<s>[INST] Undetermined coefficients in a perturbative expansion In order to familiarize myself with perturbation methods, I've been trying to derive the Lorentz transformations, given by
\begin{align*}
x \rightarrow \frac{x + vt}{\sqrt{1 - v^2}} & = (x + vt)(1 + \frac{1}{2}v^2 + \frac{3}{8}v^4 + ...) \\
& = (x + vt + \frac{1}{2}v^2x + \frac{1}{2}v^3t + \frac{3}{8}v^4x + \frac{3}{8}v^5t + ...)
\end{align*}
\begin{align*}
t \rightarrow \frac{t + vx}{\sqrt{1 - v^2}} & = (t + vx)(1 + \frac{1}{2}v^2 + \frac{3}{8}v^4 + ...) \\
& = (t + vx + \frac{1}{2}v^2t + \frac{1}{2}v^3x + \frac{3}{8}v^4t + \frac{3}{8}v^5x + ...)
\end{align*}
using a perturbative approach which entails starting with the Gallilean transformations (i.e. $x \rightarrow x + vt$, $t \rightarrow t$) and in each step adding variations $\delta x$ and $\delta t$, both of order $n$ in $v$ and linear in $x$ and $t$, such that the spacetime interval $t^2 - x^2$ is invariant with respect to the transformations up to order $n+1$ in $v$. However, I keep running into the problem that, in each step of the process, some of the coefficients in my expansions for the transformations are undetermined. For example, for $\delta x$ and $\delta t$ of order $2$ in $v$,
$$x \rightarrow x + vt + \delta x$$
$$t \rightarrow t + vx + \delta t$$
$$\delta x = v^2(a_1x + b_1t) + \mathcal{O}(v^3)$$
$$\delta t = v^2(a_2x + b_2t) + \mathcal{O}(v^3)$$
\begin{align*}
t^2 - x^2 & = (t + vx + \delta t)^2 - (x + vt + \delta x)^2 \\
& = (t + vx)^2 + 2(t + vx)\delta t - (x + vt)^2 - 2(x + vt)\delta x + \mathcal{O}(v^4) \\
& = t^2 + 2vxt + v^2x^2 + 2v^2(t + vx)(a_2x + b_2t) \\
& \qquad - x^2 - 2vxt - v^2t^2 - 2v^2(x + vt)(a_1x + b_1t) + \mathcal{O}(v^4) \\
& = t^2 + v^2x^2 + 2v^2a_2xt + 2v^2b_2t^2 \\
& \qquad - x^2 - v^2t^2 - 2v^2a_1x^2 - 2v^2b_1xt + \mathcal{O}(v^3) \\
& = t^2 - x^2 + v^2[(2b_2-1)t^2 + 2(a_2-b_1)xt + (1-2a_1)x^2] + \mathcal{O}(v^3)
\end{align*}
I end up with $a_1 = b_2 = \frac{1}{2}$ and $a_2 = b_1 \equiv c_1$, which is so far an undetermined coefficient.
I know from the Taylor expansions of the full transformations above that $c_1$ should be equal to $0$. Is there a way to mathematically determine this coefficient?
 [/INST] What you're trying to do here cannot possibly work. The parameter $v$ in the Lorentz transformation is an arbitrary parameter that you can redefine at will. Your approach has just as much reason to yield the coefficients in the expansion in $2v$ or $\sin v$ as it does to yield the coefficients in the expansion in $v$, so it's not surprising that it has a lot of degrees of freedom.
</s>