June 4, 2022
Up until now, all the mechanics derived were based on the assumptions that time for a person moving WRT someone behaves in the same way as it does for someone at rest and that length of objects for a person moving WRT someone is the same as for someone at rest. However, if we think about how time and length are actually measured we see that our assumptions may not be correct, no matter how familiar, intuitive or genuine they may seem. This is because when we measure time, we need some periodic motion to occur, a minute hand circling around the clock or the earth revolving around the sun for example. In other words, anything that we can know about time comes from studying the properties of physical systems, thus the properties of time must also come from them. This way, we will achieve an objective understanding of time without any preconceived notions and assumptions. In a similar way, the only way to know the length between two points is to measure it, thus the properties of length must also be connected to the system we are making use of to measure the length. Thus the task at hand is to understand the nature of physical systems when they are moving versus when they are not.
For the discussion at hand, we will assume two postulates to be true, which themselves come from experiments. The first is the principle of relativity, that is laws of physics take the same form in all inertial systems, in other words, all inertial systems are equivalent, there is not a single inertial system that we can say is at absolute rest, they all are on equal footing as far as laws of physics are concerned. The second postulate is that speed of light is the same for all observers of the inertial frame (inertial observers), that is it does not matter if an inertial observer is moving with some speed $v$ WRT to some other inertial observer, they both will detect the same speed of light (SOL) which we will denote by $c$. The second postulate is in direct conflict with the Galilean transforms that we studied in Newton’s Laws. Thus we will be needing a new type of transformation (a fancy way of saying a set of equations relating the motion of a particle in one coordinate system to another).
A drastic example of the postulates at hand would be as follows: Consider a train moving from left to right at some constant velocity. If we stand in the middle of the train and turn a lantern on, the light reaches both ends of the train at the same time, namely, $t=L/c$, if the length of the train was $2L$. Since this was an inertial frame all of our above postulates are met. Now consider a person at rest on the platform watching the train go by, according to them the back of the train is rushing towards the light emitted by the lantern, whereas the front of the train is going away. Thus there would be a delay between the light hitting the backs of the train. This all happens because of the finite speed of light because if we (wrongly) apply Galilean transformations we reach the conclusion that even from the perspective of someone on the platform, both light rays hit at the same time due to the addition of the velocity of the train to the velocity of light, which is against our second postulate. Thus what may be simultaneous to one inertial observer may not be to another.
The Lorentz Transformation:
Consider two inertial systems S with coordinates of time and space as $(x,y,z,t)$ and a frame S’ moving WRT S at a velocity of $v$ (entirely in x-direction), with coordinates $(x,',y',z',t')$, such that $x\parallel x',\,\ y\parallel y',\,\ z\parallel z'$ and also that the coordinates of both frames meet at $t=t'=0$. The most general transformation between these two frames take the following form
$$ x'=Ax+Bt \\ \\ y'=y \\ \\z'=z \\ \\ t'=Cx+Dt \tag{8.1,\quad 8.2,\quad 8.3,\quad 8.4} $$
The linearity is maintained because otherwise, we will get a non-zero acceleration in one reference frame even if velocity was constant in another. Another reason for this is that if the relationship was of quadratic form for example then change in positions would be given by $\Delta x= x'^2_2-x'^2_1$, thus if at time $t=t_0$ the values of $x_1'$ and $x_2'$ are 1 and two then we get $\Delta x=3$, however, if the values are 4 and 3 then we get $7$. Thus the length of the object depends upon the position of the object, which is against the isotropy of space. Similar considerations can be made in support of the linearity of the time component to preserve its isotropy.
To evaluate the constants consider the following 4 events
$$ S:(x=0,t);\quad\quad S':(x'=-vt',t') \tag{8.5} $$
Thus from equations 8.1 and 8.4, we get