What Survives Inside a Black Hole? (script) (Patreon)

We’ve established by now that black holes are weird. The result of absolute gravitational collapse of a massive body: a point of hypothetical infinite density surrounded by an event horizon. At that horizon time is frozen and the fabric of space itself cascades inwards at the speed of light. Nothing can travel faster than light, and so nothing can escape from below the event horizon- not matter, not light, not even information. 

These ideas are pretty mind-blowing, but as crazy as black holes are, they’re also kind of … simple.  Don’t get me wrong; the math is complicated. But the objects themselves are simple. In fact every black hole in the universe, no matter how it formed or what happened to it afterwards, can be perfectly described with only three properties. Those properties are mass, angular momentum, and electric charge. Or at least this is the proposition behind the famous no-hair conjecture, or no-hair theorem. 

Today we’re going to see why black holes are bald. In an upcoming episode, we’ll combine the no-hair conjecture with everything we learned recently about the conservation of quantum information. The result will be the black hole information paradox, one of the greatest unanswered questions in modern physics, and the gateway to black hole thermodynamics and the holographic principle.

Jakob Bekenstein was a graduate student at Princeton when he proposed that no other properties besides mass, electric charge, and angular momentum should emerge from beneath the event horizon. Or as Bekenstein put it, “black holes have no hairs”. This is why you don’t let graduate students name things. Bekenstein’s advisor, the legendary John Archibald Wheeler, took up the phrase, so now we’re stuck with it. 

The basic reason for the no-hair conjecture is straightforward enough – the interior of a black hole is cut off from the external universe. It’s causally disconnected. Nothing beneath the event horizon can influence the exterior universe because no signal can escape the event horizon to carry that influence. The black hole could have formed from a collapsed star, or entirely out of anti-matter, or photons, or monkeys. The only things we can know about the material that went into the black hole are its mass-energy content, electric charge, and angular momentum.

Given the one-way nature of the event horizon, the loss of this information makes some intuitive sense. Perhaps more mysterious are the three exceptions; the three remaining hairs around our otherwise-bald black hole. After all, the forces involved – namely the gravitational and electromagnetic forces – are only communicated at speed of light. So how does mass, electric charge and angular momentum communicate their influence across the uncrossable horizon?

Let’s think about this in terms of fields, and we’ll start with the gravitational field. Gravity may be caused by mass, but a gravitational field is a very real thing all on its own. In Einstein’s general theory of relativity we think of the gravitational field as curvature in the fabric of spacetime. In the good-old rubber sheet analogy, a heavy ball stretches the sheet downwards, affecting the motion of objects moving on that sheet. We can think of the gravitational field at any point as being caused by the gravitational field at the surrounding points. Each point on the rubber sheet doesn’t actually “see” the source of the gravitational field. Instead it stretches and curves in response to the behavior of the surrounding patches. 

The Earth orbits the Sun, but more directly it orbits the Sun’s gravitational field. You could change anything about the Sun other than its mass and Earth would continue in the same orbit. If the Sun were to suddenly vanish, Earth would continue to orbit in the existing gravitational for eight minutes. The spacetime at the location of Earth’s orbit would remain curved until the elastic fabric straightened itself out at the speed of light.  

In the case at the event horizon, the outside universe can’t see the mass inside the black hole, but that mass is remembered in the gravitational field - the curvature of spacetime above the event horizon. In fact the space at the event horizon is  already falling inwards - it doesn’t need to be told to continue doing that. And it drags on the layer directly above it, continuing the cascade even if the central singularity is lost to all causal influence.

There’s an extremely important law in physics that describes how the universe remembers the contents of a region of space. It’s Gauss’s law – which applies to both gravitational and electric fields. Gauss’s law of gravity states that the total gravitational field added up over any closed surface is proportional to the amount of mass and energy contained by that surface. The distribution of the matter inside that surface is irrelevant. The mass could all be located at a single point within that surface, or it could be evenly distributed across the surface – the resulting summed gravitational field would be the same. 

In the case of a black hole - at least a non-rotating one – the event horizon is a closed, spherical surface centered on the singularity. That surface remembers the internal mass as though it were spread evenly across the surface. Now the original Gauss’s law applies to the electric field. In fact it’s one of Maxwell’s equations, which together give the classical description of electromagnetism. Gauss’s law for the electrostatic field says that the total electric flux passing through a closed surface depends on the amount of charge interior to that surface.  The universe remembers the total charge that exists in the region of space enclosed by the surface. This means that the electric field above the event horizon of a black hole remembers all of the electric charge that fell through that surface. Black holes act as though their charge is spread across the event horizon.

If you’ve studied some introductory physics, you might remember that the gravitational and electric fields have something in common. Both obey an inverse square law. Newton’s law of universal gravitation and Coulomb’s law for electrostatics say that the strength of the force produced by these fields drops off with the square of the distance from the source of the field. This comes from the fact that if you draw increasingly large spheres around a point-like field source, the intensity of those forces get spread out over an increasingly large area. You can think of lines of force getting further and further apart for larger spheres. The area of that sphere is proportional to the square of its radius, so the density of the lines of force decreases with distance squared.

In fact Gauss’s law is a more general form of these inverse square laws. It applies to any shape surface surrounding any shape mass or charge. But both Gauss’s law and the inverse square law work because of a key similarity between gravity and the electric field: they are infinite in range, and they arise from conserved quantities of mass and charge. The lines of force never fade, never vanish, and never merge into each other. That means all lines of force created within a closed region must eventually cross that region’s surface.

OK, so the gravitational and electromagnetic forces have infinite range, and so Gauss’s law demands that the mass and charge contents of any region of space are remembered in the gravitational and electric fields on the surface surrounding that region. Because mass and charge are fundamentally conserved quantities, the only way to change the contents of a region of space is for more mass or charge to cross its surface. In the case of the black hole, nothing can get out but new, infalling material will adjust the black hole’s external gravitational and electric fields on its way in.

By the way, it’s worth mentioning that real black holes out there in the universe are never going to have a net electric charge. A black hole with non-zero charge will quickly attract particles with the opposite charge until positive and negative charges within the black hole balance out. 

Okay, so we’ve covered gravity and the electrostatic force. What about the magnetism part of electromagnetism? And what about angular momentum? Well you get magnetism for free – a changing electric field produces a magnetic field, so if the black hole is spinning or racing past you you’ll see that magnetic field.

In a similar way, you can see a black hole’s rotation in its gravitational field. In Einstein’s general relativity, a spinning mass drags the fabric of spacetime around with it in a phenomenon known as frame dragging.  The incredibly subtle frame dragging due to Earth’s rotation has been detected, but it required the incredible precision and ingenuity of Gravity Probe B. The frame dragging around a rotating black hole is rather stronger. It dramatically changes the shape of the event horizon and the orbits of anything nearby.  If material with angular momentum falls into a black hole – whether it’s a spinning star or a whirlpool of gas – it will either add or subtract from this flow of space above the event horizon. Its angular momentum is remembered in the frame dragging, as though the entire black hole were spinning. 

I hope I’ve given you a sense of why mass, charge, and angular momentum are remembered by the space outside a black hole. It’s because these properties are conserved and are communicated by gravity and electromagnetism, which have infinite range. As a result they are remembered in the fields surrounding the black hole.  However there are some important conserved properties that are entirely local. If matter with these properties falls into a black hole, information about those properties is lost to the outside universe. An example would be the number of particles of different types – like the balance between quarks and anti-quarks, represented by baryon number. Baryon number is a conserved quantity, but there’s no way to know the baryon number of a black hole. This doesn’t sound like a big problem – so what if the universe forgets what type of particles a black hole is made of? 

Actually, it should care a great deal. We discussed recently why quantum mechanics demands the conservation of quantum information. It’s fundamental to quantum mechanics that the universe keeps track of its quantum states – which also means the types of particles it contains. Yet Stephen Hawking showed that black holes may break this rule, revealing a conundrum that we now call the information paradox. The solution to the information paradox is highly speculative, but may may reveal that black holes are more hairy than we thought. We delve into all of that in an upcoming episode of Space Time.