4:48 AM
(cont.) We can then use a similar idea, to comprehend special relativity phenomenon:
Take muon lifetime as an example
In the frame of the muon, time as measured by the observer in the muon's frame ticks normally because it is proper time. However, space get contracted in the direction of travel (and time elsewhere is dilated as well)
In Earth's frame, the surrounding space is normal (because that's the proper distance). The time of the muon becomes dilated (and if a muon does have some notion of size, should look squished in the direction of travel)
In usual explanations, the muon's ability to survive to the earth's surface despite having a lifetime too short to do so can be explained in these two frames as follows:
1. In the muon's frame, the distance required to reach the ground is shortened
2. In Earth's frame, the muon's time get dilated, thus the onset of the decay effectively occured later
However, we can say something more than that (and perhaps also a bit more intuitive(?))
If we use our intuition that the muon can survive reaching the surface of the earth because time slows down at the muon in Earth's frame, and wrongly conclude that the time dilation is just an apparent effect explained by at the muon's frame, time flows normally but only a much shorter distance is needed to travel through
(thus in a relative sense, time seemed to slow down while it is ticking normally in the muon's frame, is because for each tick, a larger distance is being covered as seen from the Earth's frame)
then this will not work. This is because:
We would have expected the time elapsed at earth to be much faster when we switched to the muon's frame from the earth's frame, but what is observed is that time is also dilated in Earth's frame. Likewise, length get shorten but not expanded at the muon as seen from Earth's frame
This relationship where for A and B travelling relative to each other, A looks shorten and time dilated to B and B looks shorten and time dilated to A, suggests the transformation that relate these two frames cannot be galelian and more generally, cannot be expressed as a (I have no idea what is the terminology) mapping, since in such maps, if A > B as seen from space A, then B < A as seen from space B
Therefore, it must be a (I have no idea what that terminology is) mapping, which in the case of special relativity, is a hyperbolic rotation, such that if A > B as seen from space A, then B > A as seen from space B
typo: is that time is also dilated at Earth as seen from the muon's frame
This is a particular solution to a differential equation of an isotropic oscillator with the potential energy $V = \dfrac{1}{2} k(x^2 + y^2)$
$$\vec r(t) = \vec r_0\cdot cos(\omega t) + \dfrac{\vec v_0}{\omega}\cdot sin(\omega t)$$
According to this derivation of the equation of motion for thi...
