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.
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?
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.