I'll try to keep this understandable, but can expand or ELI5 bits of it if that would help you.
Physically, local flatness is a statement about the local validity of Special Relativity. Practically, a failure of the local validity of Special Relativty -- a Local Lorentz Invariance violation (often abbreviated LLI violation or local LIV or local LV) -- would be apparent in stellar physics and the spectral lines of white dwarfs and neutron stars and close binaries of them. Certainly we haven't been able to generate local LIV in our highest-energy particle smashers, so the Lorentz group being built into the Standard Model is on pretty safe footing.
(For example, we need tests of Special Relativity -- and notably those of the Standard Model, which bakes in the group theory of Special Relativity -- to work for material bodies in free-fall, even if that free-fall is an elliptical path around and close to a massive object. That's everything from our atomic-clock navigation satellites to gas clouds and stars near our galaxy's central black hole or distant quasars.)
It wasn't a piece of math, which would involve writing out an Einstein-Cartan or Palatini action that let one break out the local Lorentz transformations and diffeomorphisms into a mathematical statement, as one can find in modern (particularly post-Ashtekar in the late 1980s) advanced graduate textbooks. Nobody wants that scribbled out in pseudo-LaTeX here on HN. :-)
The choice quote from part 1: "[our] final interpretation says that every spacetime is locally approximately flat in the sense that near any point of any spacetime (or near sufficiently small segments of a curve), there exists a flat metric that coincides with the spacetime metric to first order at that point (or on that curve) and approximates it arbitrarily well," [emphasis mine].
You might prefer to emphasise "approximately" in that quote, but the approximation is much better than that of, say, a square millimetre of your floor.
Next, from a historical perspective: General Relativity was built with making gravitation Special-Relativistic, following Poincaré's 1905 argument about the finite-speed propagation of the gravitational interaction. Einstein (and others) had several false starts marrying gravitation and Special Relativity in various ways before ultimately arriving at spacetime curvature. (At that point, in the 1920s, one finally had the vocabularly to describe Special Relativity's Minkowski spacetime as flat; the Lorentz group theory came later). But making sure Special Relativity didn't break on around Earth -- where it had been tested aggressively for two decades -- was terribly important to Einstein. Additionally, he did not want to break what Newtonian gravitation got right. The mathematics follow somewhat from this compatibility approach where Newtonian gravitation and Special Relativity are correct in the limit where masses are moving very slowly compared to the speed of light and are not compact like white dwarfs or denser objects.
The regions in which there is no hope in many many human lifetimes for finding a deviation from Local Lorentz Invariance are huge (there are interplanetary tests with space probes in our solar system, and interstellar tests using pulsar timing arrays), even if General Relativity turns out to be slightly wrong. This is an area which invites frequent experimental investigation: <https://duckduckgo.com/?t=ffab&q=local%20lorentz%20invarianc...>.
Finally, it is precisely your intuition that big curvature must be built up from small curvature that is the point of investigating local LIV. So far, and to great precision, those intuitions are wrong. Nature builds up impressive spacetime curvature (e.g. in white dwarfs and neutron stars) without showing any signs of softening the local validity of Special Relativity (i.e., the interactions of matter within those compact stars). And that's part of why quantum gravitation is nowhere near decided.
> The choice quote from part 1: "[our] final interpretation says that every spacetime is locally approximately flat in the sense that near any point of any spacetime (or near sufficiently small segments of a curve), there exists a flat metric that coincides with the spacetime metric to first order at that point (or on that curve) and approximates it arbitrarily well," [emphasis mine].
This statement is a mathematical conclusion from GR of a similar nature to noting that for any point on a sphere, there is a map projection onto the plane where distances on the sphere coincide to first order to distances on the plane.
This no more or less means that space-time is locally flat than it means that a sphere is locally flat. To a first order approximation, it is. But when we calculate the curvature tensor, we find that it isn't flat at all.
How do you propose to measure the ultralocal Riemann curvature? I agree with you that it can be there in an exact solution, but I don't really agree that it's necessarily physical (there are many many exact solutions which aren't, for starters). Even with a broader definition of local, a Synge 5-point like process is not going to find a failure of parallel transport in a sufficiently small region. Would you be satisfied with "effectively flat" (in an EFT sense) or even "FAPP flat"?
> Nature builds up impressive spacetime curvature (e.g. in white dwarfs and neutron stars) without showing any signs of softening the local validity of Special Relativity (i.e., the interactions of matter within those compact stars).
I.e., you need to be near huge, dense masses, or on/in them to see LIV violations, but we can't see them from observing those masses.
> And that's part of why quantum gravitation is nowhere near decided.
There are also problems with quantizing curved spacetime.
> You need to be near huge, dense masses, or on/in them to see LIV violations
A sufficient LLI violation in a compact object is likely to lead to a difference in pressure/contact line broadening, thanks to a modified dispersion relation. Ok, there's optical depth issues there, but looking at metal-rich WDs is a start (it gets you to your "or on...them", at least). Neutrino fluxes probably carry some Lorentz-symmetry-related information from multmessenger events too.
Additionally, binaries and multiples might show various equivalence breakdowns if there are LLI violations, with enhanced ellipticities or periastron precessions (by altering orbital polarization and spin precession parameters in the PPN).
But also there are plenty of theories which slightly violate local Lorentz-invariance in the Newtonian limit, bulding up over distances, and PTA data and GRB data are already constraining those.
I'll try to keep this understandable, but can expand or ELI5 bits of it if that would help you.
Physically, local flatness is a statement about the local validity of Special Relativity. Practically, a failure of the local validity of Special Relativty -- a Local Lorentz Invariance violation (often abbreviated LLI violation or local LIV or local LV) -- would be apparent in stellar physics and the spectral lines of white dwarfs and neutron stars and close binaries of them. Certainly we haven't been able to generate local LIV in our highest-energy particle smashers, so the Lorentz group being built into the Standard Model is on pretty safe footing.
(For example, we need tests of Special Relativity -- and notably those of the Standard Model, which bakes in the group theory of Special Relativity -- to work for material bodies in free-fall, even if that free-fall is an elliptical path around and close to a massive object. That's everything from our atomic-clock navigation satellites to gas clouds and stars near our galaxy's central black hole or distant quasars.)
It wasn't a piece of math, which would involve writing out an Einstein-Cartan or Palatini action that let one break out the local Lorentz transformations and diffeomorphisms into a mathematical statement, as one can find in modern (particularly post-Ashtekar in the late 1980s) advanced graduate textbooks. Nobody wants that scribbled out in pseudo-LaTeX here on HN. :-)
Here is an interesting and very slightly contrarian (they do arrive at Theorem 1: it and most of the following text explaining it is beautifully stated orthodoxy -- and note Corollary 4) view by a pair of philosophers of mathematics (they both have also done physics, they are not cranks) at <https://philosophyofphysics.lse.ac.uk/articles/10.31389/pop....> (their rather orthodox part 2 is at <https://philosophyofphysics.lse.ac.uk/articles/10.31389/pop....>).
The choice quote from part 1: "[our] final interpretation says that every spacetime is locally approximately flat in the sense that near any point of any spacetime (or near sufficiently small segments of a curve), there exists a flat metric that coincides with the spacetime metric to first order at that point (or on that curve) and approximates it arbitrarily well," [emphasis mine].
You might prefer to emphasise "approximately" in that quote, but the approximation is much better than that of, say, a square millimetre of your floor.
Next, from a historical perspective: General Relativity was built with making gravitation Special-Relativistic, following Poincaré's 1905 argument about the finite-speed propagation of the gravitational interaction. Einstein (and others) had several false starts marrying gravitation and Special Relativity in various ways before ultimately arriving at spacetime curvature. (At that point, in the 1920s, one finally had the vocabularly to describe Special Relativity's Minkowski spacetime as flat; the Lorentz group theory came later). But making sure Special Relativity didn't break on around Earth -- where it had been tested aggressively for two decades -- was terribly important to Einstein. Additionally, he did not want to break what Newtonian gravitation got right. The mathematics follow somewhat from this compatibility approach where Newtonian gravitation and Special Relativity are correct in the limit where masses are moving very slowly compared to the speed of light and are not compact like white dwarfs or denser objects.
The regions in which there is no hope in many many human lifetimes for finding a deviation from Local Lorentz Invariance are huge (there are interplanetary tests with space probes in our solar system, and interstellar tests using pulsar timing arrays), even if General Relativity turns out to be slightly wrong. This is an area which invites frequent experimental investigation: <https://duckduckgo.com/?t=ffab&q=local%20lorentz%20invarianc...>.
Finally, it is precisely your intuition that big curvature must be built up from small curvature that is the point of investigating local LIV. So far, and to great precision, those intuitions are wrong. Nature builds up impressive spacetime curvature (e.g. in white dwarfs and neutron stars) without showing any signs of softening the local validity of Special Relativity (i.e., the interactions of matter within those compact stars). And that's part of why quantum gravitation is nowhere near decided.