Yesterday astronaut Scott Kelly returned from nearly a year in free fall aboard the International Space Station to join his identical twin brother Mark back on Earth. Due to their different spacetime paths, I estimate that Scott aged about 9 ms less than his brother, and therefore travelled about 9 ms into the future, becoming one of Earth’s most accomplished time travelers.
The familiar Pythagorean line element dl2 = dx2 + dy2 + dz2 (and corresponding metric) describes the geometry of Euclidean space. The Lorentzian line element – dτ2 = ds2 = – dt2 + dx2 + dy2 + dz2 = – dt2 + dl2 (and corresponding semimetric) describes the flat spacetime of special relativity, where space is measured in light-years and time in years. The Einsteinian line element dτ2 = gμνdxμdxν describes the curved spacetime of general relativity, with an implied sum over the indices. From third semester physics, in flat spacetime the proper time increment dτ = √(dt2 – dl2) = dt√(1 – v2) = dt/γ, where the relativistic stretch γ ≥ 1 regulates the time dilation. More generally, the length Δτ = ∫ dτ of a spacetime worldline is the proper time or aging along it (which is most evident in the observer’s rest frame).
In the curved, approximately Kerr spacetime of the rotating Earth, clocks tick faster with increasing altitude but slower with increasing speed. In low Earth orbit aboard the ISS, the speed effect dominates the altitude effect, sending Scott Kelly about 10 ms – 1 ms = 9 ms into the future. Furthermore, in curved spacetime, multiple free fall or geodesic paths between the same two events can have different lengths or aging, which can desynchronize clocks or twins without proper (as opposed to coordinate) acceleration — and without paradox.