Worked construction of a future-oriented inertial frame from an arbitrary orthonormal basis

By [L1] and [L3], $\eta (f_0,f_0)=\eta (-f_0,-f_0)=-1$, so $f_0,-f_0\in T$. They lie in opposite components: if both lay in the same component, its convexity [L2] would put $0=\frac{1}{2}{f}_0+\frac{1}{2}(-f_0)$ in $T$, contradicting $\eta (0,0)=0$. Thus $g_0:=f_0$ when $f_0\in C^{+}$ and $g_0:=-f_0$ otherwise is always future. Set $g_i:=f_i$ for $i=1,2,3$. Retaining $f_0$ retains the basis, while negating it preserves its norm and all vanishing off-diagonal pairings, since $\eta (-f_0,f_i)=-\eta (f_0,f_i)=0$; hence $(g_0,g_1,g_2,g_3)$ is $\eta$-orthonormal. Its change-of-basis matrix is the identity when $f_0$ is future and $\mathrm{diag}(-1,1,1,1)$ when $f_0$ is past. Their determinants are $1$ and $-1$, respectively, so [L4] shows that the orientation is retained in the first case and reversed in the second. In particular, past-directed negative input becomes future-directed positive input.

In the two-dimensional illustration, $f_0=(-1,0)$ has $\eta (f_0,f_0)=-1<0$ and $f_0^0=-1<0$, so $f_0$ is past timelike by the explicit two-dimensional convention in the Example; $-f_0=(1,0)$ has $\eta (-f_0,-f_0)=-1$ by [L3], $(-f_0{)}^0=1>0=\mid (-f_0{)}_{\mathrm{v}\mathrm{e}\mathrm{c}}\mid$, so $-f_0$ is future timelike by that same convention; and $\eta (-f_0,f_1)=0$, $\eta (f_1,f_1)=1$ are direct evaluations. Hence $(-f_0,f_1)$ is an $\eta$-orthonormal pair with future first vector and with change-of-basis determinant $-1$, so it is positively oriented exactly when the original pair was negatively oriented.

If $(g_0,g_1,g_2,g_3)$ is positively oriented, retain it. Otherwise replacing $g_1$ by $-g_1$ preserves $\eta$-orthonormality by the sign computation of step 1.1, leaves the future first vector $g_0$ unchanged, and has change-of-basis matrix $\mathrm{diag}(1,-1,1,1)$ of determinant $-1$. There are exactly two orientation classes by [L4], so reversing the negative class gives the positive class. Interchanging $g_1$ and $g_2$ also has determinant $-1$ and preserves orthonormality and $g_0$, giving the same alternative correction.

In every initial time-orientation and orientation case, steps 1.1 and 2.1 produce an $\eta$-orthonormal, future, positively oriented ordered basis, which together with the chosen origin $o$ is an inertial frame by [L5]. Step 1.2 verifies the displayed two-dimensional illustration. Thus the recipe realizes the construction of Existence of future-oriented inertial frames at every event from arbitrary orthonormal input, including the past-directed negative example.