First, the idea of describing it that way comes from Veritasium. Take complaints to him. See https://www.youtube.com/watch?v=XRr1kaXKBsU for the video where he does it.

Yes I know it is handwavy and misleading. But I consider it less misleading than most attempts at visualizing it.

> Ok, how does this sketch work for a low-ellipticity eccentric orbit?

At what point in the orbit does it not work as a description of what's going on locally where the orbiting body is?

> Sure, it's a set of events in a region of the whole spacetime. If we take "Big Bang" colloquially enough to include the inflationary epoch, always assuming GR is correct, then at every point in that "Big Bang" region of the whole spacetime there is a small patch -- a subregion -- of exactly flat spacetime. However, these small patches must be small because most choices of initially-close pairs of test objects can only couple to timelike curves that wildly spread in one direction (and focus in the other).

No, there is no requirement of any region of locally flat spacetime existing. It is required (outside of singularities) that, when measuring things to first order, things are flat. However in curved space-time, the curvature can be theoretically revealed in any region, no matter how small, by measurements that are sufficiently precise to show the second order deviations from flatness that we call curvature.

> I don't know how to understand your two final sentences: how do you connect the period just before the end of inflation and the expansion history during the radiation and matter epochs?

I'm just referring to the fact that the Hubble parameter is believed to have been higher in the early universe than it is today. I'm not referring to periods such as the hypothesized inflation where the behavior is not described by GR.

> See https://www.youtube.com/watch?v=XRr1kaXKBsU for the video where he does it.

I'd be grateful if you take me to the time stamp, because in a casual watch of the video there was nothing like your:

  Gravity pulls things in by causing space-time
  to accelerate in a particular direction. In
  other words we accelerate towards the Earth
  at 9.8 meters per second per second
  because that is what space-time itself does.

The closest thing I noticed was just after the 10 minute mark, where he points at Christoffel symbols and essentially says that you can choose a set of accelerated coordinates such that to remain at the spatial origin you have to undergo proper acceleration. "Your acceleration must be equal to this curvature term ... in curved spacetime you have to accelerate just to stand still". Which is totally fine, and even finer if he made it clear that you are "standing still" at some spatial coordinate after an arbitrary splitting of spacetime into space + time. But I don't know how or if those approx 90 seconds connect to what I quoted from you above.

(Even finer still if he removed the coordinates and completed the equation: see e.g. slide 20/50 at https://slideplayer.com/slide/12694784/ - his whiteboard is the term labelled in cyan, adapted. approx 10m45s "You don't have to worry about the details here. The point is...". The point is that I'm not his intended audience, and his presentation is fine enough, so there's no good reason for me to take you up on your suggestion to contact him.)

> At what point in the orbit does it not work as a description of what's going on locally where the orbiting body is?

I don't know that it doesn't work - it's just that to me it's such an odd way of putting it that the consequences of your "describing it that way" are unclear to me. An obvious probe is solving for an eccentric orbit.

Shoot a slower timelike observer on a ~secant line hyperbolic trajectory across the quasicircular orbit, comparing proper-time-series accelerometer and chronometer logs from their first kiss before the latter's periastron to their last kiss after. How does your "accelerated spacetime" vary by position and initial velocity? How does it work as we take v->c?

I don't know enough physics to know whether the parent knows what they are talking about, but there is one piece of math that makes me think they do not.

> at every point in that "Big Bang" region of the whole spacetime there is a small patch -- a subregion -- of exactly flat spacetime

Can explain how you get a non-empty region of exactly flat space time around every point?

Patching together curved things out of not curved things happens all of the time. The Earth looks flat around the point you are standing. I'm worried that just because it looks flat in my city doesn't mean it is actually flat in my city, if I measure carefully.

> I don't know enough about physics

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.

> 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.