Physlib.Relativity.Special.TwinParadox.Basic

In a twin paradox scenario $T$, the proper time experienced by Twin A traveling at a constant speed from the starting spacetime point $p_{\text{start}}$ to the ending spacetime point $p_{\text{end}}$ is defined as the Minkowski interval between these points, given by $\sqrt{\langle p_{\text{end}} - p_{\text{start}}, p_{\text{end}} - p_{\text{start}} \rangle_{\eta}}$, where $\langle \cdot, \cdot \rangle_{\eta}$ denotes the Minkowski inner product.

In the twin paradox scenario, the total proper time experienced by Twin B is the sum of the proper times for the two legs of their journey. If $p_{\text{start}}$, $p_{\text{mid}}$, and $p_{\text{end}}$ denote the starting, intermediate, and final spacetime points respectively, the total proper time $\Delta \tau_B$ is defined as: $$\Delta \tau_B = \tau(p_{\text{start}}, p_{\text{mid}}) + \tau(p_{\text{mid}}, p_{\text{end}})$$ where $\tau(q, p)$ represents the proper time between two spacetime points $q$ and $p$, calculated as the Minkowski interval $\sqrt{\langle p - q, p - q \rangle_{\eta}}$.

In a twin paradox scenario $T$, the age gap is defined as the difference between the proper time experienced by Twin A ($\Delta \tau_A$) and the total proper time experienced by Twin B ($\Delta \tau_B$), given by the formula $\Delta \tau_A - \Delta \tau_B$.

In the twin paradox scenario with instantaneous acceleration, the age gap between Twin A and Twin B is non-negative. Specifically, if $\Delta \tau_A$ is the proper time elapsed for Twin A traveling at a constant speed directly between two spacetime points, and $\Delta \tau_B$ is the proper time for Twin B traveling between the same points via an intermediate spacetime point, then the age gap $\Delta \tau_A - \Delta \tau_B \geq 0$, which implies that Twin A is older than Twin B upon their reunion.

This definition provides a specific example of the instantaneous twin paradox scenario in $1+3$ dimensional Minkowski spacetime. The scenario is characterized by the following three spacetime coordinates $(t, x, y, z)$: - The starting point for both twins is the origin $P_{\text{start}} = (0, 0, 0, 0)$. - The meeting point at which the paradox is evaluated is $P_{\text{end}} = (15, 0, 0, 0)$. - Twin B makes a detour through an intermediate point $P_{\text{mid}} = (7.5, 6, 0, 0)$. Twin A travels at a constant velocity directly from $P_{\text{start}}$ to $P_{\text{end}}$. Twin B travels at a constant velocity from $P_{\text{start}}$ to $P_{\text{mid}}$, and then at a different constant velocity from $P_{\text{mid}}$ to $P_{\text{end}}$. The definition includes proofs that $P_{\text{mid}}$ causally follows $P_{\text{start}}$ and $P_{\text{end}}$ causally follows $P_{\text{mid}}$ (and $P_{\text{start}}$), ensuring that the trajectories are physically realizable within the future light cone.

In the specific twin paradox scenario (Example 1) where Twin A travels at a constant velocity directly from the starting point $P_{\text{start}} = (0, 0, 0, 0)$ to the ending point $P_{\text{end}} = (15, 0, 0, 0)$ in Minkowski spacetime, the proper time $\Delta \tau_A$ experienced by Twin A is equal to $15$.

In the specific twin paradox scenario where Twin B travels from the start point $P_{\text{start}} = (0, 0, 0, 0)$ to an intermediate point $P_{\text{mid}} = (7.5, 6, 0, 0)$ and finally to the end point $P_{\text{end}} = (15, 0, 0, 0)$ in $1+3$ dimensional Minkowski spacetime, the total proper time experienced by Twin B, calculated as the sum of the proper times for each leg of the journey $\Delta \tau_B = \tau(P_{\text{start}}, P_{\text{mid}}) + \tau(P_{\text{mid}}, P_{\text{end}})$, is equal to $9$.

In the specific twin paradox scenario (Example 1) with spacetime points $P_{\text{start}} = (0, 0, 0, 0)$, $P_{\text{mid}} = (7.5, 6, 0, 0)$, and $P_{\text{end}} = (15, 0, 0, 0)$, where the proper time experienced by Twin A is $\Delta \tau_A = 15$ and the total proper time experienced by Twin B is $\Delta \tau_B = 9$, the resulting age gap—defined as the difference $\Delta \tau_A - \Delta \tau_B$—is equal to $6$.