Why are acceleration, jerk, etc not relative, just like velocity?

[Ahmed1029]

TL;DR Summary

It's taken as a axiom that there are inertial systems that experience no acceleration, but a reference frame can only measure the accelerations of other bodies, while the other bodies will only measure its acceleration with respect to themselves, so how can this measuring device know it's accelerating absolutely when it can only compare accelrations?

I've been thinking about this for a while, and thought it would be nice if someone could guide me to an answer.
In Newtonian mechanics, an inertial frame is coordinate system that's able to make measurements with respect to some imaginary axes attached to it.
It's a well known fact that velocity is relative, but why not all higher derivatives of the position vector? A measuring device can never tell if it's moving at all, accelerating, jerking, etc. It's a point after all, one that than can make measurements.
You might say it's going to measure ficticious forces, but I'd argue that if you cannot know discern absolute acceleration, how are you sure that the forces you measure are the right ones?

 

[PeroK]

For Newtonian physics we have the Galilean transformation between inertial reference frames.$$x' = x - vt$$ Where $v$ is the relative velocity between the frames. A body with trajectory in one IRF might be:$$x = x_0 + u_0 t + \frac 1 2 at^2$$Which transforms to $$x' = x +vt = x_0 + (u_0 - v)t + \frac 1 2 at^2$$And we see that the acceleration $a$ is the same in both frames.

This is the basis of Newton's second law, $F = ma$, which can only be valid in all IRFs if force, mass and acceleration are invariant across all IRFs.

 

[Nugatory]

if you cannot know discern absolute acceleration, how are you sure that the forces you measure are the right ones?

Ah, but we can measure absolute (the more standard term is "proper") acceleration.

Proper acceleration is measured with an accelerometer. Imagine a box containing a weight suspended by springs from all six inside faces; any proper acceleration of the box will measurably stretch the springs. The fictitious acceleration produced by fictitious forces will not stretch the springs.

 

[Ahmed1029]

Ah, but we can measure absolute (the more standard term is "proper") acceleration.

Proper acceleration is measured with an accelerometer. Imagine a box containing a weight suspended by springs from all six inside faces; any proper acceleration of the box will measurably stretch the springs. The fictitious acceleration produced by fictitious forces will not stretch the springs.

Perfect! Thank you!

 

[DrGreg]

In relativity, there is a difference between proper acceleration, which can be measured directly by an accelerometer (as described by Nugatory) without any knowledge of coordinates, and coordinate acceleration ($\text{d}^2 x / \text{d}t^2$) which can vary from one coordinate system to another. There are formulas to calculate proper acceleration from coordinate acceleration and coordinate velocity, but the answer you get for the proper acceleration is the same in every coordinate system.P.S. What PeroK showed in post #2 was that in Newtonian physics, which is a low-speed approximation to relativity, proper acceleration is the same as coordinate-acceleration-relative-to-any-inertial-frame.

 

[FactChecker]

You should always distinguish between the second derivative, which is relative, versus the physics "acceleration", $a$ of $F=ma$, which is always associated with a force.
This is especially true for a rotating coordinate system. Suppose you attach a "pilot's head" coordinate system to the head of a fighter pilot with the X-axis pointing where he faces and the Y-axis out his right ear. If he is going 500 mph forward and then rotates his head left in one second, then the X velocity goes from 500 to 0 in one second and his Y velocity goes from 0 to 500 in one second. So the second derivatives are huge even though there is no force and nothing has really happened.

 

[Dale]

a reference frame can only measure the accelerations of other bodies,

This isn’t true. The acceleration of a reference frame itself can be measured by an accelerometer which is at rest with respect to the frame.

It's a well known fact that velocity is relative, but why not all higher derivatives of the position vector?

I don’t think that we have a good “why” for this. It is an observed fact that we incorporate into our models. We can show how this fact is encoded into the models, but not really explain why those models work and not others.

In the current models this is encoded in the fact that spacetime is represented as a manifold. Locally there is no preferred timelike direction, so all straight worldlines are equivalent. In contrast, worldlines that are not straight are distinguishable by how tightly they curve and any change in direction of curvature.

A measuring device can never tell if it's moving at all, accelerating, jerking, etc.

Accelerometers are measuring devices that can tell if they are accelerating, jerking, etc.

You might say it's going to measure ficticious forces, but I'd argue that if you cannot know discern absolute acceleration, how are you sure that the forces you measure are the right ones?

Accelerometers do not measure fictitious forces. They only measure real forces. So you can tell that the forces you measure are the right ones precisely because they are the ones that you do measure.

 

[Ahmed1029]

This isn’t true. The acceleration of a reference frame itself can be measured by an accelerometer which is at rest with respect to the frame.

I don’t think that we have a good “why” for this. It is an observed fact that we incorporate into our models. We can show how this fact is encoded into the models, but not really explain why those models work and not others.

In the current models this is encoded in the fact that spacetime is represented as a manifold. Locally there is no preferred timelike direction, so all straight worldlines are equivalent. In contrast, worldlines that are not straight are distinguishable by how tightly they curve and any change in direction of curvature.

Accelerometers are measuring devices that can tell if they are accelerating, jerking, etc.

Accelerometers do not measure fictitious forces. They only measure real forces. So you can tell that the forces you measure are the right ones precisely because they are the ones that you do measure.

It's all good, but my overthinking mind is thinking that a properly accelerating accelerometer, the box with the springs, could say that there is a peculiar force acting downwards, like gravity, and I as an observer will say the box is accelerating in the opposite direction. So he thinks I'm accelerating, while I think he's accelerating. Also, what if all points of the box are accelerating at the same rate? There would be zero measured acceleration, not?

 

[Dale]

a peculiar force acting downwards, like gravity

Those are fictitious forces. Since the accelerometer doesn’t measure it then everyone agrees that it is fictitious.

So he thinks I'm accelerating, while I think he's accelerating.

Both of you agree on who is undergoing proper acceleration. There is no ambiguity. Furthermore, for any coordinate acceleration that either is undergoing, they know how much is due to real forces and how much is due to fictitious forces.

Also, what if all points of the box are accelerating at the same rate? There would be zero measured acceleration, not?

Yes, such local forces are called fictitious forces.

 

[Ahmed1029]

Those are fictitious forces. Since the accelerometer doesn’t measure it then everyone agrees that it is fictitious.

Both of you agree on who is undergoing proper acceleration. There is no ambiguity. Furthermore, for any coordinate acceleration that either is undergoing, they know how much is due to real forces and how much is due to fictitious forces.

Yes, such local forces are called fictitious forces.

But gravity just behaves the same way and it isn't a ficticious force.

 

[DrGreg]

But gravity just behaves the same way and it isn't a ficticious force.

In General Relativity, gravity is a fictitious force.

 

[PeroK]

But gravity just behaves the same way and it isn't a ficticious force.

Oh, yes it is!

 

[Nugatory]

but my overthinking mind is thinking that a properly accelerating accelerometer, the box with the springs, could say that there is a peculiar force acting downwards, like gravity, and I as an observer will say the box is accelerating in the opposite direction.

You are confusing coordinate acceleration and proper acceleration. We have a bunch of older threads explaining the difference between the two…. Find some and read through them.

 

[PeterDonis]

velocity is relative, but why not all higher derivatives of the position vector?

In relativity, ordinary 3-velocity is not a derivative of the position vector. What you think of as ordinary 3-velocity is the (hyperbolic tangent of the) angle in spacetime between two worldlines. There are no "higher derivatives" of this because it isn't a derivative in the first place.

In relativity, there is 4-velocity, which is the normalized tangent vector to a worldline. This can be thought of as the derivative with respect to proper time of a "position vector" once you have chosen a coordinate chart, but it is still an invariant geometric object independent of any choice of coordinates. The 4-acceleration vector is the derivative with respect to proper time of the 4-velocity, and you can continue to take higher derivatives with respect to proper time, but all of these things are invariants just like the 4-velocity.

 

[Dale]

But gravity just behaves the same way and it isn't a ficticious force.

Locally gravity is a fictitious force. That is Einstein’s equivalence principle.

 

[FactChecker]

But gravity just behaves the same way and it isn't a ficticious force.

Einstein's theory of General Relativity explains gravity as a distortion of a straight (geodesic), unaccelerated path near a large mass. There are no forces involved. IMO, General Relativity is the greatest single intellectual accomplishment by a human.

 

[vanhees71]

Ah, but we can measure absolute (the more standard term is "proper") acceleration.

Proper acceleration is measured with an accelerometer. Imagine a box containing a weight suspended by springs from all six inside faces; any proper acceleration of the box will measurably stretch the springs. The fictitious acceleration produced by fictitious forces will not stretch the springs.

Nowadays such a thing is built in our smart phones. It can be used to measure accelerations:

https://www.scientificamerican.com/article/science-with-a-smartphone-accelerometer/

 

[robphy]

If I drop a wii remote,
its acceleration sensors read zero.
It is in free fall (experiencing geodesic motion)
…until it hits the ground.

 

[najoop]

Acceleration is also relative, Consider the case of 2 Para jumper, when falling down freely, one of them opens up his parachute, he feels that he has stabilized from his free fall , but the other Para jumper feels that he is accelerating towards the sky.

 

[Ibix]

Acceleration is also relative, Consider the case of 2 Para jumper, when falling down freely, one of them opens up his parachute, he feels that he has stabilized from his free fall , but the other Para jumper feels that he is accelerating towards the sky.

You aren't distinguishing between coordinate acceleration, of which your scenario is an example, and proper acceleration, which is the quantity measurable in a closed box. The former is relative. The latter is not.

This has already been pointed out in this thread.

 

[Dale]

Acceleration is also relative, Consider the case of 2 Para jumper, when falling down freely, one of them opens up his parachute, he feels that he has stabilized from his free fall , but the other Para jumper feels that he is accelerating towards the sky.

No, it is not relative. An attached accelerometer can clearly distinguish between the two.

 

[najoop]

No, it is not relative. An attached accelerometer can clearly distinguish between the two

attached to where : to the free fall(guy) or to the stable one(parachute opened)

 

[Ibix]

attached to where : to the free fall(guy) or to the stable one(parachute opened)

You need one attached to each. The accelerometer attached to the free falling parachutist will read zero (assuming we can neglect air resistance here) and the accelerometer attached to the one with the open chute will read whatever the local $g$ is.

 

[Dale]

attached to where : to the free fall(guy) or to the stable one(parachute opened)

I would attach the accelerometers to their chest or abdomen on each. As close as possible to their respective centers of mass.

 

[PeroK]

... and the accelerometer attached to the one with the open chute will read whatever the local $g$ is.

And he'll feel the upwards pull of the parachute straps.

 

[Ibix]

And he'll feel the upwards pull of the parachute straps.

Real men don't use accelerometers - they are accelerometers!

 

[FactChecker]

, but the other Para jumper feels that he is accelerating towards the sky.

The question is whether he feels a force (ignoring air resistance). He does not because he is in free fall. So he feels just like nothing is happening.

 

[PAllen]

I would attach the accelerometers to their chest or abdomen on each. As close as possible to their respective centers of mass.

A possibly funny practical observation: one of the most ubiquitous accelerameters in daily life is the bathroom scale. To work for a parachutist, though, it should be balanced on their head, unattached to anything. This, of course, is just a limitation in the design of the scale as an accelerometer.

 

[Sagittarius A-Star]

A possibly funny practical observation: one of the most ubiquitous accelerameters in daily life is the bathroom scale.

The bathroom scale is only the spring part of a mass-spring-system as accelerometer.

To work for a parachutist, though, it should be balanced on their head, unattached to anything. This, of course, is just a limitation in the design of the scale as an accelerometer.

Yes, it is only the spring part. To get a mass-spring-system as accelerometer, a person could for example stand on a bathroom scale, which is placed on a board (with negligible mass), which is connected symmetrically to the ropes of the parachute.

 

[andresB]

Summary: It's taken as a axiom that there are inertial systems that experience no acceleration, but a reference frame can only measure the accelerations of other bodies, while the other bodies will only measure its acceleration with respect to themselves, so how can this measuring device know it's accelerating absolutely when it can only compare accelrations?

I've been thinking about this for a while, and thought it would be nice if someone could guide me to an answer.
In Newtonian mechanics, an inertial frame is coordinate system that's able to make measurements with respect to some imaginary axes attached to it.
It's a well known fact that velocity is relative, but why not all higher derivatives of the position vector? A measuring device can never tell if it's moving at all, accelerating, jerking, etc. It's a point after all, one that than can make measurements.
You might say it's going to measure ficticious forces, but I'd argue that if you cannot know discern absolute acceleration, how are you sure that the forces you measure are the right ones?

In the most basic version of Newtonian mechanics, true forces are interaction between at least two object. So, by Newton 3rd law, when an object feels a true force you are sure that somewhere else in the universe there is another object that feels the same force but in the opposite direction.

On the contrary, with fictitious forces there is no second object to obey the 3rd law.

In view of interaction via fields, a similar reasoning works, but instead of using Newton's 3rd law we have to use conservation of momentum.