like a tangled nightmare  of strings and branes, like the hairball coughed up from some hyperdimensional quantum cat.

A fuzzball.

Just as the neutron star’s gravitational field is so intense that atomic nuclei are crushed into a soup of neutrons, a star collapsing  into a fuzzball will see its constituents crushed into a soup of elementary strings.

The most amazing element of the fuzzball paradigm  is the discovery that quantum gravity effects might not just be important at the center of the black hole but instead may pile up to the horizon scale.

The ability for strings to maintain structure in this extreme gravity comes down to an effect specific to string theory called fractionation.

The tension of a string is inversely proportional  to its length: tight rubber bands are naturally small because they are difficult to stretch and vice versa.

But in string theory, when you merge a large number of strings, the tension of the resulting object is effectively smaller than for each individual string - proportional to the number of strings merged.

That means by crushing an enormous number of  strings together, the resulting monstrosity can be stretched to enormous scales.

The normally Planck-length strings can therefore  pile up into fuzzballs with sizes ranging from kilometers to light years.

The resulting fuzzball doesn’t have a local event horizon - in that there’s no single surface of no-return.

But from a distance, fuzzballs would look like black holes.

Light trying to escape would still be massively  redshifted - sapped of energy by the gravitational field - rendering the object effectively black.

It would still cause massive gravitational lensing, time dilation, etc.

All classical effects of black holes from general relativity would be preserved.

But if you approach the fuzzball you start to see this surface of stringy material with a thickness of about a Planck length.

And what happens if you punch through that surface?

Well, that’s where things get properly weird.

In fact there is no interior.

A fuzzball is really a fuzzsphere, or fuzzshell.

It’s not that the interior is empty, but rather that space and time literally end at the surface.

As a fuzzball is forming, all of the matter - now dissolved into stringy mess, is pushed up to the surface and the interior grid of spacetime is deleted from the universe.

This last thing is the weirdest and coolest thing about fuzzballs, so let’s explore a bit further by dropping a few dimensions.

Instead of the 4 spatial dimensions of a   Strominger-Vafa or the 3 dimensions  of regular black holes, let’s think about a 1-D black hole.

One dimension means a line, so a 1-D black hole is just a segment of a line with a point of infinite density on it.

The event horizon is just the pair of equidistant  points on the line inside of which escape is impossible.

Now let’s try a string-theoretic version.

In string theory we have extra compact dimensions  - spatial dimensions that are coiled up on the Planck scale so we can’t see them.

Adding a single coiled dimension to our 1-D black hole turns our line into a drinking straw.

The extra dimension is the distance around the straw.

For a fuzzball, spacetime closes on itself at the event horizon.

The extra compact dimensions contract  and pinch off, so all spatial dimensions end in that direction.

As a side benefit, this eliminates the problem  of the gravitational singularity at the center of a black hole.

There is no central singularity  because there is no center.

The emergence of fuzzballs in string theory radically changed our vision of what a black hole could be and provided a satisfying  potential resolution to several paradoxes.

However, this picture is not yet complete.

The model has only been carried out for  simplified and nonrealistic-cases like the Strominger-Vafa black hole.

Constructing fuzzballs that more closely match  real astrophysical black holes remains a major theoretical effort.

I should also add that fuzzballs are not the only quantum extension of the GR black hole.

We can come back to those another time.

Black holes, or something  like them, definitely exist.

And they’re one of our best  hopes for understanding the   union of general relativity with the quantum.

Perhaps that’ll come when we find evidence of fuzziness surrounding those distant holes in the fabric of space time.