I posted two papers to the arXiv in August 2026. Both are about the same thing: the region immediately outside a black hole’s event horizon, and how little freedom nature has in choosing what goes there.
This post is my attempt to explain what they say to someone who has not spent a decade with a copy of Wald on their desk. If you want the general background on black holes first, I have a separate post on that. The papers themselves are linked at the bottom.
The one-sentence version
Black holes are far more rigid objects than they look. If you zoom right in on the horizon of a certain kind of black hole, the geometry there is so tightly constrained that in one case you can check a condition at a single point and thereby determine the entire spacetime, and in another case the horizon is forced to have a rotational symmetry that nobody put in by hand — a symmetry strong enough to rule out a shape that people had been searching for.
Everything below is the unpacking of that sentence.
What a horizon actually is
The event horizon is not a surface in any physical sense. Nothing is there. You could cross the horizon of a large black hole without noticing — no wall, no jolt, no local measurement that would tell you. It is defined entirely by what happens afterwards: the horizon is the boundary of the region from which no signal can ever reach the outside universe. It is a statement about the future, not about the present.
That makes it a slippery thing to study, and it is also what makes it interesting. The horizon is where the global structure of a spacetime becomes visible in local geometry.
Extremal black holes, and the trick of zooming in
Real black holes have a mass, and they can also carry spin and electric charge. There is a limit to how much: spin a black hole too fast, or charge it too much, and the horizon would disappear entirely, leaving the singularity exposed. A black hole sitting exactly at that limit is called extremal.
Extremal black holes have a remarkable property. If you zoom in on the horizon — take a limit in which you magnify the region around it indefinitely — the geometry does not blur or become featureless. It settles down into a well-defined limiting shape, called the near-horizon geometry, which is a perfectly good solution of Einstein’s equations in its own right.
This is the central trick of the field. Rather than trying to classify entire black hole spacetimes, which is extraordinarily hard, you classify the near-horizon geometries, which is merely very hard. And because every extremal black hole has one, any near-horizon geometry you can rule out rules out a whole family of black holes at once.
An analogy: instead of cataloguing every possible mountain, you catalogue every possible summit. Fewer things to classify, and if a shape cannot be a summit, no mountain has it.
Supersymmetry, and why a theorist would bother
My work is in supergravity, which is what you get when you combine Einstein’s general relativity with supersymmetry — a hypothesised symmetry relating the two families of fundamental particles, the matter particles and the force carriers. Supergravity theories arise naturally as the low-energy limits of string theory, and eleven-dimensional supergravity is the low-energy limit of M-theory, the framework that unifies the various string theories.
Two honest caveats. First, supersymmetry has not been observed in nature; the Large Hadron Collider has looked hard and found nothing. Second, we are talking about black holes in ten or eleven dimensions, not the four we live in.
So why do it? Because supersymmetry is a calculational gift. In a supersymmetric solution there is a spinor field — think of it as a kind of directional quantity attached to every point of spacetime — that satisfies a first-order equation called the Killing spinor equation. First-order equations are enormously more tractable than Einstein’s second-order ones. Squaring the spinor gives you back the metric and the fluxes, so anything you learn about the spinor is something you have learned about the geometry. Supersymmetric black holes are the ones where the mathematics is sharp enough to prove theorems rather than run simulations, and the structural lessons — that horizons are rigid, that they carry more symmetry than you assumed — have repeatedly turned out to generalise beyond the supersymmetric case.
The horizon conjecture, in one paragraph
My PhD thesis was about a pattern that keeps recurring. Take a supersymmetric black hole preserving some amount of supersymmetry. Zoom in on the horizon. The near-horizon geometry preserves twice as much. Not sometimes, not generically — always, across every supergravity theory anyone checked. The horizon is more symmetric than the black hole it belongs to, and the extra symmetry appears in the limit for free. This is the horizon conjecture, and proving it theory by theory has been a programme running for well over a decade.
The two new papers are about what else falls out of the same machinery.
Paper one: how one point can determine an entire spacetime
The warp factor of supersymmetric D = 11 near-horizon geometries
Near a horizon, time runs at different rates in different places. There is a function — the warp factor, usually written $\Delta$ — that records how much time is stretched at each point of the horizon cross-section. There is also a one-form $h$ that measures rotation: how much the horizon is dragging space around with it. Between them, these two objects encode most of what distinguishes one horizon from another.
The paper starts from an identity relating the warp factor, the rotation, and the lengths of the Killing spinors. That identity is not new in itself. What is new is refusing to simplify it — previous work reduced it by assuming the spinor had constant length, which is convenient and true in many cases but not something you are entitled to assume. Keeping the unreduced identity turns out to buy three things.
First, a single-point rigidity theorem. If the warp factor and the rotation both vanish at one point of the horizon, then the flux — the eleven-dimensional analogue of the electromagnetic field — vanishes everywhere, the horizon cross-section is Ricci-flat, and the whole geometry is the trivial one. One point. Not a condition checked across the entire horizon, which is what earlier arguments needed; a condition checked at a single location, which then propagates to the whole spacetime.
That is the kind of statement that is surprising even to people in the field. Local information almost never determines global structure in general relativity. Here it does, and the reason is that the supersymmetry conditions are so tightly interlocking that a single zero cannot be an isolated accident.
Second, a perfect square. In eleven dimensions the flux naturally decomposes according to a group called $\mathrm{Spin}(7)$ — the same exceptional structure that shows up in the geometry of certain string compactifications. When you do that decomposition, the rotation $h$ is fixed algebraically, and the warp factor comes out as a perfect square: an explicit quantity squared, so its positivity is automatic rather than something you have to argue for.
Perfect squares are the most useful thing in this business. When a quantity you need to be non-negative turns out to be manifestly a square, you get positivity for free and you can then ask when it vanishes — usually the road to a classification.
Here there is a genuine subtlety, and I want to be honest about it because it is the difference between a nice result and an oversold one. The square is degenerate, not definite: it vanishes not just at the origin but along a whole linear subspace. Only two of the components of the flux reach the warp factor at all, and they can cancel each other. So the positivity is real, but it does not deliver the clean list of cases you would hope for. Saying so explicitly is part of the point of the paper.
Third, a budget. The two results combine into a pointwise constraint that reads like an accounting statement: at every point on the horizon, a fixed amount of supersymmetry is shared between two things — how far the horizon is from being static, and how far the flux is from being aligned with the $\mathrm{Spin}(7)$ structure. Each is separately bounded by that constant. Spend more on rotation and you have less available for flux misalignment, and vice versa. There is a ledger, and it does not overdraw.
Paper two: a symmetry nobody asked for
A second rotational Killing field on gauged D = 5 vector-multiplet horizons
The second paper is in five dimensions, in a theory with extra fields called vector multiplets whose scalar components are known as moduli — numbers that can vary from place to place and that control the local physics, rather like a set of dials distributed over the horizon.
Five dimensions is where black holes get strange. In four dimensions, a stationary black hole must have a spherical horizon: that is a theorem, and it is why the astrophysical picture is as tidy as it is. In five dimensions this fails. There are black rings — horizons with the topology of a doughnut, $S^1 \times S^2$, discovered in 2001 — which coexist with spherical black holes of the same mass and spin. Uniqueness, the property that made four-dimensional black hole physics so clean, is simply false one dimension up. Whether supersymmetric black rings exist in anti-de Sitter space, which is the setting relevant to the AdS/CFT correspondence and hence to the holographic counting of black hole entropy, has been an open question for years. People have looked. Nobody has found one.
This paper proves two things.
A hidden symmetry. Supersymmetry hands you one rotational symmetry of the horizon essentially for free. The paper shows that on a compact horizon cross-section without boundary there must always be a second rotational Killing field, independent of the first, defined over the whole cross-section. Nothing about this was assumed. No symmetry ansatz was imposed, and no assumption was made about where the frame constructed from the Killing spinors degenerates — which matters, because those degeneration points are exactly where earlier arguments tended to quietly break down.
This continues a long line of “rigidity” results in general relativity, where symmetry that was not put in turns out to be forced by the equations. It is one of the recurring surprises of the subject: horizons are not free-form surfaces, they are extremely constrained ones.
And a no-go. With that second symmetry in hand, the horizon reduces to a manageable classification problem. Where the moduli are constant, you recover geometries already known from earlier work — but now as a conclusion rather than an assumption, which is the stronger logical position. Where the moduli genuinely vary, the geometry is cohomogeneity-one under a two-torus action, and the possible topologies are $S^3$ or $S^1 \times S^2$. The paper then shows that the $S^1 \times S^2$ case — the black ring — is excluded whenever a particular function in the horizon data is not identically zero.
So a supersymmetric anti-de Sitter black ring with varying moduli does not exist, at least on that branch. There is a reason nobody has found one.
There is a further technical point I am pleased about. The whole analysis requires only that a certain superpotential is nowhere zero, which is a weaker hypothesis than the non-negativity of the scalar potential that the earlier literature assumed. Weakening a hypothesis and keeping the conclusion is the quiet, unglamorous way that this kind of result actually advances.
Why negative results are the point
Both papers are, in a sense, about things that cannot happen. That can look like a strange thing to spend ninety pages on.
But black hole classification is precisely the business of narrowing the space of possibilities. In four dimensions, the endpoint of that process is the no-hair theorem: a stationary black hole is completely specified by mass, charge and angular momentum, and nothing else. That single statement is what makes it possible for the Event Horizon Telescope to compare an image against a prediction at all — there is only a two-parameter family of things it could be looking at.
No comparable statement exists in higher dimensions. Rings, spheres and stranger objects coexist. Every no-go theorem shrinks the space; every rigidity theorem says a dial you thought you could turn is welded in place. The eventual goal is a classification as sharp as the four-dimensional one, and it is assembled the way a wall is: one result at a time.
There is also a concrete payoff in the AdS case. The entropy of a black hole in anti-de Sitter space can be counted microscopically using the dual field theory, and the counting has to match the horizon area. Knowing exactly which horizons exist tells you exactly which states the field theory must have. Ruling out the ring is information about the field theory as much as it is information about the geometry.
How the calculations were actually done
Neither paper could have been produced with pen and paper alone. Eleven-dimensional supergravity involves spinors with 32 components and a four-form flux; the Killing spinor equation integrability conditions expand into expressions with thousands of terms, and single sign errors are both easy to make and fatal to the conclusion.
The work was done in Cadabra2, a computer algebra system built specifically for field-theory calculations — it understands abstract indices, spinors, gamma matrix algebra, and the Fierz rearrangements that make these manipulations tractable. I have a companion project that automates the integrability analysis across several supergravity theories, with the derivations reproducible from a driver script rather than reconstructed from a notebook.
I think this matters beyond convenience. A ninety-page calculation that only one person has ever verified is a weaker piece of evidence than one where the derivation runs end to end from a script anyone can execute. Reproducibility is not just for the experimental sciences.
Where this goes next
The obvious continuations: extend the single-point rigidity theorem to other supergravity theories and see whether it is a general feature or an eleven-dimensional accident; close the remaining branch of the five-dimensional classification to make the no-go unconditional; and work out what the $\mathrm{Spin}(7)$ perfect square means for the moduli space of M-theory horizons.
None of this will be tested by a telescope. But the questions the observers are asking and the questions I am asking are the same question approached from opposite ends. They point an array of radio dishes at a horizon and ask what it is. I sit with the equations and ask what it could have been. The answer, increasingly, is: much less than you would think.
The papers
- The warp factor of supersymmetric $D = 11$ near-horizon geometries: single-point rigidity, the $\mathrm{Spin}(7)$ perfect square, and global constraints — arXiv:2608.29879
- A second rotational Killing field on gauged $D = 5$ vector-multiplet horizons, and a no-go for varying-moduli black rings — arXiv:2608.22509
Both are sole-authored. Comments and corrections are genuinely welcome — my full publication list is here, and my CV is here.