Twins in relativistic spacetimes: dispelling some misconceptions
In contrast to Newtonian physics, there is no absolute time in relativistic (Lorentzian) spacetimes. This immediately implies that two twins may, in general, age at different rates. For this to happen, there must be, of course, some asymmetry between their worldlines, along which the elapsed proper times are evaluated; such asymmetry might not be, however, so intuitively apparent. Our primary objective is to present, in a concise and didactic manner, some lesser-known results and derive novel ones from a modern, geometrical, and covariant standpoint, which aims to clarify the issue and dispel related misconceptions. First, we recall that: (i) the original ``twin paradox'' may be perfectly dealt with in special relativity (physics in a flat spacetime) and does not necessarily involve an accelerated twin. We then explore the issue of differential aging in general relativity (physics in a curved background) in the prototypical case of the vacuum Schwarzschild spacetime, considering several pairs of twins. In this context, we show that: (ii) it is not true that a twin which gets closer to the Schwarzschild horizon, by being subject to a stronger gravitational field, where time sort of slows down, should always get younger than a twin that stays further away, in a region of weaker gravitational field, and (iii) it is also false that an accelerated twin always returns younger than a geodesic one. Finally, we argue that (iv) in a generic spacetime, there is no universal correlation between the phenomena of differential aging and the Doppler effect. Two particularly pedagogical resources provided are a glossary of relevant terms and supplementary Python notebooks in a GitHub repository.
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Stephen Moreton-Howell
Jeffrey Roe