Time Dilation and Desynchronization 

Previous section: Spacetime Trigonometry.

Next section: Lorentz 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

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


Vcτ ct O 2ct History of the solar system Vcτ ～ X₁ X₂ Astronaut outbound Astronaut return X₁ X₂ = Xb X₀ = Xa Light Cone Light Cone ∆τ₀ ∆τ₂ ∆τ₁ Twin at Home Parable of the Twins Longest Path Between Two Points Separated by Timelike Interval



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 section: Spacetime Trigonometry.

Next section: Lorentz Contraction.


© David Hestenes 2005, 2014