[geordief (OP)]

If we can say this ,I am intrigued by the notion of "falling".

When we "fall" towards Earth (or any massive body) are we also "falling through local spacetime"?
Also,do we also fall through spacetime  when no massive bodies affect our movement?

Would  this concept apply to the twin "paradox"? Do both twins "fall" through the same spacetime interval but follow ("fall through"?) different paths?

Could  two hosepipes  with ends that meet at beginning and end serve as anything like a useful analogy?

[Halc]

If we can say this ,I am intrigued by the notion of "falling".

When we "fall" towards Earth (or any massive body) are we also "falling through local spacetime"?
Also,do we also fall through spacetime  when no massive bodies affect our movement?

There is no movement through spacetime.  An object has a worldline through spacetime and its presence at one event is never replaced by its presence at a subsequent event.  The object is in fact present at both events, and no change to spacetime takes place.

Could  two hosepipes  with ends that meet at beginning and end serve as anything like a useful analogy?

Sort of.  The hosepipes could be worldlines, except there is nothing moving through the pipes.  The twins are the hoses, not the contents of them.

[geordief (OP)]

Sort of.  The hosepipes could be worldlines, except there is nothing moving through the pipes.  The twins are the hoses, not the contents of them.

Can we extract three  properties of the hosepipe scenario that correspond to
(1) the spacetime interval (the same for both hosepipes)
(2) The Space constituent of (1)
(3) The Time constituent of (1)               ?

Or does that stretch the analogy too far?

That is what happens to the twins though ,isn't it ? The distance between their (same) beginning and end points is the same although they take different paths through spacetime..

(I think you may have addressed this in the first part of your answer ,so perhaps I am repeating myself)

[Halc]

Sort of.  The hosepipes could be worldlines, except there is nothing moving through the pipes.  The twins are the hoses, not the contents of them.

Can we extract three  properties of the hosepipe scenario that correspond to
(1) the spacetime interval (the same for both hosepipes)

Interval is a measure between events, not of a worldline.  The two can be the same if the worldline is straight, but for an accelerating twin, it isn't. One can still integrate over a curved worldline and get the interval for it.
So you can consider events A, B, and C which are departure, turnaround (at the traveling twin) and arrival.
The interval between A and C corresponds to the worldline of the twin that stays home, and is X.  The interval between A and B equals the interval between B and C, and these two correspond to the 'hose' of the moving twin assuming instant acceleration and same speed both ways.  Thus sum of these two intervals is Y, and Y is less than X.

(2) The Space constituent of (1)
(3) The Time constituent of (1)

These definitely apply to the hoses, but their values are frame dependent. The intervals mentioned above are not frame dependent.

That is what happens to the twins though ,isn't it ? The distance between their (same) beginning and end points is the same although they take different paths through spacetime.

Just so, yes. Their worldlines trace different paths through spacetime, but their beginning and end events (A and C) are the same events.

[geordief (OP)]

That is what happens to the twins though ,isn't it ? The distance **between their (same) beginning and end points is the same although they take different paths through spacetime.

Just so, yes. Their worldlines trace different paths through spacetime, but their beginning and end events (A and C) are the same events.

I thought you just said that the spacetime interval of the stay-at-home twin was greater than that of the travelling  twin .

Events A and C are the same  beginning and endpoints for both but you just said that "The sum of these two intervals is Y, and Y is less than X" and so I deduce that their respective spacetime intervals between the two boundary events are different.

**I meant "spacetime distance",did you perhaps take me to have meant "spatial distance"?

[Halc]

I thought you just said that the spacetime interval of the stay-at-home twin was greater than that of the travelling  twin .

Sorry, I read your comment wrong.  The intervals of each twin is different.  The one at home has a longer interval.

[geordief (OP)]

No problem ,Can I also ask why apparently we can move ,albeit only in relation to another body in Space  but not when  we consider the changes of position in terms of spacetime?

How does one "follow a geodesic " in spacetime without a concept of motion ? Is "follow" to be considered  as a mathematical term restricted to the model? (I was hoping that "motion" would allow me to  use the idea of "falling" but apparently  this is "ruled out" ,although we do talk about "gravity wells" which  might seem to point to some  idea of "falling" )

[Halc]

Can I also ask why apparently we can move ,albeit only in relation to another body in Space  but not when  we consider the changes of position in terms of spacetime?

Motion in spacetime is relative to a reference frame, which is essentially the direction you choose to assign to time.  So you have this arbitrary time line, and objects have worldlines that trace lines relative to that.  If they're parallel to the time line, then the object is stationary in that frame.  If they're at any other angle, they're moving.  If two objects have different angles relative to each other, they're in motion relative to each other, regardless of choice of orientation of the time dimension.  If the worldlines of objects are parallel, they're stationary relative to each other, again, regardless of frame.

How does one "follow a geodesic " in spacetime without a concept of motion ?

One draws a line that is locally straight.  That makes for a straight line in Euclidean space, but spacetime isn't Euclidean, especially at medium scales.  So two straight parallel lines will meet if space has positive curvature.  If you doubt this, consider the space of the surface of Earth which isn't Euclidean.  If you draw two parallel chalk lines on the parking lot, they'll meet eventually in both directions, because they follow a geodesic.

Yes, falling objects trace (follow) a geodesic on curved spacetime, as Einstein modeled it. Others model it as a force with the falling objects curving due to the force, but that's not a spacetime model.

Is "follow" to be considered as a mathematical term restricted to the model?

'Follow' is fine.  I also used the word 'trace' above.  A road follows a route through the mountains, but it doesn't imply that the road actually is moving along that path.  Thus a road makes a pretty good model of a worldline.

(I was hoping that "motion" would allow me to  use the idea of "falling" but apparently  this is "ruled out" ,although we do talk about "gravity wells" which  might seem to point to some  idea of "falling" )

'Falling' typically means not being acted upon by any force other that gravity, and in the spacetime model, gravity is curvature, not a force. So your examples of falling are all fine. Earth is falling, as is the ISS.  But I am not since the chair under my butt is exerting a force on me, and my worldline curving (not following a geodesic) as a result.

[geordief (OP)]

Thanks for all that.There is quite a lot there for me to (hopefully) digest