## Time Dilation and Desynchronization

Previous section: Spacetime Trigonometry.

Next sectionLorentz Contraction.

The basic idea is illustrated by the parable of the twins. As illustrated in the spacetime map (below) for their histories, the astronaut twin travels to a distant star with velocity

$\Large{V\,\,\, = \,\,\,\gamma \,\left( {1\,\, + \,\,\frac{{\bf{v}}}{c}} \right)\,\,,\,}$

and returns with velocity

$\Large{\widetilde V\,\,\, = \,\,\,\gamma \,\left( {1\,\, - \,\,\frac{{\bf{v}}}{c}} \right)\,\,.\,}$

The algebraic equation relating the home twin's aging to the astronaut twin's aging is

$\Large{2ct\,\,\, = \,\,\,Vc\tau \,\, + \,\,\widetilde Vc\tau \,\,\, = \,\,\,2\gamma c\tau }$

Exercises:

(1) Compare ages of the twins when the trip is over. Discuss implications of this result.

(2) Prove that the longest path between two points separated by a timelike interval is a
straight line.
(
See 2nd figure above)

(
Click here for solutions to these exercises.)

Previous sectionSpacetime Trigonometry.

Next section: Lorentz Contraction.