Twin Paradox - Difference in Elapsed Times: How To Calculate IT From The Ship

Difference in Elapsed Times: How To Calculate It From The Ship

In the standard proper time formula

Δτ represents the time of the non-inertial (travelling) observer K' as a function of the elapsed time Δt of the inertial (stay-at-home) observer K for whom observer K' has velocity v(t) at time t.

To calculate the elapsed time Δt of the inertial observer K as a function of the elapsed time Δτ of the non-inertial observer K', where only quantities measured by K' are accessible, the following formula can be used:

where aτ is the proper acceleration of the non-inertial observer K' as measured by himself (for instance with an accelerometer) during the whole round-trip. The Cauchy–Schwarz inequality can be used to show that the inequality Δt > Δτ follows from the previous expression:

\begin{align} \Delta t^2 & = \left \,\left \\ & > \left^2 = \left^2 = \Delta \tau^2. \end{align}

Using the Dirac delta function to model the infinite acceleration phase in the standard case of the traveller having constant speed v during the outbound and the inbound trip, the formula produces the known result:

In the case where the accelerated observer K' departs from K with zero initial velocity, the general equation reduces to the simpler form:

which, in the smooth version of the twin paradox where the traveller has constant proper acceleration phases, successively given by a, −a, −a, a, results in

where the convention c = 1 is used, in accordance with the above expression with acceleration phases Ta = Δt/4 and coasting phases Tc = 0.

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