Driving Toward Infinity
Drive Toward Infinity
What if singularities aren’t failures of physics, but clues about what comes next?
So apparently I wasn’t finished.
I published a piece about the recent Navier–Stokes singularity result and ended up going down a particular rabbit hole.
The basic thought was this:
If an extraordinarily successful physical theory predicts infinity, perhaps the interesting question isn’t:
How do we get rid of the infinity?
Maybe it’s:
What assumption broke immediately before we got there?
For Navier–Stokes, this gives us a rather nice experiment.
The equations treat fluids as continuous.
But fluids aren’t continuous.
They’re made of molecules.
So if Navier–Stokes constructs some increasingly violent little vortex and eventually says:
$$u\rightarrow\infty$$
we don’t actually have to follow it all the way to infinity.
Stop before.
Then closer.
Then closer again.
At each point, compare the continuum prediction against kinetic theory or an actual molecular simulation.
Eventually, presumably, the molecules disagree.
And that disagreement might tell us exactly what piece of microscopic physics the continuum equation has stopped capturing.
I liked this idea.
Quite a lot, actually.
Then about ten minutes later another thought arrived.
Oh.
Fuck.
Black holes.
We’ve seen this movie before
General relativity is one of the most successful theories humanity has ever produced.
Einstein writes down:
$$G_{\mu\nu}+\Lambda g_{\mu\nu}=\frac{8\pi G}{c^4}T_{\mu\nu}$$
and spacetime becomes dynamical.
Matter tells spacetime how to curve.
Curved spacetime tells matter how to move.
And somehow this beautiful geometric idea goes on to predict things that sound completely insane and then turn out to exist.
Gravitational lensing.
Gravitational time dilation.
Black holes.
Gravitational waves.
GPS satellites quietly correcting their clocks because Einstein would otherwise have you driving into the wrong street.
The theory works.
Ridiculously well.
And then you follow it into a black hole.
And eventually…
infinity.
Again.
First: the event horizon is not the singularity
This distinction matters.
A lot.
The event horizon is basically a causal boundary.
Cross it and you can no longer send information back outside.
For a sufficiently large black hole, there needn’t be some dramatic local event as you cross the horizon.
No wall.
No flashing sign.
No little booth saying:
WELCOME TO THE BLACK HOLE
RETURNS NOT ACCEPTED
The actual classical pathology lies deeper inside.
For the simplest non-rotating black hole, the Schwarzschild solution has a metric that can be written as:
$$ds^2=-\left(1-\frac{2GM}{rc^2}\right)c^2dt^2+\left(1-\frac{2GM}{rc^2}\right)^{-1}dr^2+r^2d\Omega^2$$
At
$$r=\frac{2GM}{c^2}$$
some parts of these coordinates look badly behaved.
But that’s not a physical singularity.
Change coordinates and the apparent disaster disappears.
The event horizon survives.
Spacetime is perfectly capable of continuing through it.
But keep going toward
$$r\rightarrow0$$
and something genuinely nastier happens.
Curvature invariants blow up.
For the Schwarzschild black hole, for example, the Kretschmann scalar behaves as:
$$R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}=\frac{48G^2M^2}{c^4r^6}$$
So:
$$r\rightarrow0\quad\Rightarrow\quad R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\rightarrow\infty$$
Now we have an actual physical problem.
Not merely ugly coordinates.
General relativity is telling us that spacetime curvature itself becomes unbounded.
And once again nature apparently hands us:
∞
which should immediately make us suspicious.
Maybe infinity isn’t the answer
This is where the Navier–Stokes thought came rushing back.
With fluids, the continuum equation says:
smaller
↓
smaller
↓
smaller
↓
smaller
↓
∞
But eventually reality says:
Mate. Molecules.
There simply isn’t an infinitely divisible fluid underneath the continuum description.
So perhaps the Navier–Stokes singularity isn’t a place where water achieves infinite velocity.
Perhaps it’s where the mathematical language of a continuum keeps talking after the physical system has stopped answering in continuum.
Now look at gravity:
ordinary curvature
↓
stronger curvature
↓
stronger
↓
stronger
↓
∞
And the obvious question becomes:
What if spacetime is doing the same thing?
Maybe asking what classical spacetime does at arbitrarily tiny scales is like asking what the velocity field of water is between two molecules.
Perhaps the problem isn’t that we don’t know the answer.
Perhaps eventually the question itself stops making physical sense.
Except there’s one rather enormous difference
For Navier–Stokes, we know what’s underneath.
Molecules.
That’s almost embarrassingly convenient.
We can say:
NAVIER–STOKES
|
↓
kinetic theory
|
↓
atoms and molecules
We know the microscopic theory.
Or at least enough of it to run the experiment.
With gravity:
GENERAL RELATIVITY
|
↓
?
|
↓
?
|
↓
quantum gravity?
Ah.
Slight problem.
We don’t know what spacetime is made of.
We don’t even know whether “made of” is the right phrase.
There are plenty of candidates and ideas.
Strings.
Loops.
Spin networks.
Holography.
Causal sets.
Asymptotic safety.
Emergent spacetime.
And various approaches where asking what spacetime is “made of” may itself be embarrassingly classical thinking.
But there is no equivalent of saying:
Fine. Fire up the molecular dynamics simulator.
We don’t possess the accepted microscopic theory.
Which initially seems to kill my analogy.
Except…
I think it makes the singularity even more interesting.
Run the experiment backwards
With Navier–Stokes, our proposed experiment is:
$$\text{known microscopic physics}\rightarrow\text{discover correction to macroscopic theory}$$
But perhaps gravity lets us ask the inverse question:
$$\text{known macroscopic theory}\rightarrow\text{singularity}\rightarrow\text{infer constraints on microscopic theory}$$
In other words:
Don’t ask what happens at the singularity.
Ask what must begin happening before it.
Suppose general relativity is an effective description of some deeper structure.
Then we expect GR to work beautifully when curvature is small compared with whatever fundamental scale governs the deeper theory.
And then, eventually, corrections appear.
Schematically:
$$G_{\mu\nu}+\epsilon H_{\mu\nu}=\frac{8\pi G}{c^4}T_{\mu\nu}$$
Far away from extreme curvature:
$$\epsilon H_{\mu\nu}\approx0$$
and Einstein wins.
As he should.
But perhaps as curvature approaches some fundamental scale:
$$\ell_P^2R\sim1$$
the correction stops being negligible.
Now the question becomes:
What can (H_{\mu\nu}) possibly be?
And suddenly the singularity becomes a constraint generator.
Don’t patch Einstein
This is exactly the trap we talked about with Navier–Stokes.
You can always modify an equation so it doesn’t blow up.
That’s easy.
Add enough smoothing.
Introduce a cutoff.
Make gravity repulsive at high curvature.
Put a little mathematical traffic cone around infinity.
Congratulations.
You have successfully prevented the thing you designed your equation to prevent.
That tells us almost nothing.
The interesting correction has to survive everywhere else.
If we modify Einstein’s equations, the new theory still needs to reproduce all the places where Einstein was spectacularly right.
Weak-field gravity.
Planetary orbits.
Gravitational lensing.
Binary pulsars.
Gravitational waves.
Black-hole exterior geometry.
Cosmology.
Every precision test we can throw at it.
So we’re looking for something rather peculiar:
ordinary universe
correction ≈ NOTHING
↓
stronger gravity
correction ≈ STILL NOTHING
↓
black-hole exterior
correction ≈ BASICALLY NOTHING
↓
extreme curvature
correction wakes up
↓
classical singularity
correction says:
"Nope."
That is an enormously constrained search space.
Which is good.
Physics loves constraints.
And black holes have already been leaking clues
This is where the story gets much stranger than the fluid analogy.
Black holes already behave as if they know about microscopic physics.
In the 1970s, Jacob Bekenstein realized that black holes appear to have entropy.
Then Hawking showed that quantum fields around a black hole imply that black holes have a temperature.
And eventually we get the extraordinary relation:
$$S_{\mathrm{BH}}=\frac{k_Bc^3A}{4G\hbar}=\frac{k_BA}{4\ell_P^2}$$
Look carefully at that.
The entropy of the black hole is proportional to its surface area.
Not its volume.
Area.
Which is weird.
If I give you a box of gas, you’d normally expect the number of microscopic degrees of freedom to scale roughly with the volume of the box.
Make the box twice as large and you can fit more stuff inside.
Reasonable.
But black holes seem to be telling us:
Nope. Count the boundary.
That observation eventually became one of the great clues behind holographic thinking in quantum gravity.
Maybe the information contained in a region of spacetime doesn’t work anything like our naïve picture.
Maybe volume isn’t fundamental.
Maybe locality isn’t fundamental.
Maybe spacetime itself isn’t fundamental.
We don’t know.
But the black hole is clearly trying to tell us something.
And, as usual, we’re staring at the equation going:
AREA?
...why the fuck area?
Then Hawking made everything worse
Quantum mechanics says information shouldn’t simply disappear.
General relativity says things can cross an event horizon and never communicate with the outside again.
Hawking then tells us that black holes radiate.
Slowly.
Very slowly.
And eventually, at least in the semiclassical picture, can evaporate.
So now ask:
Where did the information go?
Welcome to the black-hole information problem.
This isn’t some annoying technicality sitting beside the singularity problem.
It’s another place where our two enormously successful descriptions of nature —
quantum mechanics
and
general relativity
— start giving us answers that are deeply uncomfortable when forced into the same room.
So black holes are basically physics’ most productive argument.
GENERAL RELATIVITY
"Spacetime is geometry."
↕
QUANTUM MECHANICS
"Information is unitary."
↕
THERMODYNAMICS
"Excuse me, that object has entropy."
↕
BLACK HOLE
*sits quietly*
And somewhere inside this mess is presumably the microscopic description of spacetime.
So let’s steal our Navier–Stokes idea
Imagine we could follow an infalling observer.
Not merely across the horizon.
Further.
Into increasingly strong curvature.
At each stage we calculate every invariant we can think of.
Curvature.
Tidal forces.
Quantum-field fluctuations.
Entanglement.
Stress-energy.
Entropy.
Correlation functions.
Whatever survives as meaningful physics.
Then instead of asking:
$$\text{What happens at }r=0?$$
ask:
$$\boxed{\text{Where should classical GR first stop being sufficient?}}$$
That’s a different question.
And I suspect a much more useful one.
The singularity itself is the endpoint of the failure.
I want the first deviation.
The equivalent of our molecular stress residual in the fluid problem.
For Navier–Stokes we imagined:
$$\Delta\sigma=\sigma_{\mathrm{MD}}-\sigma_{\mathrm{NS}}$$
For gravity we don’t yet have the microscopic answer to subtract.
But perhaps we can search for the structure of the residual indirectly.
What corrections are compatible with everything we already know?
What kinds of new degrees of freedom could appear?
Which modifications preserve known symmetries?
Which ones maintain causality?
Which ones reproduce black-hole thermodynamics?
Which ones recover Einstein at ordinary curvature?
Which ones resolve rather than merely hide the singularity?
And which make predictions somewhere else?
Because again:
somewhere else is the test.
This is where AI becomes interesting again
And no, I don’t mean:
Ask ChatGPT to invent quantum gravity.
Please don’t.
I mean something much closer to what happened with the Navier–Stokes mathematics.
Use AI as an enormous search engine over mathematical possibility.
Suppose we specify brutal constraints:
recover GR at low curvature
preserve covariance
respect conservation laws
reproduce known black-hole solutions
remain compatible with gravitational-wave data
reproduce black-hole thermodynamics
behave sensibly semiclassically
avoid pathological ghosts / instabilities
regularize extreme curvature
make falsifiable predictions
Then let enormous populations of mathematical agents explore candidate effective theories.
Most will be garbage.
Excellent.
Kill them.
Thousands more.
Kill those too.
What remains?
Maybe nothing.
Maybe known effective-gravity corrections reappear.
Maybe families of higher-curvature theories emerge.
Maybe the constraints reveal that certain apparently attractive ideas are impossible.
Or maybe the search discovers mathematical structures humans haven’t seriously explored.
The point isn’t to ask AI for the answer.
The point is to use AI to explore a mathematical space so enormous that no collection of humans could exhaustively walk through it.
Exactly the sort of thing these systems are becoming rather good at.
But don’t let the AI cheat
This part matters enormously.
If the objective function is:
MAKE SINGULARITY GO AWAY
then the machine will make the singularity go away.
Wonderful.
So will deleting the equation.
The objective needs to be much nastier.
Something like:
$$\text{new theory}=\text{GR everywhere GR works}+\text{minimal new physics where GR fails}$$
And then we punish complexity.
Hard.
If some correction requires forty-seven arbitrary fields, eighteen tunable constants and a small goat sacrifice every Tuesday…
perhaps keep searching.
Nature has surprised us before.
But simplicity remains a useful knife.
And most importantly:
the candidate theory needs to predict something we didn’t use to construct it.
Otherwise we’re just curve-fitting the universe.
There’s another possibility that I find much more disturbing
What if there is no correction to Einstein’s equation?
At least not in the way we’re imagining.
What if the equation itself is emergent?
Think about temperature.
Temperature is real.
You can measure it.
It has equations.
It predicts things.
But asking:
What is the temperature of one molecule?
is already starting to misuse the concept.
Temperature emerges from a collective description of many microscopic degrees of freedom.
Likewise fluid velocity.
A molecule has velocity.
But the smooth field
$$u(x,t)$$
is a coarse-grained object.
Now imagine spacetime itself is like that.
The metric
$$g_{\mu\nu}(x)$$
might be the gravitational equivalent of temperature.
Perfectly real.
Perfectly measurable.
Extraordinarily useful.
But emergent.
Then asking what happens to
$$g_{\mu\nu}$$
at arbitrarily tiny scales might be like asking about the temperature field at half the diameter of an atom.
There isn’t necessarily a better value.
The variable itself has stopped being fundamental.
And suddenly the singularity looks very different.
Not:
spacetime becomes infinitely curved.
But:
the concept of classical spacetime has been extrapolated beyond the scale where it exists.
That possibility is wild.
And entirely different from merely adding another term to Einstein’s equations.
Which gives us a bigger pattern
We started with fluids.
Now we’re talking about spacetime.
And I’m beginning to wonder whether singularities should be treated as a general research tool.
Whenever a spectacularly successful effective theory says:
$$\text{something}\rightarrow\infty$$
don’t immediately ask:
How do we regularize this?
Ask:
What assumptions allowed infinity to appear?
Then drive toward the singularity.
Not into it.
Toward it.
Watch the hierarchy of assumptions.
See which one becomes questionable first.
For fluids:
continuum
↓
local equilibrium
↓
constitutive relation
↓
molecular physics
For gravity perhaps:
smooth spacetime
↓
local quantum fields
↓
semiclassical gravity
↓
???
↓
quantum gravity
The infinity may be less interesting than the road leading to it.
There is something wonderfully arrogant about infinity
Infinity usually appears because our mathematics allows us to keep asking the same question forever.
Smaller?
Yes.
Smaller?
Yes.
Smaller?
Yes.
Again?
Sure.
Again?
Of course.
Again?
Mathematics doesn’t care.
Nature might.
Nature contains scales.
Atoms.
Quantum wavelengths.
Interaction lengths.
Relaxation times.
The speed of light.
Planck’s constant.
Perhaps the Planck length.
Perhaps something deeper still.
At some point reality may simply refuse the premise of the next question.
And our equation — dutiful little bastard that it is — keeps calculating anyway.
Until eventually it prints:
$$\infty$$
Maybe that’s not nature giving us an answer.
Maybe it’s nature’s equivalent of a stack trace.
So I’d like to revise my position on singularities
I used to think of them mostly as ugly endpoints.
Places where the equations break and physics shrugs.
Now I’m beginning to think that framing is backwards.
A singularity may be one of the most informative things an effective theory can give you.
It’s telling you:
Follow this road and eventually one of my assumptions becomes impossible.
Wonderful.
Which one?
Where?
At what scale?
What observable changes first?
What deeper theory would reproduce everything before that point while naturally taking over afterward?
Those are experimentally and mathematically interesting questions.
The singularity is just the giant red arrow.
And yes, there is a research programme hiding in here too
Take families of black-hole interior solutions approaching extreme curvature.
Do not obsess over the formal endpoint.
Map the approach.
Identify the hierarchy of scales where known effective quantum corrections become relevant.
Generate broad classes of generally covariant corrections.
Constrain them against every successful test of GR.
Demand consistency with black-hole thermodynamics and quantum theory.
Use symbolic mathematics, formal verification and AI-assisted search to explore the surviving theory space.
Then look for predictions away from the singularity.
Gravitational-wave ringdowns.
Black-hole merger dynamics.
Early-universe cosmology.
Primordial black holes.
Tiny deviations in horizon-scale physics.
Anything observable.
Because if your theory only explains the place from which no information can reach us…
that’s rather inconvenient.
The correction becomes interesting when it leaves fingerprints somewhere we can actually look.
Drive toward infinity
So now I apparently have a philosophy.
This is what happens when I am allowed near a keyboard unsupervised.
When a theory predicts infinity:
Don’t worship it.
Don’t panic about it.
Don’t immediately patch it.
Drive toward it.
Measure everything along the way.
Ask which assumption fails first.
With Navier–Stokes, perhaps the molecules can tell us.
With black holes, we don’t yet know what lies underneath spacetime.
But maybe the singularity is precisely the clue we need to start asking the right question.
Because perhaps singularities aren’t places where nature becomes infinite.
Perhaps they’re places where our equations keep going…
after reality has changed languages.