The universe has two surfaces where the speed of light goes to zero ── and one of them has been called a fluid for forty years
Episode 5 showed that the boundary for holography in flat spacetime is three-dimensional and carries a "geometry with zero speed of light." It may have seemed a strange conclusion ── does such a geometry really turn up in physics?
It does. And not in only one place. Because any null hypersurface is necessarily a Carroll manifold. And the universe has two physically important null hypersurfaces.
This episode goes to look at the other one ── a black-hole horizon. Waiting there is a curious fact known long before Carroll geometry was discovered: "a horizon behaves like a viscous fluid."
Episode 5 gave the reason the metric on \(\mathscr{I}\) degenerates. The \(u\) direction is null, so its length is zero.
That argument used nothing about \(\mathscr{I}\). It used only nullness. So it holds generally.
null hypersurface \(\;\Longrightarrow\;\) degenerate metric \(\;\Longrightarrow\;\) Carroll manifold
So where are the physically important null hypersurfaces? There are two.
| surface | boundary of what | Carrollian | symmetry |
|---|---|---|---|
| \(\mathscr{I}\) (null infinity) | the outside of the universe | ○ | BMS (supertranslations, superrotations) |
| the event horizon | the inside of a black hole | ○ | horizon supertranslations, superrotations |
Episode 5 dealt with only one of them. It follows that the universe is sandwiched between two Carroll surfaces ── the far edge, and the rim of a hole.
This is a calculation, not a figure of speech. In the Schwarzschild metric, follow light travelling radially.
Light has \(ds^2=0\). Radially, this gives
$$0=-\Bigl(1-\frac{r_s}{r}\Bigr)c^2dt^2+\frac{dr^2}{1-\dfrac{r_s}{r}}$$solving
$$\frac{dr}{dt}=\pm\,c\Bigl(1-\frac{r_s}{r}\Bigr)$$Far away (\(r\gg r_s\)) it is \(\pm c\). But at the horizon \(r\to r_s\) ── it becomes zero.
Recall the slider from Episode 5. That was a hypothetical, "what if the speed of light were different." At a horizon, that is what actually happens.
The radial coordinate near the horizon plays the part of an "effective speed of light" heading to zero.
The limit of approaching the horizon = the ultra-relativistic limit \(c\to0\) = the Carroll limit.
See the figure. The same "light cone collapsing" phenomenon as in Episode 5 now happens not by fiddling with coordinates but merely by approaching the hole.
Far away it is the familiar 45° light cone. As you approach the horizon it closes up, and at \(r\to r_s\) it collapses completely into a vertical line ── the same picture as Episode 5's "Carroll's country."
From here comes the most interesting part of the episode.
In 1979, Damour noticed something strange. Project the Einstein equations onto the horizon and you get the equations of a fluid.
Thorne and others systematized this as the "membrane paradigm" (1986). Regard the horizon as a two-dimensional viscous membrane with the following properties, and the physics seen from outside is correctly reproduced.
| membrane property | value |
|---|---|
| shear viscosity \(\eta\) | \(1/16\pi G\) |
| bulk viscosity \(\zeta\) | \(-1/16\pi G\) ── negative! |
| entropy density | \(1/4G\) (per unit area) |
| surface resistivity | \(377\;\Omega\)／square ── the impedance of the vacuum \(Z_0=\mu_0c\) itself |
That the surface resistivity equals the impedance of the vacuum is a fact worth remembering for the pleasure of it. And the bulk viscosity is negative ── impossible for an ordinary fluid. It is a manifestation of the horizon's teleological character, "reacting in anticipation of the future."
This correspondence was long thought to be an analogy. Useful, but with no explanation of why it should hold. There was nothing beyond "a horizon resembles a fluid."
What Donnay–Marteau (2019) showed is this.
The two equations governing horizon dynamics ── the null Raychaudhuri equation and the Damour equation ── are obtained as the ultra-relativistic (\(c\to0\)) limit of the conservation of the energy-momentum tensor.
That is, they are the conservation laws of a Carrollian fluid.
Furthermore, as a timelike surface is brought towards the horizon, its induced geometry degenerates from Lorentzian to Carrollian and the induced Einstein equations drop to the conservation equations of a Carrollian fluid ── the limit has been exhibited in controlled form.
So something that was "an analogy that happens to work" for forty years has acquired a geometric pedigree. That the horizon looked like a fluid was no accident: it was because that surface is a Carroll manifold.
If \(\mathscr{I}\) is Carrollian and has BMS symmetry, the horizon should have the same structure. It does.
Donnay–Giribet–González–Pino (PRL 2016) computed the asymptotic symmetries of a black-hole horizon and found horizon supertranslations and superrotations. A structure isomorphic to \(\mathscr{I}\)'s BMS exists at the rim of the hole as well.
The form is the same ── the operation of shifting "time" independently at each point of the horizon. As we saw in Episode 5, this is the inevitable consequence of ultra-locality.
Recall Bonus ③. Why was soft hair disqualified?
Bousso–Porrati's criticism ── "that is not hair, it is a wig." The supertranslation charge is fixed by asymptotic data at infinity, so it can be changed simply by emitting one soft graviton far from the black hole. It cannot distinguish what fell in.
Horizon symmetry is in a structurally stronger position against this criticism.
| soft hair at \(\mathscr{I}\) | horizon symmetry | |
|---|---|---|
| where the charge is fixed | at infinity | on the horizon itself |
| changeable from far away? | yes (a wig) | localized |
| is the state count \(e^{A/4G}\)? | does not match | unsolved |
The locality problem is solved. But the counting problem remains exactly as it was.
Let us draw the overall picture.
The universe has two surfaces with degenerate metric.
The outer \(\mathscr{I}\) ── if a Carrollian field theory sits there, you get holography for flat spacetime.
The inner horizon ── if a Carrollian field theory sits there, you get black-hole entropy.
The same mathematics acts on two utterly different problems.
And both are stuck at the same place ── there is no independent definition of Carrollian field theory.
Episode 5 said "the place we are looking became the right one." What this episode adds is that there are two such places, and they are the same one. If either is solved, the other is likely to move.
Carroll geometry was not a special circumstance. Any null hypersurface necessarily has a degenerate metric and is a Carroll manifold ── the \(\mathscr{I}\) argument had used only nullness. And the universe has two physically important null hypersurfaces: null infinity, and the event horizon.
Approaching the horizon is taking \(c\to0\). The radial coordinate speed of light is \(dr/dt=c(1-r_s/r)\), which is zero at the horizon. Episode 5's slider was a hypothetical; at a horizon it actually happens.
The pedigree of the membrane paradigm came out. Since 1979, "a horizon behaves like a viscous fluid" had been no more than a useful analogy. The null Raychaudhuri equation and the Damour equation were the conservation laws of a Carrollian fluid ── they come out of the \(c\to0\) limit. A fit became a derivation.
The defect in soft hair is only half filled. Horizon supertranslations are by definition localized there, so they dodge the "wig" criticism. But whether the state count gives \(e^{A/4G}\) is unsolved, and there are several competing claims about what the near-horizon symmetry algebra is.
And the picture came into view. The universe is sandwiched between two Carroll surfaces. Put a theory on the outer one and you get holography for flat spacetime; on the inner one, black-hole entropy ── the same mathematics acts on two utterly different problems. And both are stuck on the same single point: there is no independent definition of Carrollian field theory.
This document is Episode 7 of the "Lattice We Build" series, a reading piece for high-school and university students who love physics. Where the sister series "That Clicks" explains known physics, this series shows the work itself.
Established material: that the induced metric on a null hypersurface degenerates; the radial coordinate speed of light \(dr/dt=c(1-r_s/r)\) in Schwarzschild coordinates; the membrane paradigm (Damour 1979, Thorne–Price–Macdonald 1986) and its transport coefficients; the Carrollian geometry of the horizon and that the Damour and null Raychaudhuri equations are Carrollian conservation laws (Donnay–Marteau, Class. Quantum Grav. 36, 165002, 2019); and horizon supertranslations and superrotations (Donnay–Giribet–González–Pino, PRL 116, 091101, 2016).
On the other hand, whether the state count of a Carrollian field theory on the horizon reproduces \(A/4G\) is unsolved. Several proposals for the near-horizon symmetry algebra exist in the literature and the matter is unsettled. And Carrollian holography is an ongoing research programme; an independent construction of the boundary theory does not yet exist. The light cones in the figure are schematic, based on the coordinate speed of light in Schwarzschild coordinates; the speed measured in a local inertial frame is always \(c\).
Main series: Episode 1｜Episode 2｜Episode 3｜Episode 4｜Episode 5｜Episode 6 ｜ bonus: ①／②／③ ｜ sister series: Black Holes That Click ── to print, use your browser's "Print" → "Save as PDF."
Print / PDF: ⌘+P (Ctrl+P on Windows). Move the slider in the figure all the way right and the light cone collapses.