The discussion centers on the nature of non-inertial frames and the concept of proper acceleration. Proper acceleration is defined as the acceleration measured by an accelerometer and is invariant, meaning it does not depend on any reference frame. Inertial frames are characterized by having a proper acceleration of zero, and the acceleration of particles is invariant under Galilean transformations between inertial frames. The conversation emphasizes that while kinematic quantities are often frame-dependent, proper acceleration is a fundamental invariant in both classical and modern physics.
PREREQUISITESStudents of physics, educators teaching classical mechanics, and anyone interested in the foundational concepts of motion and reference frames in both classical and modern physics.
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.feynman1 said:
The proper acceleration of an observer is an invariant. It is not "with respect to" anything.feynman1 said:
I prefer to define it as the quantity that is measured by an accelerometer. But the two definitions are equivalent.etotheipi said:
If This acceleration is most naturally quantified with respect to inertial frames, then are inertial frames absolute and are they always inertial?etotheipi said:
Um, yes, inertial frames are always inertial.feynman1 said:
Not all kinematic quantities are relative. Proper time is not, nor is proper acceleration.feynman1 said:
Then this reference frame has an acceleration=0, then how is this acceleration measured, is it measured with respect to a reference frame?Dale said:
It is measured with an accelerometer. As I said above.feynman1 said:
Okay thanks but can we speak only Newtonian mechanics and forget about special relativity?Dale said:
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.feynman1 said:
Got it thanks. So is my question unanswerable at all within the Newtonian framework?Dale said:
Which of your many questions are you specifically referring to?feynman1 said:
the original postDale said:
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?etotheipi said:
I already answered it in post 4, which is valid in Newtonian physics toofeynman1 said:
Is proper acceleration even defined in classical physics/introductory physics before relativity?Dale said:
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...!)feynman1 said:
An odd complaint, since the person who dragged in SR was...you.feynman1 said:
Proper acceleration is just the acceleration measured by an accelerometer. It is perfectly compatible with Newtonian physics.feynman1 said:
No.feynman1 said:
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.feynman1 said:
Yes. It’s essential to understanding what’s going on with centripetal and centrifugal forces when you’re swinging a weight on a string in a circle, for example.feynman1 said:
How so? You can do all the analysis simply with the definition of acceleration with respect to a frame ##F## ##\mathbf{a} \big{|}_{F} = \frac{\mathrm{d^2} \boldsymbol{x}}{\mathrm{d}t^2} \big{|}_F## and if you're using a non-inertial frame then the additional relationship ##\frac{\mathrm{d} \mathbf{u}}{\mathrm{d}t} \big{|}_{Oxyz} = \frac{\mathrm{d} \mathbf{u}}{\mathrm{d}t} \big{|}_{O'x'y'z'} + \boldsymbol{\omega} \times \mathbf{u}## for any arbitrary vector ##\mathbf{u}##, which is required in order to obtain the equation of motion in the non-inertial frame (either by finding the Lagrangian in the rotating frame, or by deriving the relationship between ##\mathbf{a} \big{|}_{Oxyz}## and ##\mathbf{a} \big{|}_{O'x'y'z'}## and then substituting for ##\mathbf{a} \big{|}_{Oxyz}## in Newton's equation, where ##Oxyz## constitute inertial co-ordinates).Nugatory said:
The term may not be used, but accelerometers are classical devices and their readings are perfectly compatible with classical mechanics. There is nothing inherently non-classical about proper acceleration, and borrowing the term aids greatly in clarity.etotheipi said:
The ##\left.\mathbf{a}\right|_F## you defined is coordinate acceleration, and physical accelerometers will not measure this in general. So you clearly have two concepts here - coordinate acceleration and whatever it is the accelerometers are measuring. The latter is proper acceleration, whether it's called that or not, and I would say that it's actually the truly important concept because it's where the maths corresponds to something directly measurable. (Edit: there's a gag about using the proper terms for things here somewhere, but it'd add more confusion than it'd be worth.)etotheipi said:
But that can just be stated: let ##K## be any inertial reference system, then what you call the proper acceleration is what I would denote ##\mathbf{a} \big{|}_K## - accelerometers are a nice heuristic, but you can't really define an accelerometer mathematically. And I don't see the point of introducing extraneous terminology.Ibix said: