Galilean relativity in terms of homogenity for car example

Question

I have a question related to Landau & Lifshitz's book. In that, he says:

If we were to choose an arbitrary frame of reference, space would be inhomogeneous and anisotropic. This means that, even if a body interacted with no other bodies, its various positions in space and its different orientations would not be mechanically equivalent. The same would in general be true of time, which would likewise be inhomogeneous; that is, different instants would not be equivalent. Such properties of space and time would evidently complicate the description of mechanical phenomena. For example, a free body (i.e. one subject to no external action) could not remain at rest: if its velocity were zero at some instant, it would begin to move in some direction at the next instant.

It is found, however, that a frame of reference can always be chosen in which space is homogeneous and isotropic and time is homogeneous. This is called an inertial frame. In particular, in such a frame a free body which is at rest at some instant remains always at rest.

As an example, I'd like to bring a car and the ball hung inside the car and we can look at it from 2 different frames of reference.

• Frame of reference is me (I'm inside the car): If car moves with constant speed, nothing happens to the ball, but if car accelerates, ball starts to swing back or forward (depending on acceleration forward or backward). You could say that in acceleration mode, space inside the car is inhomogeneous, anistropic as ball behaves differently.

When we choose a frame or reference as the person outside the car, to him, wouldn't the ball inside car behave the same way as observed from my reference frame? It's like things are kind of mixed up in my head and can't put them in order.

Would appreciate the complete, good explanation of Landau's thoughts(I think he is repeating Galillei's relativity, but still) in terms of my example.

Answer 1

For example, a free body (i.e. one subject to no external action) could not remain at rest: if its velocity were zero at some instant, it would begin to move in some direction at the next instant.

I believe what they are referring to here is a non-inertial (accelerating) frame of reference. That would be the frame for the observer in the accelerating car.

It is found, however, that a frame of reference can always be chosen in which space is homogeneous and isotropic and time is homogeneous. This is called an inertial frame.

Here they are referring to the inertial (non accelerating) frame of the observer on the road.

The different observations for the car reference frame and the road reference frame are due to the fact the car reference frame is a non-inertial (accelerating) inertial frame whereas the road (approximates) an inertial frame.

In the frame of the car the movement of the ball backward when accelerating appears to be due to some force. But no actual physical force is involved. It is referred to as a "pseudo" or "fictional" force.

In the frame of the road the ball does not appear to move backwards but instead initially appears to remain in place when the car accelerates. This is due to the inertial of the ball, not a physical force acting on the ball.

This is illustrated in the figures below (for a ball hanging by a string on the roof of the car).

from the road reference(me looking at car movement and ball hung on a car), when car accelerates, ball will definitely accelerate in the opposite direction. If you call "road frame" as inertial frame, then from the definition, the acceleration of ball must be explained by real force. What's the real force ? if you mean it's a string force, then why wasn't it the case from car frame?

Look again at FIG 2. The “x” position of the ball in A and B are essentially the same due to its inertia while the car accelerates forward. Only when the horizontal component of the tension in the string builds will the ball experience be a real horizontal force accelerating the ball along with the car.

A better illustration of this than a ball hung from a string would be if the ball was simply on top of a frictionless horizonal roof. It would then remain at the same horizontal location with respect to the ground as the car accelerates.

Hope this helps.