# Whether a non-inertial frame is absolute

• I
If a frame is a non inertial frame, then it must have an acceleration. Then which reference frame is this acceleration with respect to? If this acceleration varies with the reference frame this acceleration is calculated with respect to, is this non inertial frame absolute?

## Answers and Replies

etotheipi
This acceleration is most naturally quantified with respect to inertial frames. Exercise, for you: prove that the acceleration of a particle is invariant under a general Galilean boost.

• atyy and vanhees71
Ibix
Science Advisor
2020 Award
Then which reference frame is this acceleration with respect to?
Proper acceleration can be measured by a spring balance with no reference to any frame. A reference frame is inertial or not depending on whether objects defined to be at rest in the frame measure proper acceleration.

• vanhees71, Twigg, hutchphd and 2 others
Dale
Mentor
2020 Award
Then which reference frame is this acceleration with respect to?
The proper acceleration of an observer is an invariant. It is not "with respect to" anything.

The acceleration of a non-inertial frame is given as the proper acceleration of observers at rest throughout the non-inertial frame. Therefore, it is also not "with respect to" anything. It is invariant, and depending on the details of the specific non-inertial frame, it can vary from location to location throughout the frame.

Edit: thus also an inertial frame is inertial in an invariant sense and not "with respect to" anything, since inertial means roughly that the proper acceleration of a stationary object is 0 everywhere

Last edited:
• cianfa72 and vanhees71
etotheipi
Here is a question. How is the 'proper acceleration' of a point in classical mechanics to be defined, if not as the acceleration of that point with respect to an inertial co-ordinate chart (one of a class of co-ordinate charts defined by the restriction that a Galilean transformation of the co-ordinate space applied to all points in a mechanical system gives world-lines of the same system except with new initial conditions)?

• vanhees71 and Dale
Dale
Mentor
2020 Award
Here is a question. How is the 'proper acceleration' of a point in classical mechanics to be defined, if not as the acceleration of that point with respect to an inertial co-ordinate chart (one of a class of co-ordinate charts defined by the restriction that a Galilean transformation of the co-ordinate space applied to all points in a mechanical system gives world-lines of the same system except with new initial conditions)?
I prefer to define it as the quantity that is measured by an accelerometer. But the two definitions are equivalent.

• robphy, vanhees71 and etotheipi
This acceleration is most naturally quantified with respect to inertial frames. Exercise, for you: prove that the acceleration of a particle is invariant under a general Galilean boost.
If This acceleration is most naturally quantified with respect to inertial frames, then are inertial frames absolute and are they always inertial?

I don't understand why the acceleration can be invariant. Aren't all kinematic quantities measured w.r.t a reference frame? Then the magnitude of the acceleration should also be relative to some frame?

Dale
Mentor
2020 Award
If This acceleration is most naturally quantified with respect to inertial frames, then are inertial frames absolute and are they always inertial?
Um, yes, inertial frames are always inertial.

I don't understand why the acceleration can be invariant. Aren't all kinematic quantities measured w.r.t a reference frame? Then the magnitude of the acceleration should also be relative to some frame?
Not all kinematic quantities are relative. Proper time is not, nor is proper acceleration.

In fact, it is necessary that proper acceleration be invariant. Otherwise you could not unambiguously define an inertial frame.

• vanhees71
Um, yes, inertial frames are always inertial. Rainy afternoons are always rainy. And magical fairies are always magical.
Then this reference frame has an acceleration=0, then how is this acceleration measured, is it measured with respect to a reference frame?

• vanhees71
Dale
Mentor
2020 Award
Then this reference frame has an acceleration=0, then how is this acceleration measured, is it measured with respect to a reference frame?
It is measured with an accelerometer. As I said above.

• vanhees71
Um, yes, inertial frames are always inertial.

Not all kinematic quantities are relative. Proper time is not, nor is proper acceleration.

In fact, it is necessary that proper acceleration be invariant. Otherwise you could not unambiguously define an inertial frame.
Okay thanks but can we speak only newtonian mechanics and forget about special relativity?

Dale
Mentor
2020 Award
Okay thanks but can we speak only newtonian mechanics and forget about special relativity?
No, I don’t think so. Newton didn’t have a very clear concept of reference frames at all, and some of his key ideas on the matter were wrong. The modern clarity on reference frames was largely due to relativity forcing us to confront and fix some Newtonian errors.

In any case, Newtonian physics has more invariants, not more relative quantities

No, I don’t think so. Newton didn’t have a very clear concept of reference frames at all, and some of his key ideas on the matter were wrong. The modern clarity on reference frames was largely due to relativity forcing us to confront and fix some Newtonian errors.

In any case, Newtonian physics has more invariants, not more relative quantities
Got it thanks. So is my question unanswerable at all within the newtonian framework?

etotheipi
The question is quite answerable completely within the classical framework. Did you try to prove the invariance of acceleration under a general Galilean transformation, as in post #2?

• vanhees71
Dale
Mentor
2020 Award
Got it thanks. So is my question unanswerable at all within the newtonian framework?
Which of your many questions are you specifically referring to?

Which of your many questions are you specifically referring to?
the original post

The question is quite answerable completely within the classical framework. Did you try to prove the invariance of acceleration under a general Galilean transformation, as in post #2?
As per the galilean transformation, two systems need to move relative to each other with a constant speed. Then what if there’s a relative acceleration between them and how to apply the galilean transformation?

• vanhees71
Dale
Mentor
2020 Award
the original post
I already answered it in post 4, which is valid in Newtonian physics too

I already answered it in post 4, which is valid in Newtonian physics too
Is proper acceleration even defined in classical physics/introductory physics before relativity?

etotheipi
As per the galilean transformation, two systems need to move relative to each other with a constant speed. Then what if there’s a relative acceleration between them and how to apply the galilean transformation?
The point is that in classical mechanics there are certain distinguished frames of reference, so-called inertial frames, where Newton's law of inertia holds (free particles move at constant velocities). A Galilean transformation maps from one inertial frame to a different inertial frame. (And of course a transformation between two frames undergoing non-zero relative acceleration is not Galilean...!)

Take, for instance, a particle with a trajectory ##\mathbf{x} = \mathbf{X}(t)## in one inertial frame. It's acceleration in this frame is nothing but ##\mathbf{a}(t) = \ddot{\mathbf{X}}(t)##. Under a Galilean transformation, for example to a second inertial frame moving at a constant velocity ##\mathbf{v}## with respect to the first and with initial displacement ##\mathbf{s}##, the trajectory in this new inertial frame is ##\mathbf{x}' = \mathbf{X}'(t) = \mathbf{X}(t) - \mathbf{v}t - \mathbf{s}##, which again implies ##\mathbf{a}'(t) = \ddot{\mathbf{X}}(t)##. The acceleration of the particle as measured by any inertial frame is then the same, i.e. ##\mathbf{a}(t) = \mathbf{a}'(t)##.

That's partly why inertial frames are distinguished, because the accelerations of particles as measured by any frame in that class are invariant. Whilst, by a suitable choice of non-inertial frame, you can make the acceleration whatever you want Last edited by a moderator:
• vanhees71
Vanadium 50
Staff Emeritus
Science Advisor
Education Advisor
Okay thanks but can we speak only newtonian mechanics and forget about special relativity?
An odd complaint, since the person who dragged in SR was...you.

I count five answers to your original question. Clearly these answers don't satisfy you. It might help if you explained how and why they don't satisfy you. Otherwise we will keep going around in circles.

Dale
Mentor
2020 Award
Is proper acceleration even defined in classical physics/introductory physics before relativity?
Proper acceleration is just the acceleration measured by an accelerometer. It is perfectly compatible with Newtonian physics.

PeterDonis
Mentor
2020 Award
Aren't all kinematic quantities measured w.r.t a reference frame?
No.

A.T.
Science Advisor
I don't understand why the acceleration can be invariant. Aren't all kinematic quantities measured w.r.t a reference frame? Then the magnitude of the acceleration should also be relative to some frame?
Proper acceleration can be thought of as relative to a local free falling frame. But this boils down to what an accelerometer measures, so you don't need to introduce that frame.