> It seems like this is just the “you need to study the problem and it’s dependencies for many years to understand the physics and mathematical details” problem. Which does not actually seem like much of a problem to be solved, just a reality.

Which is exactly the "draw the rest of the owl" problem. Something that takes substantial time, skill and knowledge to achieve is demonstrated as "three easy steps."

I wonder if we’re in agreement?

The key word is ‘numerical’.

Maybe you’ve yet to see or do numerical analysis of both general relativity and quantum field theory but they are in fact both linear algebra.

Which as you point out is “fundamentally more accessible”.

As a physicist who used to work on unifying QFT and gravity, no, QFT is not accessible at all, to anyone. General relativity is kind of accessible to a non-physics student if they otherwise have a strong math background, which is like 0.001% of the population, optimistically.

All these topics are so advanced that even for someone trained in one it would take years to become proficient in the other.

Source: I'm a Physics PhD

Fun fact: When I was in a highly respected grad school in the 1980s (multiple Nobel Prize winners in the department), the QFT course was presented in such an obscure way, I stopped attending lectures, and I doubt anyone passed. The next year under "QFT" they taught Ashcroft -- i.e. a Solid State Physics course. That way requirements were met.

I took QFT in college and then grad school in two of the leading institutions in HEP. I can’t say the instruction was very enlightening in either place. Whichever path you take (not sure what you had in the 80s, I’m thinking of Peskin & Schroeder / Schwartz / Weinberg, maybe even supposedly easier textbooks like Nutshell) it’s just a very dense subject.

Now, GR or at least the core of it is a lot easier as long as you’re mathematically prepared for differential geometry.

Btw, I took particle physics from the Peskin in Peskin & Schroeder when I was an undergrad. Everyone in the class (like four of us by the third week?) was super duper lost by midterm.

yeah like this for anything. String theory is easy if you're mathematically prepared by being a field's medalist

Having attended many IAS talks, I don’t think string theory is easy for the likes of Ed Witten.

0.001% is already about 800_000 people :-)

80,000. And imagine, QFT is even harder than percentages!

I wonder how many people on earth can compute 8_000_000_000 * 0.001% 'cos obviously I'm not in that part :-)

General relativity is not linear algebra. Just glancing at the underlying equations, you can see they are non linear. [https://en.wikipedia.org/wiki/Einstein_tensor]

The same is true for quantum field theory. I'm curious why you're so confident that they are "in fact both linear algebra"

they involve matrices

They also involve addition, but that doesn't make them arithmetic. Knowing linear algebra gets you perhaps 10% of the way to having the math background for a first course in GR.

General relativity is linear algebra? Has anyone called up Tullio Levi-Civita and told him that his services will no longer be necessary?

I’ve seen this sort of reactionary response in mathematics and music as well.

When talking to a musical educator complaining about struggling students just starting out I’ll say “I found it extremely helpful to paint the C major scale on my guitar when learning music theory.”

And they respond “Well what about chromatic 12 pitch atonal music?! What would Anton Webern say?!”

As someone who has studied GR (at the grad school level) and music, no, that's a terrible analogy.

In GR terms, Levi-Civita is more like "This is a major scale and you need to know it" than serialism.

Some subjects are just plain hard and only accessible to smart persistent people.

GR and QFT definitely qualify - but they seem to be warm up exercises compared to whatever Quantum Gravity will eventually become.

Eh, no. The problem with your analogy is that the “not linear algebra” bits aren’t esoteric, rarely practiced or used edge cases: they’re the core of the mathematical model.

I guess you are mistaking the “local linear nature” of tensors for differential geometry. GR is certainly not “linear algebra”.

>Maybe you’ve yet to see or do numerical analysis of both >general relativity and quantum field theory

I call you out. Don’t think you’ve done either one after reading your comments.

No, just because general relativity and quantum field theory use linear algebra does not mean they are linear algebra.

Numerical relativity is not just or even mostly linear algebra. Even in the weak-field limit, where linearized GR is qualitatively correct, you still have to actually solve the resulting PDE.

They're both what now?

A tensor is a differential structure with pointwise linear properties. But the important thing is not the pointwise behavior but the differential nature of the “family of tensors in non-Euclidean space”.

This is too strong. There are important fields whose points are not tensors (e.g. spinor fields) and treating them like tensor fields risks introducing all sorts of subtle confusions.

Of course you are right but I was just trying to explain to OP that certainly General Relativity is not just “linear algebra”.

That's very weak