I am trying to solve problem 2.1 in Schwartz, which is to derive the transformations $x \rightarrow \frac{x+vt}{\sqrt{1-v^2}}$ and $t \rightarrow \frac{t+vx}{\sqrt{1-v^2}}$ in perturbation theory. Start with Galilean transformation $x \rightarrow x + v_gt$. Add a transformation $t \rightarrow t + \delta t$ and solve for $\delta t$ assuming it is linear in $x$ and $t$ and preserves $t^2 - x^2$ to $O(v^2)$ with $v_g = v$. Repeat for $\delta t$ and $\delta x$ to second order in $v$ and show that the result agrees with the second-order expansion of the full transformations for some function $v(v_g)$.
Could someone explain to me what "...for some function $v(v_g)$ means?"
Expanding $x$ and $t$ as perturbative series in $v$:
$$x' = \sum_{n=0}^{\infty} v^n (a_nx+b_nt)$$ $$t' = \sum_{n=0}^{\infty} v^n (c_nx+d_nt)$$
Where $a_0 = 1$, $b_0 = 0$, $c_0 = 0$, $d_0 = 1$, $b_1 = 1$ to conform with Galilean transformation.
Requiring Lorentz invariance (*):
$$x'^2 - t'^2 = \sum_{n,m=0}^{\infty} v^{n+m} \bigg((a_nx+b_nt)(a_mx+b_mt)-(c_nx+d_nt)(c_mx+d_mt) \bigg)=x^2-t^2$$
Order $O(v^0)$: $$(a_0x+b_0t)^2-(c_0x+d_0t)^2 = x^2 - t^2$$ $$(a_0^2-c_0^2)x^2+2(a_0b_0-c_0d_0)xt+(b_0^2-d_0^2)t^2=x^2-t^2$$ which is consistent with the required $0$ order terms above.
Order $O(v^1)$: $$(a_0x+b_0t)(a_1x+b_1t)-(c_0x+d_0t)(c_1x+d_1t)=0$$ $$x(a_1x+b_1t)-t(c_1x+d_1t)=0$$ which gives $a_1 = d_1 = 0$ and using $b_1 = 1$ we find $c_1 = 1$
Order $O(v^2)$: $$2x(a_2x+b_2t)-2t(c_2x+d_2t)+t^2-x^2=0$$ $$(2a_2-1)x^2+(1-2d_2)t^2+2(b_2-c_2)xt=0$$
Now $a_2 = d_2 = \frac{1}{2}$, and here is where the trouble is $b_2 = c_2$, which is completely undetermined.
Order $O(v^3)$: $$x(a_3x+b_3t)-t(c_3c+d_3t)+t(a_2x+b_2t)-x(c_2x+d_2t)=0$$ $$x^2(a_3-c_2)+xt(b_3-c_3+a_2-d_2)+t^2(-d_3+b_2)=0$$ and now the rest of the series depends on the $b_2$ term, which is undetermined.
Why doesn't Lorentz invariance uniquely fix all the coefficient terms, despite the fact that we are considering $SO(1,1)$, with an assumed origin (so no translations allowed)? Is this a mathematical artefact of perturbative series?