Summarizing Simultaneity and Time Dilation

Simultaneity And Time Dilation Summary

• Two events are defined to be simultaneous if an observer measures them as occurring at the same time. They are not necessarily simultaneous to all observers—simultaneity is not absolute.
• Time dilation is the phenomenon of time passing slower for an observer who is moving relative to another observer.
• Observers moving at a relative velocity \(v\) do not measure the same elapsed time for an event. Proper time \(\Delta {t}_{0}\) is the time measured by an observer at rest relative to the event being observed. Proper time is related to the time \(\Delta t\) measured by an Earth-bound observer by the equation

  \(\Delta t=\cfrac{{\Delta t}_{0}}{\sqrt{1-\cfrac{{v}^{2}}{{c}^{2}}}}={\gamma \Delta t}_{0},\)

  where

  \(\gamma =\cfrac{1}{\sqrt{1-\cfrac{{v}^{2}}{{c}^{2}}}}.\)

• The equation relating proper time and time measured by an Earth-bound observer implies that relative velocity cannot exceed the speed of light.
• The twin paradox asks why a twin traveling at a relativistic speed away and then back towards the Earth ages less than the Earth-bound twin. The premise to the paradox is faulty because the traveling twin is accelerating. Special relativity does not apply to accelerating frames of reference.
• Time dilation is usually negligible at low relative velocities, but it does occur, and it has been verified by experiment.

Glossary

time dilation

the phenomenon of time passing slower to an observer who is moving relative to another observer

proper time

\(\Delta {t}_{0}\). the time measured by an observer at rest relative to the event being observed: \(\Delta t=\cfrac{{\Delta t}_{0}}{\sqrt{1-\cfrac{{v}^{2}}{{c}^{2}}}}={\gamma \Delta t}_{0}\), where \(\gamma =\cfrac{1}{\sqrt{1-\cfrac{{v}^{2}}{{c}^{2}}}}\)

twin paradox

this asks why a twin traveling at a relativistic speed away and then back towards the Earth ages less than the Earth-bound twin. The premise to the paradox is faulty because the traveling twin is accelerating, and special relativity does not apply to accelerating frames of reference