= Solution
Write $\Gamma_a(v)=(1-v^2/a^2)^{-1/2}$. The first proposed <Lorentz transformation> is
$$
x'=\Gamma_a(v)(x-vt),\qquad
t'=\Gamma_a(v)\left(t-\frac{vx}{a^2}\right).
$$
For the next transformation, write $\Gamma_b(u)=(1-u^2/b^2)^{-1/2}$ and substitute the first pair into $x''=\Gamma_b(u)(x'-ut')$ and $t''=\Gamma_b(u)(t'-ux'/b^2)$. The result is
$$
\begin{aligned}
x''&=\Gamma_a(v)\Gamma_b(u)
\left[\left(1+\frac{uv}{a^2}\right)x-(u+v)t\right],\\
t''&=\Gamma_a(v)\Gamma_b(u)
\left[\left(1+\frac{uv}{b^2}\right)t
-\left(\frac{v}{a^2}+\frac{u}{b^2}\right)x\right].
\end{aligned}
$$
The direct transformation between $S$ and $S''$ must belong to the same standard boost family with the same calibration of space and time units. Such a transformation has one common factor multiplying the coefficient of $x$ in $x''$ and that of $t$ in $t''$. Equality of these coefficients for generic $uv\ne0$ requires
$$
1+\frac{uv}{a^2}=1+\frac{uv}{b^2},\qquad
\boxed{a^2=b^2.}
$$
The case $uv=0$ alone cannot constrain the scales, but a transformation law must work for arbitrary successive boosts. Once the scales agree, their composed <velocity> is $V=(u+v)/(1+uv/a^2)$ and the common factor is $\Gamma_a(V)$. This establishes the <universality of an invariant speed under boost composition> within the proposed family.
\b[Finiteness needs physical input.] Sending $a^2$ to infinity gives the consistent <Galilean transformation> $x'=x-vt$, $t'=t$. Its boosts compose with $V=u+v$ and satisfy a relativity principle. Thus the composition calculation itself does \i[not] prove that $a^2$ cannot be infinite. The finite propagation speed of light in vacuum and its invariance, supplied by the second of the <postulates of special relativity>, rule out this limit for physical spacetime. Under a <Galilean transformation>, a ray with laboratory speed $c$ would have speed $c-v$ in the moving frame, in conflict with that postulate.
Finally, the common invariant scale must be identified with the physical <speed of light>, rather than an arbitrary speed measured in one frame. Requiring a ray $x=ct$ also to obey $x'=ct'$ gives
$$
\frac{c-v}{1-vc/a^2}=c
\quad\Longrightarrow\quad a^2=c^2\qquad(v\ne0).
$$
Hence \b[the physical scale is $a=c$], taking the positive speed. The postulate about light identifies and requires a finite invariant scale; closure alone only establishes its universality. This distinction is necessary for a logically valid answer to the question's request to exclude infinity.
Back to article page
