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According to the principle of relativity, the laws of nature should be independent of the relative movement of different frames. My doubt is the meaning of "laws of nature".

So, suppose a spaceship starting from earth with zero velocity, and (except for a few minutes after lauching) keeping acceleration $g$ until point $x$. Then inverting the direction of the engines, keeping $-g$ until stop after reaching $2x$, and returning to $x$. Finally, inverting again the engines, keeping $g$ until stop at the Earth.

In spite of pendulum with the same lenght $l$ at the ship and on the Earth have the same period $$T = 2\pi \sqrt{\frac{l}{g}}$$ the total number of oscillations after the trip is different, because less time passed for the ship. It could be argued that the length $l$ changes between frames, but a quartz clock doesn't depend on such a length and also the number of oscillations would be different.

Apparently number of oscillations is not included in the expression "laws of nature". That means: there are properties that change only because of relative movement, even when the physical environment that should affect the outcome is not different from the point of view of each frame.

It is possible also to compare different ships, one as before, and another going to a distance $y<x$ and repeating the sequence several times, until they meet after some time. Again the number of oscillations are different, even having not only the same acceleration, but also avoiding the difference of potential well (uniform acceleration and gravity).

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    $\begingroup$ What is the question? $\endgroup$
    – Dale
    Sep 19 at 1:59
  • $\begingroup$ The number of oscillations of a pendulum is certainly not a law of Nature. $\endgroup$ Sep 19 at 3:20
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A pendulum on the ship will have the same proper period as it would on Earth, but the ship's pendulum will have a shorter period in the Earth's reference frame. As a consequence, the number of oscillations the pendulum undergoes is frame invariant, since it is given by the ratio of the proper time of the trajectory to the proper period of the pendulum.

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You can think of the laws of nature as being rules about the relationships between different physical quantities. The principle of relativity is that the rules should appear to hold regardless of the reference frame you chose to label positions in time and space.

Clearly that doesn't mean that the values of physical properties don't change from one reference frame to another. Measured relative to my chair, I have zero momentum- measured relative to a passing plane I have a large momentum.

Likewise with the passage of time. In my rest frame time passes at a certain rate. On the passing plane time also passes at a certain rate. However, the rate at which time on the plane happens appears to be reduced when I measure it in my frame.

The relevant law of nature, as far as we know, is that the relationships between times and distances in different inertial reference frames are governed by the Lorenz transformations. It doesn't matter which reference frames you pick- the transformations will always hold.

By a circular argument I can answer your question 'what is a law of nature' by saying that it is any fixed rule governing the relationship ship between quantities that applies independently of the reference frame chosen to represent the quantities.

You can then see that you have to be careful with your definitions of the laws if you want to be rigorous.

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  • $\begingroup$ In one hand, I agree on the idea of the relationship between variables in any frame. But on the other hand, as the constancy of the speed of light is at the heart of relativity, it is clear that an atomic clock, that uses EM radiation as input, should follow Lorenz transformations. But a simple mechanical device like a pendulum is another stuff. I can't see why it should be affected. $\endgroup$ Sep 20 at 0:15
  • $\begingroup$ Because time itself is affected. Whether the device is a pendulum, a Caesium clock, a bird singing...they all occur through time, and as time is relative they are all subject to the transformations. $\endgroup$ Sep 20 at 6:30

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