Are objects accelerating together in the same direction inertial to each other?

[A.T.]

Once again, proper velocity might do the trick.

What trick? It's still frame dependent, just like velocity.

 

[jaketodd]

So, what does this have to do with the rate at which sand drops off of accelerating planets?

Nope. The name "proper velocity" for this quantity is misleading; it is not an invariant the way proper acceleration or proper jerk are invariants. Don't rely on Wikipedia as a source for learning physics.

Yes, it is called inertia, and it behaves the way @jbriggs444 described, not the way you described.

Well, on stepping back a moment, it doesn't matter whether we use proper velocity, or coordinate velocity, or whatever. The fact is that inside my three body system, explained in detail above, the planets start moving first, with non-zero jerk, so their acceleration will always be more than the observer, leading to more sand flying off them to the observer, than compared to the reference frames of the two planets. However, the "jerk-ometers" would always read the same, for all three bodies.

 

[Dale]

If jerk and acceleration can be invariant, in the context of spacetime geometry, then so can velocity,

This is incorrect. Since you mention geometry, let’s discuss the geometrical meaning of these concepts.

In spacetime, velocity is the slope of a worldline, and proper acceleration is its curvature. Different frames are different rotations of the axes.

If you draw a curve on a piece of paper then at each point along the curve there is a certain radius of curvature and a certain slope. If you rotate the paper the curvature at each point is unchanged, but the slope is changed.

So, geometrically speaking, you are wrong. The invariance of proper acceleration does not imply the invariance of velocity.

 

[jbriggs444]

The fact is that inside my three body system, explained in detail above, the planets start moving first, with non-zero jerk, so their acceleration will always be more than the observer, leading to more sand flying off them to the observer, than compared to the reference frames of the two planets. However, the "jerk-ometers" would always read the same, for all three bodies.

The relativistic invariant is the total quantity of sand that flies off between some defined starting event and some defined ending event.

You appear to be speaking of the rate at which sand flies off at a particular coordinate time -- and then starting the coordinate time clocks at different times.

 

[Ibix]

leading to more sand flying off them to the observer

I have no idea how you reach this conclusion. The observer is utterly irrelevant to the physics of sand coming off an accelerating planet. Why would you think what the observer is doing could have any effect?

 

[jaketodd]

This is incorrect. Since you mention geometry, let’s discuss the geometrical meaning of these concepts.

In spacetime, velocity is the slope of a worldline, and proper acceleration is its curvature. Different frames are different rotations of the axes.

If you draw a curve on a piece of paper then at each point along the curve there is a certain radius of curvature and a certain slope. If you rotate the paper the curvature at each point is unchanged, but the slope is changed.

So, geometrically speaking, you are wrong. The invariance of proper acceleration does not imply the invariance of velocity.

I'm talking about acceleration, and jerk. I've shown how acceleration can vary, depending on reference frame, while jerk is invariant.

 

[jbriggs444]

I'm talking about acceleration, and jerk. I've shown how acceleration can vary, depending on reference frame, while jerk is invariant.

You have not shown how proper acceleration can vary depending on reference frame. It does not. We all agree that coordinate acceleration can vary.

 

[Ibix]

I'm talking about acceleration, and jerk. I've shown how acceleration can vary, depending on reference frame, while jerk is invariant.

No you haven't. Coordinate acceleration and coordinate jerk are frame dependent. They're related to the change of angle between a worldline and some chosen reference worldline, so depend on the choice of reference worldline. Proper acceleration and proper jerk are related to the curvature of the worldline (edit: loosely, the angle a worldline makes with itself as it was a moment earlier) and rate of change thereof, so are invariant.

 

[A.T.]

I've shown how acceleration ...

Without specifying what type of acceleration you mean, it's not clear what you are talking about.

 

[Dale]

so their acceleration will always be more than the observer,

Certainly. Their acceleration will always be more than the observer’s acceleration. However, since it is proper acceleration and since that is invariant then at any point on the planet’s worldline the planet and the observer will agree on the value of that acceleration, and hence on the amount of sand falling off. The observer’s acceleration is simply not relevant

I've shown how acceleration can vary, depending on reference frame, while jerk is invariant.

Proper acceleration is invariant, as is its derivative wrt proper time (jerk).

 

[PeterDonis]

on stepping back a moment, it doesn't matter whether we use proper velocity, or coordinate velocity, or whatever

Yes, it does. You continue to ignore the key fact that has already been explained to you several times: any actual observable must depend only on invariant quantities. Your scenario does not disprove this at all; your repeated attempts to show that it does only show that you still don't understand the key fact I just stated. You need to stop trying to convince anyone else of your claims, and take a step back and consider that key fact carefully, and continue doing that until you understand what it means.

At this point I am closing the thread since you are continuing to repeat your errors without responding to the corrections you have been given.