Curt Jaimungal: General Relativity Is NOT Deterministic (Here's the Proof)

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You've probably heard that Einstein's theory of general relativity, the theory of gravity

is a deterministic theory.

Technically speaking, this is false.

The reason is quite subtle, but many physicists like Penrose and others that have spoken to

know about this intimately.

There's a severe failure of determinism in general relativity that cannot be taken lightly.

Einstein's general theory of relativity is not a deterministic theory.

There is a lack of predictability.

And these go on and on as you'll see and hear over the next few minutes.

Today we'll learn what precisely is GR, so general relativity, beyond the bowling ball

on a rubber sheet, which is actually a circular argument, if you think about it, will also

learn what is determinism precisely, which also has two subclasses, a local and a global

determinism.

And then also, what is this global hyperbolicity that you've heard about or may have heard

about?

It's not just over scrupulous, abstract math, it's actually extremely important and we'll

talk about what these three have to do with one another, of course.

So I'll speak at two levels simultaneously.

One is for the person who wants a rigorous technical definition, that's where all the

sources come in, and another is for those who just want the crux of the argument.

For that latter person, you also have to keep in mind that almost any claim when simplified

is replete with nuances that challenge it.

That's why on this podcast I attempt to be as technical and rigorous as I can, to ameliorate

some of the predicaments of this compressed message, I'll put a link on screen about

my opinions against this whole hey, explain it like I'm five bro, or you don't understand

it.

So why the heck is saying general relativity is deterministic is technically a false

claim that needs to be caveated.

And I'm not speaking about false in some punctilious sense that only philosophers care

about.

So false in that there exist perfectly valid solutions to Einstein's field equations,

where specifying the complete state at one time doesn't uniquely determine the future.

I'm not talking about singularities, although there's that as well.

I'm talking about regular, smooth regions of space time where your future simply isn't

determined yet.

I find this super interesting because unlike quantum mechanics where you at least get

probabilities, in some sense, the indeterminism in gravity in GR is worse.

You've heard of Einstein's field equations, but what is the relationship between that

and GR?

Well, GR general relativity is a fivefold package.

It comprises some theoretical principles like the equivalence principle.

You may have heard it stated informally that physics in a free falling frame is locally

going to be reduced to special relativity or that physics doesn't depend on your coordinate

choice.

Another is the mathematical scaffolding of certain types of Ramani and geometry called

pseudo-ramani and geometry on a four manifold.

And of course, there's the field equations of Einstein and then a geodesic equation that

talks about how free particles move in that straightest possible path you've heard

of in curved space time.

And then there's also the physical interpretation that curvature of space time is gravity.

Not that gravity causes curvature, but that they're identical.

This is why that Wheeler statement is a bit misleading that space time tells matter

how to move and matter tell space time how to curve because it gives the sense that there

is some causal path that ticks forward.

But actually, as you can see, these are dynamically coupled equations.

It's not like there are time steps of causation here.

But anyhow, these form a package deal.

This is all part of general relativity.

You're not just referring to one element here.

I have sub-stack notes on this here, which is what you're seeing on screen.

And this is the coordinate free notation, one that I prefer because I don't like my physics

with arbitrary choices and not these coordinate index gymnastics.

You probably have some intuitive sense of what determinism is.

It's something like, look, if you know everything about the universe right now, whatever that

means, then you can predict everything about the future.

But let's be precise about what physicists and philosophers actually mean by that term.

A theory is deterministic if the complete state of the system at any given time combined

with the laws governing it uniquely specifies all future states.

Now notice the key words, complete states uniquely specifies all future states.

Each of these matters.

So then what counts as the system?

And what if there is no such thing as the complete state at a given time?

This brings us to distinction that most discussions gloss over.

Social determinism is if you specify initial data in a small region of space time, so technically

some open set.

The equations uniquely determine what happens in the immediate future of that region.

I'll put some mathematical jargon on screen here.

Again, this is written in my sub-stack in more detail.

And there are other sources in the description.

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theories of everything, all one word.

Global determinism, on the other hand, is if you specify initial data across the entire

universe at one moment, then the equations uniquely determine the entire future of that

universe.

Then there's some more mathematical jargon about Cauchy surfaces, but it is important,

so I'll just say it's a space-like slice that every causal curve hits exactly once.

The hope is that this uniquely determines the space time thereafter.

You might ask, are these two the same?

The local and the global?

Isn't the global just made up of many locals taken as a totality?

In flat space time, this is super trivial, yes, but it becomes tricky with curve

curvature.

There are solutions to the field equations of Einstein, where you literally can't define

a moment across the entire universe, and it's not because you're not clever enough, it's

that you can prove that they don't exist.

In general relativity, there may not be a way to slice space time into all of space at

time t.

And without that, you can't even formulate what global determinism even means, though

you can retain local determinism, which brings us to our third concept.

You're likely thinking, okay, Kurt, who cares?

Some space times don't have this property that you say about slicing, whatever slicing

means, into a space at a certain time.

Big deal.

Let's just work with the ones that do.

Great.

Okay, that's what physicists often do.

So this condition is called global hyperbalicity.

It means a space time is globally hyperbolic if it admits, so a mathematician's word for

allows for the existence of a koshi surface, which is a space like slice that every inextendable

causal curve intersects it exactly once.

Again, I'm speaking at two levels, one technical, and then to the other probably sounds like

gibberish.

So let me unpack that.

A koshi surface is like a snapshot of the universe at one time.

And what's interesting, when you even just take a polaroid photo of something, that's

a spatial slice, you're not looking at a time slice of something, provided your polaroid

was instantaneous.

You're looking at space.

So this is a spatial slice, and then you can recall those particle whirl lines that

you've seen, every one of those whirl lines, every light ray, they all cross this surface

exactly once.

And I say surface it is slightly more abstract sense.

Now in the polaroid case, that's a 2D surface, that's fine.

But when it comes to space time, you have four dimensions.

So a surface is a hyper surface, you just minus one dimension.

That's our three dimensions of space.

Anyhow, there's no looping back of these whirl lines.

They cross it exactly once.

They don't miss it entirely.

The existence of this guy, this 3D spatial guy, is what even allows you to say the state

of the universe at time t.

Now if your space time is globally hyperbolic, you're golden, you know, there's a theorem

that says if you have initial data on a Cauchy surface like that, then you uniquely determine

the maximal globally hyperbolic development.

Try saying that three times fast.

So what's the problem?

Well, not all solutions to Einstein's equations are globally hyperbolic.

And I'm not talking about exotic mathematical curiosity is that only sadists who read counter-examples

and topology would find pre-possessing.

Some of the most physically interesting space times violate global hyperbolicity.

So charged black holes, for instance, rotating black holes.

Anti-deciter space has a time-like infinity that you can reach in finite time.

Your whirl line just ends.

Gertel universes actually contain close time-like curves where you can travel to your own past.

This is something I referenced in this video I made about misconceptions about girdles

in completeness theorem that somehow went viral.

In these space times, knowing everything about the now actually doesn't tell you everything

about the later.

And not because you're missing information, but because that information literally does

not exist yet.

So what happens?

And some of these, it seems like your future would just stop, but Einstein equations actually

have multiple incompatible slash in equivalent solutions beyond that surface.

It says, if the surface reaches a point and says, I have no idea what's going to happen

next, pick any of these infinite options.

There's no probability distribution.

There's no selection principle.

There's just ambiguity.

See, quantum theory is famous or infamous for its uncertainty principle.

It at least gives you probabilities, though.

GR, which is supposedly a paragon of determinism, can leave you with genuine ambiguity.

In other words, Schrodinger's cat doesn't know if it's alive or dead, but it at least

knows the odds.

And observe, or crossing a Koshi horizon on the other hand, God doesn't even play dice.

He just shrugs.

Now you can see how these three concepts play together.

General relativity has these field equations, and they're perfectly deterministic locally.

And there's a theorem in PDEs about this, which means any small patch of space time

if you know the conditions there, then you can evolve them forward uniquely.

However, when you zoom out to a global picture, there can be problems.

So without caveatting by restricting yourself to global hyperbolicity, you can't even define

what the state of the universe at time t meets.

And when global hyperbolicity fails, such as when there exists Koshi horizons, which are

different than Koshi's surfaces, by the way, then the equations give you various inequival

and answers for what occurs beyond that surface.

Let's take a specific example, let's say the charged black hole, and observer would

see the entire future history of the outside universe compressed into a finite time.

Beyond that point, the Einstein equations become ill-posed as an initial value problem.

Now you can extend the space time, yes, but the cost is that there aren't just many

ways to do it.

There are infinitely many ways to do it.

All of them are equally valid mathematically.

So then you wonder, what the heck would this feel like physically?

I don't know.

I'll let you know when Musk sends some minions there.

As far as I can tell from the safety behind my LCD screen here, information would emerge

from nowhere, and I'm not talking about quantum uncertainty again where at least you get

these born probabilities.

Instead is just new information, appearing without a cause.

You'll often hear some physicists dismiss these examples as pathological or unphysical.

Only some do this though.

Most relativists that I know are sharp enough to realize that there's no rigorous definition

of pathological.

It's basically saying, I don't like this solution, and when I look out my window and

astrophysically, I don't see this.

Now recall, black holes were said to be pathological before.

Even the big bang as a solution Einstein didn't want that.

We can't a priori dismiss something as being unphysical.

Now there are some attempts at rigorous.

So for instance, some of these solutions are unstable, and maybe we just say unstable

solutions are pathological.

Although as I mentioned, pathological seems to be more of the times slash an opinion.

Several universes are unstable under perturbations, under small changes.

You can think of this as, yes, a pencil upside down is a solution to classical physics,

but it's just extremely sensitive.

However, even this objection about unstable as a synonym for pathological has some problems

as you can have Cauchy horizons, which are stable in certain contexts, like charged

black holes with the cosmological constant.

This is work by Cardoso.

To be clear, a Cauchy surface is that initial data that you evolve forward, whereas a Cauchy

horizon is where that data breaks down.

They both happen to have the same name as that guy that you've heard from differential

equation courses.

The Cauchy horizon is the boundary of the region that the Cauchy surface can predict.

So beyond the Cauchy horizon, determinism fails because new information can quote leak

in and quote in a sense from somewhere.

It's all subtle and quite odd.

I talk much more about various sorts of space times and indeterminism with Professor

J.B. Manchak here, so subscribe to get notified for that.

It may already be out in the link is in the description if so.

Then you could also say, well, if there's a solution that violates energy conditions,

then it's pathological, except quantum fields violates these routinely.

Dark energy violates the strong energy condition.

So double oops, then you could say, well, cosmic censorship saves us.

This is Penrose's conjecture that nature censors naked singularities behind event horizons.

It's unproven and has potential counter examples.

He knows this, of course, and I spoke to him about this personally here.

The truth is that we have no principled way to exclude these non-deterministic solutions.

The space of solutions of Einstein's field equations is infinite dimensional.

As far as I know, we don't even have a natural measure to say something like most solutions

are globally hyperbolic.

But then, who cares about global determinism, local determinism is all we need, right?

Well, the problem is that in a space time with closed time-like curves, even local determinism

becomes suspect.

You can have a region where the future loops back to influence the past, and this creates

a consistency condition that constrains your free initial data.

The Gertel universe, as I mentioned before, have closed time-like curves through every

point.

It's quite trippy to see visualizations of this.

Here's one from Busser and Cajari and Schleich.

Hopefully, I'm not mispronouncing their names.

In this Gertel universe, both local and global determinism fail.

Another problem is that if nature allows naked singularities, then information can appear

from the singularity with no prior cause.

This violates determinism in a finite region, not at some abstract infinity.

Think of it like this.

Quantum mechanics, it has indeterminism, yes, but it's domesticated.

It's random, but it's predictable in distribution.

It's like a good boy.

Einstein's general relativity, on the other hand, has genuine indeterminism, which is

feral.

It's either unmeasured, or it's unknowable, or both.

It's like a Tarantonian raccoon.

There's no rules.

There's no remorse.

Most of my best ideas don't happen during interviews.

They come spontaneously, most of the time in the shower, actually, or while I'm walking.

Until I had plot, I would frequently lose them because by the time I write down half

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I tried voice capture before, like Google Home, and it just cuts me off in the middle.

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Look, we've spoken about quantum mechanics and quantum field theory, so perhaps the

hope here is that quantum gravity saves determinism.

After all, these classical solutions wouldn't survive quantization.

Or would they?

Well, let's think about this.

Quantum gravity will still have quantum indeterminacy, so you'll still have indeterministic

gravity.

But even disregarding that, the irony is that many approaches to quantum gravity assume

hyperbilicity from the start.

Global quantum gravity, loop quantum gravity, even many formulations of string theory,

but not all of them, require koshe surfaces to even define the theory.

Technically, yes, world-sheet amplitudes can be computed on non-globally hyperbolic

backgrounds, like girdle spaces or even orbefolds with CTCs.

It's specifically asymmetric formulations and unitary requirements that typically demand

global hyperbilicity, not the world-sheet consistency conditions themselves.

I did a three-hour iceberg into string theory explaining the math in case you were wondering

what the heck that was.

Link's are in the description.

You're correct if you notice that this is like assuming determinism to prove determinism.

Much like how John Norton shows that Newtonian physics actually has indeterminism in it

as well, unless you assume lift-sheets continuity a special condition which amounts to assuming

the very determinism you're attempting to prove.

My view is that by the strict definition of what a deterministic theory is, namely

that we always have a future being entailed uniquely by the past, then the answer is no.

GR is not a deterministic theory as such.

Of course, a more comprehensive answer would be that GR is a theory whose solution space

contains both deterministic and non-deterministic equations.

Furthermore, the physical realizability of these non-deterministic solutions is empirically

under-determined.

It's an open empirical question.

Manchak would say even when you try to formulate what determinism is in GR, the phrase

complete state at any given time has many non-equivalent rigorous translations.

So what can we definitely say?

Number one, the Einstein equations are locally deterministic.

Great.

Number two, global determinism requires global hyperbolicity.

And number three, many physically interesting solutions perfectly valid lack global hyperbolicity.

So perhaps the better way to define determinism isn't a property of theories, but a property

of specific solutions.

And in a universe described by general relativity, whether your future is determined could depend

on where you are in space-time.

Einstein said, God doesn't play dice.

Times out, in Einstein's own theory, God sometimes doesn't even show up to the table.

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