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Mathematics for Physics

205 Time Dilation

The relativistic effect that a clock moving relative to an observer ticks more slowly than an identical clock at rest in that observer’s frame, that is used to relate proper time to time in a moving frame.

Proper time as the shortest interval. The time between two events is shortest in the frame where those events occur at the same place. Proper time is the duration measured by one clock present at both events. This principle is used to fix the baseline interval for aging and decay.

Relativistic clock slowing. A clock moving relative to an observer runs slower than an identical rest clock in that observer’s frame. Coordinate time is the time difference between the events as read on two synchronized clocks at rest in the observer’s frame. This principle is used to compute the stretched duration of a moving process.

The time dilation formula is

\[ \Delta t = \gamma\,\Delta t_{0} \]

where

  • \(\Delta t\) is the coordinate time in the observer’s frame.
  • \(\Delta t_{0}\) is the proper time.
  • \(\gamma\) is the Lorentz factor.

Geometric light-path extension. Time dilation follows from a longer light path in a moving light clock together with the constancy of the speed of light. A light clock is a device that times a light pulse bouncing between two mirrors. This principle is used to derive the time-dilation formula from geometry.

Reciprocal observation of rate. Each of two observers in uniform relative motion measures the other’s clocks as running slow. Reciprocity is that mutual slowing. This principle is used to show that the slowing is a property of spacetime geometry, not of a clock’s inner parts.

205.1 References

  1. Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — source for the heading explanation.
  2. Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — \(\Delta t=\gamma\Delta t_{0}\); proper time as shortest interval.
  3. Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. — moving clocks run slow; reciprocal slowing.
  4. Emam, M. H. Covariant Physics. Oxford University Press, 2021. — time dilation formula.
  5. Shankar, R. Fundamentals of Physics I. Yale University Press, 2019. — proper time; light clock; reciprocity.
  6. Carroll, S. M. Spacetime and Geometry. Cambridge University Press. — light-clock geometry.