Principle of relativity and Galileo's group

[HighPhy]

A doubt has arisen for me about the principle of relativity, and being such a fundamental subject I think it only fair to try and clarify it. The following line of reasoning was presented to me in a lecture, but is it really correct? It seems very unstable to me.

Take two frames of reference, $K$ and $K'$, with the latter moving in uniform rectilinear motion with respect to the former. It is possible to express the position of a material point with respect to the system $K$ ($\vec{r}$) by knowing the position of the point with respect to the system $K'$ ($\vec{r'}$) and knowing the position of the origin of the coordinates of $K'$ with respect to $K$ ($\vec{r_o}$). Thus:

$$\vec{r} = \vec{r'} + \vec{r_o}$$

Deriving twice with respect to time:

$$\vec{a} = \vec{a'} + \vec{a_o}$$

But since the system $K'$ moves in a uniform straight line:

$$\vec{a} = \vec{a'}$$

That is, the acceleration of the material point is the same in the two systems. The equations of motion with respect to the two systems are therefore the same.
It would seem that we have thus demonstrated the principle of relativity, but by definition it cannot be demonstrated. I will try to make a few remarks and ask someone to help me.

First objection
In the argument it is implicitly being stated that the properties of the position vectors do not change when passing from a system of fixed to one in uniform rectilinear motion and vice versa. It can be shown that the transformations that are part of Galileo's group are simple displacements of the metric. They therefore preserve the distances between two points and the time intervals. The vectors should therefore not undergo mutation in their length and orientation.
However, I have a question to ask with respect to this group. It includes all affine transformations that preserve distances between simultaneous events and time intervals. Is it possible to prove that even in the transition from an inertial to a non-inertial system these characteristics are preserved?

Second objection
One is also implicitly assuming that time is equally measured in the two systems, since we derive it without too much trouble. However, this objection too can be dismantled by assuming a priori that the flow of time is the same in all systems.

Third objection
I would say the most important one: the statement $\vec{a} = \vec{a'}$ does not mean a priori that the laws of motion are equal in the two systems. Since in fact $f(\vec{r}, \vec{v}) = m \vec{a}$ with the reasoning described above we have shown only that the RHS of the equation is conserved, but nothing is known about the LHS.

Could you help me shed some light on these doubts of mine by explaining to me whether the reasoning presented in the lecture is correct and, if so, where I am wrong; or whether, if not, my objections make sense, or what is the right way to deal with this topic from this presentation?

 

[Hill]

Is it possible to prove that even in the transition from an inertial to a non-inertial system these characteristics are preserved?

I think so. Just integrate over infinitesimal intervals while these characteristics are preserved on each of them.

assuming a priori that the flow of time is the same in all systems.

Yes, the flow of time is assumed to be universal in pre-relativistic mechanics, implicitly and explicitly.

nothing is known about the LHS

How is the force measured? Say, for example, that there is a force meter attached to the body that shows the force applied to it. Its reading would not change if observer reads it from a moving train.

 

[HighPhy]

I think so. Just integrate over infinitesimal intervals while these characteristics are preserved on each of them.

Show me explicitly how this could be done?

How is the force measured? Say, for example, that there is a force meter attached to the body that shows the force applied to it. Its reading would not change if observer reads it from a moving train.

OK, you probably want to say that $\vec a = \vec a'$ does mean that the laws of motion hold across inertial frames. If we apply Newton's laws, we should have: $f'(\vec r', \vec v') = m \vec a'$ for $K'$ and $f(\vec r, \vec v) = m \vec a$ for $K$. So the form of the law is the same in both frames. Right?

 

[vanhees71]

Of course, only closed systems show the complete invariance under Galilei transformations. The most simple case are two interacting particles. For all symmetries (spatial, temporal translation invariance, rotation invariance, and Galile-boost invariance) the interaction force must be a central not explicitly time-dependent force, derivable from a corresponding potential:
$$m_1 \ddot{\vec{x}}_1=\vec{F}_1=-\vec{\nabla}_1 V(|\vec{x}_1-\vec{x}_2|), m_2 \ddot{\vec{x}}_2 = \vec{F}_2=-\vec{\nabla}_2 V(|\vec{x}_1-\vec{x}_2|).$$
The forces are given by
$$\vec{F}_1=-\vec{F}_2=-\frac{\vec{x}_1-\vec{x}_2}{|\vec{x}_1-\vec{x}_2|} V'(|\vec{x}_1-\vec{x}_2|).$$
These equations of motion are obviously invariant under Galilei boosts,
$$\vec{x}_1'=\vec{x}_1-\vec{v} t, \quad \vec{x}_2'=\vec{x}_2-\vec{v} t,$$
where $\vec{v}=\text{const}$ but not under arbitrary accelerated shifts of the origin.

 

[HighPhy]

Yes, the flow of time is assumed to be universal in pre-relativistic mechanics, implicitly and explicitly.

I have a question: is it correct to say that the initially presented proof applies to the statement that: "The laws of mechanics are invariant in the transition from a $K$ system to a $K'$ system moving in uniform rectilinear motion with respect to $K$"? I believe that assuming that the flow of time is the same in all systems is wrong, since the decay time of a particle in motion is longer than that of a particle at rest. How can this be justified? That was how it was presented to me in a lecture.

 

[Hill]

I have a question: is it correct to say that the initially presented proof applies to the statement that: "The laws of mechanics are invariant in the transition from a $K$ system to a $K'$ system moving in uniform rectilinear motion with respect to $K$"? I believe that assuming that the flow of time is the same in all systems is wrong, since the decay time of a particle in motion is longer than that of a particle at rest. How can this be justified? That was how it was presented to me in a lecture.

This is the assumption of pre-relativistic mechanics, i.e., Newtonian-Galilean. It has been changed in the Einstein's special relativity. It is still a good approximation when speeds in the system of interest are much slower than the speed of light in vacuum.

 

[HighPhy]

This is the assumption of pre-relativistic mechanics, i.e., Newtonian-Galilean. It has been changed in the Einstein's special relativity. It is still a good approximation when speeds in the system of interest are much slower than the speed of light in vacuum.

Exactly. So how can we say that we prove that all the laws of mechanics are the same by admitting the invariance of acceleration in the two reference systems? Isn't this a logical leap, so it must be specified that one is operating with pre-relativistic mechanics? Or is there no need and is it sufficient to say that all laws of mechanics are invariant in any inertial frame?

 

[Hill]

Exactly. So how can we say that we prove that all the laws of mechanics are the same by admitting the invariance of acceleration in the two reference systems? Isn't this a logical leap, so it must be specified that one is operating with pre-relativistic mechanics? Or is there no need and is it sufficient to say that all laws of mechanics are invariant in any inertial frame?

All laws are the same in all inertial frames - this is true in the Newtonian and in the relativistic mechanics. But the laws of Newtonian and relativistic mechanics, differ.

 

[vanhees71]

I have a question: is it correct to say that the initially presented proof applies to the statement that: "The laws of mechanics are invariant in the transition from a $K$ system to a $K'$ system moving in uniform rectilinear motion with respect to $K$"? I believe that assuming that the flow of time is the same in all systems is wrong, since the decay time of a particle in motion is longer than that of a particle at rest. How can this be justified? That was how it was presented to me in a lecture.

This is, of course, right, but it cannot be described in Newtonian mechanics, because it is a relativstic effect, i.e., it must be describe within the (special) theory of relativity. There Newton's Lex I still holds true, but in addition you need the assumption that the speed of light is independent of the motion of the light source as well as the detector relative to any inertial reference frame. This leads to an entirely different spacetime model with all the kinematic effects, which are discussed over and over again in the relativity forum ;-).

 

[Meir Achuz]

"Deriving twice with respect to time:"
I haven't read all the other posts, but there are two different times (t and t') involved.

 

[HighPhy]

"Deriving twice with respect to time:"
I haven't read all the other posts, but there are two different times (t and t') involved.

If in general your observation is correct, note that in Galilean relativity the axiom of absolute time ($t = t'$) applies. This axiom is of course discarded in SR. Or do I get something wrong?