Some Consequences of the Lorentz Equations — Question 6

Question 6

Changing inertial frames at turnaround changes the distant simultaneity assignment

A traveler turns around at Earth-frame event $t_T=5.0\,\mathrm{yr}$, $x_T=4.0$ light-years. The outgoing speed is $+0.80c$ and the incoming speed is $-0.80c$. Idealize the turnaround as instantaneous.

1. Find the Earth times simultaneous with turnaround in the outgoing and incoming inertial frames.

2. Find the jump in assigned distant Earth time and explain whether Earth physically ages discontinuously.

Question 6 - Solution

Changing inertial frames at turnaround changes the distant simultaneity assignment

(a) Method and calculation

$$\text{Outgoing: }t_E^{\mathrm{out}}=t_T-vx_T/c^2=5.0-3.2=\boxed{1.8\,\mathrm{yr}}.$$$$\text{Incoming: }t_E^{\mathrm{in}}=t_T+vx_T/c^2=5.0+3.2=\boxed{8.2\,\mathrm{yr}}.$$

(b) Interpretation and further calculation

$$\boxed{t_E^{\mathrm{in}}-t_E^{\mathrm{out}}=2vx_T/c^2=6.4\,\mathrm{yr}}.$$$$\text{Earth clock worldline is continuous; the simultaneity slice changes.}$$

Physical check

The jump is in a coordinate assignment to distant events, not in a local clock reading or a received light signal.