= Invariance of reunion ages under clock resynchronization
{title2=$s=t+\kappa x/c$}
Consider a change of distant-time convention $s=t+\kappa x/c$, where $\kappa$ is constant and $|\kappa|<1$. This changes time labels and simultaneity slices while leaving the <world lines> and the <metric tensor> as geometric objects unchanged. In one spatial dimension,
$$
ds_{\rm spacetime}^2=-c^2ds^2+2\kappa c\,ds\,dx+(1-\kappa^2)dx^2.
$$
Substitute $dt=ds-\kappa\,dx/c$ into the <Minkowski interval> to obtain the formula. For a <timelike curve>, the <proper time> calculated from this transformed <metric tensor> equals the old value. At shared departure and reunion events with $x=0$, the labels also obey $s=t$. Thus a distant synchronization convention cannot alter the <twin paradox>'s reunited <clock> readings. It would be an error to change time labels while retaining the old diagonal metric components.
Back to article page
