Black hole soft hair, the edge of the universe, and closed timelike curves
In this week’s newsletter I’m bringing together a few ideas that are interesting me: stereographic projection (and conformal mappings in general), boundaries, supertranslation symmetries, and closed timelike curves.
Mostly I’ve been thinking about asymptotically flat spacetime, that concept that as you go further away from a massive object, such as a black hole, spacetime becomes more flat. Flat spacetime is much easier to understand and work with than curved spacetime because it has symmetries. Curved spacetime near a black hole, on the other hand, has few symmetries, not easy to grasp.
About 7 years ago, the Stephen Hawking published what would be one of his last theories, along with Andrew Strominger, and Malcolm Perry.
A side note on Strominger: at the Institude for Advanced Study in Princeton, he is always showing up in my research. He’s not as well known as some other physicists, but he was one of the early string theorists in the ‘80’s and has been working on topics in quantum gravity as long as I have been alive. Really brilliant mind.
Anyway, this theory is called the “soft hair” theory of black holes. Hawking had this way of talking about black holes by saying they have “no hair” meaning that they are featureless. In reality, they have three hairs: mass, charge, and angular momentum. Those are the only quantities that you can measure about a black hole, regardless of what formed them. That means, if you have two black holes the same size, charge, and rotation, they are indistinguishable.
Hawking argued in the ‘70’s that this meant that if objects fell into black holes, the information they contained was lost to the universe. Yet, that disagrees with quantum mechanics which says that information is never lost. This became the big black hole information paradox which has been “solved” several times now.
Hawking’s own solution, however, was, to me, far more compelling than any of the other ones. He argued that you could potentially extract all the information from a black hole, but only if you were very, very far away from it.
If we imagine a black hole as being in an otherwise empty universe, the further we get from it, the flatter the universe becomes. This means that the universe is asymptotically (meaning in a limit of large distances) flat.
Asymptotically flat spacetimes are cool because they obey symmetries at their boundaries that they do not obey in their interiors.
Now, for simplicity, let’s take two spatial dimensions away and just think of one space and one time dimension. Spacetime then appears as an infinite plane. In the center of that is a line which represents a black hole. It is a line propagates in time but is localized in space.
Black holes form and they also evaporate so the line is finite in length.
Beyond that, towards infinite time and distance, everything is flat.
This plane can be thought of as an unbounded plane, but we can also take all the points that represent infinity and add them to the definition of the set of points within the plane. (Yes, we can just do that.) This becomes the extended plane. Now, those points at infinity form a boundary.
It is hard to draw infinitely large planes, so instead we map the plane, including the infinities, to a diamond which is finite in size. This is a conformal or Penrose diagram. (A conformal mapping, which this is, is just one that preserves angles.)
The way to understand a Penrose diagram is that it is like taking a photo of the infinite plane with a fisheye lens. Fisheye lenses aren’t conformal, however, so this is more like a stereographic projection of a 180 degree panorama. In any case, like the fisheye lens, this condenses down an effectively infinite plane into an ordinary size photograph, edges and all. And as in a fisheye photo, the central focus is blown up while the most distant objects are shrunk, eventually down to nothing.
In the conformal diagram, any boundary will appear as a 45 degree line. That could be infinite distance or the event horizon of a black hole. All light beams travel at 45 degree angles in the diagram and black hole singularities look like universe sized barriers. This site has some good examples as does this one.
The point here is that if you have an asymptotically flat spacetime, you can describe the boundary of that spacetime as the edges of a Penrose diagram.
If you read my recent article on Weinberg’s soft graviton theorem, the following discussion might be a repeat for you, though I’m going into a bit more detail here. Feel free to skip if you are familiar with the Poincaré group.
Inside a flat spacetime, not on the boundary, you have some symmetries that are not true near a gravitational object such as a black hole. A symmetry is simply a knob you have that you can change in your theory that doesn’t change the physical predictions that you make.
We know from Einstein that, in a flat spacetime, we can perform an experiment at one point or a point 100 km away from that point in any direction and get the same result. So those are three knobs we call spatial translations.
We also know that we can perform an experiment in the morning, afternoon, next Tuesday, or 100 billion years in the future and expect the same result. (This is NOT true in general since our universe is not flat in all four dimensions but expanding and therefore curved.) This is called time translation symmetry and is a fourth knob.
We also know that we can be rotated in whatever way we want. This is why it really makes no sense for spaceships in deep space to be oriented in any particular way. It is purely aesthetic that TVs and movies do so. We can be rotated in one of three rotational directions, pitch, yaw, and roll. Note that I said we can be “rotated” we cannot be “rotating”. The rotational symmetries give us three more knobs.
The final knobs are velocity. I can be traveling at 10 kph or 100 kph or near the speed of light relative to you and we will see the same results in our experiments. We cannot be accelerating but we can be at different fixed velocities. Since there are three directions I can move in, that is three more knobs which we call “boost” symmetries.
Altogether, that is 10 knobs: four translation, three rotation, and three boost.
Together these form what mathematicians call a “group” and this one is named after Henri Poincaré, so it is the Poincaré group.
At the boundary of flat spacetime, however, we have many more symmetries, an infinite number, in fact. These are called supertranslations.
Supertranslations can be thought of as time translations, but we have one for every direction in space (azimuth and elevation) centered on some origin (such as the black hole). Not every point in spacetime, mind you, just every direction.
That means that if we start at a black hole and we fire a light beam from near it along a particular direction and it travels into flat spacetime, at the boundary (which it will only reach after an infinite amount of time) it obeys a symmetry called a supertranslation.
This symmetry means that if I sent two light beams out in different directions, the time they began to propagate, which is called their retarded time coordinate, can be added to (or subtracted from) by arbitrary amounts. Only if they are in the same direction do we have to translate them by the same amount.
One way to understand why is because if I send light beams in different directions, after an infinite amount of time, they will be infinitely far apart from one another. Since they are infinitely far apart, their clocks cannot be synchronized in any way. They are causally unlinked. So I can set their clocks to anything I want. Only if they are sent in the same direction will they be causally linked and follow just the ordinary time translation symmetry which means that their clocks must be adjusted by the same amount.
I honestly find this explanation, while correct, is not that intuitive.
I began to search for a better way to understand it and that is where I hit upon the idea of using stereographic projection. So, you see, if we can map an infinite plane to a conformal (Penrose) diagram, we can also map it to the surface of a sphere in a conformal way as well. This is one of the great results of 19th century analysis actually and Penrose worked with this idea as well. He used the ideas a great deal in his 1984 book with Rindler on spinors in spacetime.
The sphere in question is the Riemann sphere and, using stereographic projection, you can map an infinite plane onto the sphere just as you can map it onto a diamond shaped structure.
Stereographic projection is very, very old. It goes back to Ptolemy in the year 125 in fact and was first used to make astrolabes, the devices that showed the night sky and helped navigators find their way before compasses.
The idea is that Ptolemy took the inside of a sphere, the celestial sphere or heavens, and mapped it onto a flat piece of paper using geometry. He did this by picking a point on the sphere as the projection point and then imagined the plane as sitting under it. Then he drew a line from the projection point through the line on the sphere and then down onto the plane. Wherever that point on the plane ended up, that was where he drew the point that he could see on the sphere.
The projection point Ptolemy chose was the celestial south pole, which meant the northern hemisphere’s night sky was neatly contained in a circular map that could fit on a circular piece of metal that a ship’s captain could take with them on a voyage.
You can read my article about stereographic projection here.
Riemann basically took that idea and reversed it, showing how an infinite plane could be mapped onto a sphere and help us understand things like what is infinity. He chose the North Pole as his projection point and had the plane pass through the equator of the sphere instead of being under it. This meant that the southern half of the sphere would be inside a unit circle while the northern half would be outside it.
Some cool things happen when you do this. For example, circles in the plane map to circles on the sphere, which is nice, but lines in the plane also map to circles on the sphere. They are just circles that pass through infinity, i.e., the North Pole, at one point. You can see a picture of that at this page. (The relevant image comes from Visual Complex Analysis by Tristran Needham which explains all this in great detail.)
The difference is that instead of having a linear, diamond shaped boundary for infinite future, past, and distance, the boundary becomes just one point: the North Pole. This is probably a bit less useful for some applications, but it can generate some amazing insights about what infinite time really is.
Suppose we shoot a beam of light from the infinite past and it passes into the infinite future. That line will be a circle that passes through the North Pole on the Riemann sphere for the universe. Beams of light that start in some finite place and head to infinite future will be parts of circles that also pass through the north pole.
The neat point about the Riemann sphere is that the North Pole being infinity is rather arbitrary. A circle that passes through the North Pole is indeed a straight line in the plane, but we could just rotate the sphere a bit or move the projection point, and the circle no longer passes through infinity. It becomes a circle in the plane instead of a straight line, but, on the sphere, it is the same as it was before.
This means that we can reason about light at infinite distances by thinking about particles traveling in circles in space and time. Basically, if you are familiar with Closed Timelike Curves (CTCs), these are paths that follow closed loops in space and time.
Time travel.
Here is one example: a pair of particles are created at a point out of vacuum, a particle and its antiparticle. They travel away from one another and then slow and fall back together, annihilating.
Richard Feynman reasoned that the particle and antiparticle are actually the same particle. It is just that the antiparticle is the particle as it travels backwards through time. The point of creation is the part of the circle the furthest in the past and the point of annihilation is the part furthest in the future. Thus, such a trajectory is a circle in spacetime.
As an aside, you can infer from this and the Riemann sphere why our universe contains matter and not anti-matter. If matter is created in the infinite past and flows to the infinite future, that is just a circle on the Riemann sphere passing through the North Pole. The North Pole is both the past and the future. Unlike other spacetime loops where you get both matter and anti-matter manifesting, circles that pass through the North Pole appear as straight line trajectories of a single kind of particle. Since a straight line is a circle with infinite radius it never curves backwards through time and the anti-particle can never manifest.
In order to determine that we have supertranslation symmetry, let’s do a thought experiment. Alice is at this point and sends a particle anti-particle pair out in opposite directions. At some later time, the particles come back together and annihilate.
Alice has a clock and marks when the particles came back together.
If Bob is at the same place and time as Alice, he can send out beams likewise and his clock can be offset from Alice’s, or he can send his out at an earlier time but have them travel farther out so they come back at the same time. He will observe the same annihilation of his particles as Alice does of hers.
If Bob is in a different place from Alice, however, he can’t just send out two particles symmetrically like Alice does and see the exact same annihlation. Instead, he sends out one particle at a different time from the other timed just right so that they come together at the same place as Alice’s particles and annihilate. (We assume Bob has enough time to do this.)
But Bob doesn’t just have to send them out in the same direction, he can send his particles out in different directions and still time them just right so that they annihilate at the exact same time and place as Alice’s.
The reason Bob can do this is that his pair doesn’t have to be the same as Alice’s anywhere else but the point of annihilation. Supertranslations only apply to that single point on the loop. All he needs to do is make his annihilation look like Alice’s at that one point.
This means that Bob effectively has an infinite number of knobs he can turn to make his particle-anti-particle pair annihilate at the same time and place as Alice’s.
This thought experiment, on the Riemann sphere, is the same as sending beams of light to infinite future or from infinite past. They are just circles.
There is a connection between symmetry groups and conserved quantities which is why it is important to black holes.
For every supertranslation, there is a quantity, net energy, that is conserved for every direction. These infinite numbers are almost enough to give black holes “soft” hair, meaning very, very distant, low energy hair. We need a bit more though. One proposal is that superrotations are conserved as well which would mean that we not only have conserved energies, but also angular momenta for every direction.
That would be enough, it is conjectured, to determine all the information that fell into the black hole, but only if you are infinitely far away from it. In other words, the black hole projects its information to the boundary of the universe. Or is it the other way around? There is a connection between this and holographic principle, the idea that matter is simply a projection of information existing at the boundary of the universe.
This is an ongoing story and it seems as if a lot of exciting developments are happening in this direction.
Hawking, Stephen W., Malcolm J. Perry, and Andrew Strominger. "Soft hair on black holes." Physical Review Letters 116.23 (2016): 231301.
Penrose, Roger, and Wolfgang Rindler. Spinors and space-time: Volume 1, Two-spinor calculus and relativistic fields. Vol. 1. Cambridge University Press, 1984.