Midframe Lemma and composition of velocities [closed]

Question

The Midframe Lemma states that [Rindler, 2006]

"between any two inertial frames S and S' there exists an inertial frame S" relative to which S and S' have equal and opposite velocities."

How is this compatible with the velocity addition formula?

Answer 1

This is how Rindler proves it: "For proof, consider a one-parameter family of inertial frames moving collinearly with S and S', the parameter being the velocity with respect to S. It is then obvious from continuity that there must be one member of this family with the required property (see Fig 2.4)."

So we have the two frames S and S', and between them we have the frame S''. Let $v_1$ be the velocity of S wrt S'', and $v_2$ be the velocity of S' wrt S''. Our target is to find S'' such that $v_1 = - v_2$. Let's parameterize S'' with $\lambda$, such that at $\lambda = 0$, $S'' = S$, and that at $\lambda = 1$, $S'' = S'$.

Now, when $\lambda = 0$, $v_1 = 0$ and $v_2$ = $u$, where $u$ is the velocity of S' wrt S. Conversely, when $\lambda = 1$, $v_1 = -u$ and $v_2 = 0$. So the magnitude of $v_1$ changes from $0$ to $u$ as $\lambda$ varies, and the magnitude of $v_2$ changes from $u$ to $0$. As $v_1$ and $v_2$ must change continuously as a function of $\lambda$, they must cross at a certain point. That is the S'' that we are seeking.

Finding the actual value of $\lambda$, and thus the value of $v_1$, will be rather more time-consuming. But I think this demonstrates that this frame exists.

Answer 2

The formulas for the "midframe" are used in Loedel diagrams

• https://en.wikipedia.org/wiki/Minkowski_diagram#Loedel_diagram

The corresponding formulas were derived from the Lorentz transformation by Mirimanoff in 1921:

• The Lorentz-Einstein transformation and the universal time of Ed. Guillaume

and from the velocity addition formulas by Shadowitz:

• The Electromagnetic Field (Reprint of 1975 ed.). Courier Dover Publications. p. 460

Starting with the velocity addition formula

$$w=\frac{u+v}{1+uv/c^{2}}$$

and by noticing that u=v in the midframe, we have

$$\frac{w}{c}=\frac{(v/c)+(v/c)}{1+(v/c)(v/c)}=\frac{2\beta}{1+\beta^{2}}$$