 Solution

With metric convention $(+---)$, the invariant interval is $c^2\Delta {\tau }^2=c^2\Delta t^2-|\Delta \mathbf{r}{|}^2$ along an inertially moving clock. For constant speed $v$ in the ground frame, $|\Delta \mathbf{r}|=v\Delta t$, soThe aircraft records proper time $\Delta \tau =1000\mathrm{s}$. The ground duration is measured by a network of clocks at rest in the ideal ground frame, with Einstein synchronisation. Its excess isThus the ground-frame duration is longer by about $0.50\mathrm{ns}$. This is the special-relativistic time dilation contribution under the stated constant-speed assumptions; gravitational potential differences and Earth's rotation are outside this model.