# Are objects accelerating at the same rate in the same direction considered inertial to each other?

• B
jbriggs444
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No, I don't understand what you mean. The "jerk-ometers" would all agree, yet the velocities would differ, creating two different realities, dependent on reference frame. The "jerk-ometers" are invariant because they rely on the geometry of spacetime, which is universal, as you have said.
The velocity is a coordinate dependent quantity. Accordingly, as @PeterDonis points out, no observable quantity depends on it.

• Dale
Gold Member
The velocity is a coordinate dependent quantity. Accordingly, as @PeterDonis points out, no observable quantity depends on it.
If jerk and acceleration can be invariant, in the context of spacetime geometry, then so can velocity, because the derivative of velocity, is acceleration, and the derivative of acceleration is jerk. They are all self-contained within spacetime geometry. Perhaps "proper velocity?"

A.T.
If jerk and acceleration can be invariant, in the context of spacetime geometry, then so can velocity, because the derivative of velocity, is acceleration
The derivative of velocity, is coordinate acceleration, which is frame dependent. What is frame invariant is proper acceleration.

jbriggs444
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If jerk and acceleration can be invariant, in the context of spacetime geometry, then so can velocity, because the derivative of velocity, is acceleration, and the derivative of acceleration is jerk. They are all self-contained within spacetime geometry. Perhaps "proper velocity?"
First you would have to define "proper velocity". Doing it as the integral of proper acceleration over proper time runs into problems -- you would be taking an infinite sum of infinitesimal vectors drawn from different vector spaces.

Ibix
If jerk and acceleration can be invariant, in the context of spacetime geometry, then so can velocity,
To add to A.T.'s comment, velocity turns out to be related to the angle between your worldline (your path through spacetime) and the worldline of whatever you chose as stationary. But proper acceleration (not coordinate acceleration) is a measure of the curvature of your worldline - it has no dependence on any other worldline, so is grame independent.

Gold Member
The derivative of velocity, is coordinate acceleration, which is frame dependent. What is frame invariant is proper acceleration.
First you would have to define "proper velocity". Doing it as the integral of proper acceleration over proper time runs into problems -- you would be taking an infinite sum of infinitesimal vectors drawn from different vector spaces.
Once again, proper velocity might do the trick. "Proper velocity equals velocity at low speeds." That's according to: https://en.wikipedia.org/wiki/Proper_velocity

PeterDonis
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If jerk and acceleration can be invariant, in the context of spacetime geometry, then so can velocity, because the derivative of velocity, is acceleration, and the derivative of acceleration is jerk.
Wrong. Again you are not paying careful attention to the distinction I made in my previous post. See below.

Perhaps "proper velocity?"
Here's the problem: there is no such thing as proper velocity. That is, there is no invariant quantity that is a velocity, corresponding to proper acceleration or its derivative with respect to proper time.

(There are some technicalities involved here, but they are beyond the scope of a "B" level thread.)

jbriggs444
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Once again, proper velocity might do the trick. "Proper velocity equals velocity at low speeds." That's according to: https://en.wikipedia.org/wiki/Proper_velocity
So, what does this have to do with the rate at which sand drops off of accelerating planets?

PeterDonis
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proper velocity might do the trick
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.

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

A.T.
Once again, proper velocity might do the trick.
What trick? It's still frame dependent, just like velocity.

Gold Member
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
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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.

• Ibix
jbriggs444
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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?

• jbriggs444
Gold Member
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
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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.

• Ibix
Dale
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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
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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.

• phinds and Dale