From my head to yours

Classical Mechanics Level 4

Two frames can have a relative velocity and since any velocity is allowed there is an infinite number of possible frames. Rotating frames have a necessary relative acceleration (think centripetal acceleration), so if an object is at rest in one frame it cannot remain at rest in the other. A simple example of this can be shown by holding a ball in your hand and spinning around. In this situation you define a rotating frame relative to the earth, and if you let go of the ball you will see that it flies outwards, even though there is no horizontal force on the ball. Therefore you are not an inertial reference frame. However, if you were on a train moving at a constant speed with respect to the earth and simply placed the ball on a table, the ball would not move.

How does one transform coordinates between two inertial reference frames? Let's consider two families of observers, A and B, moving with respect to each another with a relative speed $v$ . The two inertial reference frames corresponding to each family have coordinates $(t_A,x_A)$ and $(t_B,x_B)$ . Which of the following statements must be true? ( $X,T$ are arbitrary functions below.)

x_B\neq x_A

v \rightarrow 0

x_B=x_A

v\rightarrow 0

x_B=X(t_A,x_A,v), t_B=T(t_A,x_A,v)

x_B=X(x_A,v),t_B=T(t_A,v)

2 solutions

Milly Choochoo
Jun 26, 2014

How I saw this was, you want some functions $X$ and $T$ that, given information of $A$ , can tell you the corresponding information of $B$ , of course being $x_B$ and $t_b$ . Obviously $x_A = x_B$ as $v\rightarrow 0$ isn't always true - just think of two rocks sitting apart from each other. The same goes for the converse, $x_A \neq x_B$ as $v \rightarrow 0$ - what about two rocks basically in the same spot, sitting and doing nothing?

Now if you were given only $x_A$ and $v$ to calculate $x_B$ , you'd have a problem. What if $A$ was 1000 years ahead of $B$ ? Sure, $B$ will eventually reach the same point that $A$ is at at $t_a$ , and knowing the velocity $v$ will give you a function that will tell you exactly where $B$ is at some arbitrary time $t$ , but then that equation will be specific to the original constraint that they, $A$ and $B$ are 1000 years apart. If we want to calculate the location of $B$ , $x_B$ , relative to $A$ 's inertial reference frame, we must include $t_A$ . The same goes for the function for $t_B$ .

Therefore, $x_B=X(t_a,x_a,v) , t_b=T(t_a,x_a,v)$ must be correct.

Siam Habib
Jun 25, 2014

We use Lorentz Transformation to deal with such transformations in inertial reference frame. If check the equations, you will find that the answer is $x_B = X(t_A,x_A,v), t_B=T(t_A,x_A,v)$

From my head to yours

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