Galilean transformation of non-inertial frame

[A.T.]

You might consider a system of three observers in a single dimension. A is our chosen reference frame , with B moving left (negative velocity) and C moving right (positive velocity). Firstly, no one will agree on energy or momentum of A, B, C.

So A,B,C are not just frames. but massive objects at rest in the frames A,B,C respectively?

But they will agree on the total energy and momentum of the system.

No, the rest frames of A,B,C will not agree on the combined total momentum and energy of the objects A,B,C.

 

[valenumr]

So A,B,C are not just frames. but massive objects at rest in the frames A,B,C respectively?No, the rest frames of A,B,C will not agree on the combined total momentum and energy of the objects A,B,C.

That's correct, I worded that poorly. I meant they will agree on a net change in energy and momentum before and after. They would usually have different values for the totals.

 

[valenumr]

To be more clear, they would agree that momentum was conserved.

 

[A.T.]

To be more clear, they would agree that momentum was conserved.

Yes, inertial frames have momentum conserved. And non-inertial frames don't have momentum conserved, because the inertial forces there don't obey Newtons 3rd Law.

We already established that applying a Galilean transformation to a non-inertial frame will preserve the total inertial force, so the amount of momentum conservation violation is also preserved.

 

[valenumr]

Yes, inertial frames have momentum conserved. And non-inertial frames don't have momentum conserved, because the inertial forces there don't obey Newtons 3rd Law.

We already established that applying a Galilean transformation to a non-inertial frame will preserve the total inertial force, so the amount of momentum conservation violation is also preserved.

Yes, inertial frames have momentum conserved. And non-inertial frames don't have momentum conserved, because the inertial forces there don't obey Newtons 3rd Law.

We already established that applying a Galilean transformation to a non-inertial frame will preserve the total inertial force, so the amount of momentum conservation violation is also preserved.

After rereading all of this, is this basically how we do physics in Earth's gravitational frame? Or am I still misunderstanding the question?

 

[vanhees71]

In the restframe of the Earth, i.e., our usually used frame for experiments, we have an approximately inertial frame and we can take the gravitational force of the Earth into account in the approximation of $\vec{F}_G=m \vec{g}$ with $\vec{g}=\text{const}$. Equivalently you can switch to a freely falling frame within this approximation of a constant gravitational force. Then you are really in an inertial frame.

This is, however of course, an approximation, and the Earth-fixed frame is not an inertial frame, because of the rotation of the Earth around its axis, leading to nice phenomena like Foucault's pendulum.

Today the best realization of a real inertial frame is to use a freely-falling reference frame, where the cosmic-microwave background radiation is at rest, i.e., is really homogeneous and isotropic (neglecting the otherwise very important fluctuations of its temperature at the order of $10^{-5}$), but that's another story.