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Mathematics for Physics

193 Proper Time

The time interval between two events measured by a clock present at both events, equal to the elapsed time in that clock’s rest frame.

Wristwatch time. Proper time is the time between two events on one clock that is present at both events. A worldline is the path of that clock through spacetime. This principle is used to define the elapsed time experienced by a traveler.

The proper time for a clock moving at constant speed \(v\) is

\[ \Delta\tau = \Delta t\sqrt{1 - \dfrac{v^{2}}{c^{2}}} \]

where

  • \(\Delta\tau\) is the proper time.
  • \(\Delta t\) is the coordinate time in the frame where the clock moves at speed \(v\).
  • \(v\) is the speed of the clock.
  • \(c\) is the speed of light.

Invariance of proper time. Every inertial observer computes the same proper time between those two events. A Lorentz invariant is a quantity that does not depend on which inertial frame is used. This principle is used to give an objective clock time for a chosen worldline.

Interval length of a worldline. Proper time is the length of a timelike worldline as measured by the spacetime interval. A spacetime interval is the invariant four-dimensional squared separation of two events. This principle is used to convert coordinate displacements along a path into one duration.

Maximal aging on an inertial path. Among worldlines joining the same two events, the straight inertial path has the largest proper time. Unaccelerated motion is motion at constant velocity. This principle is used to compare the aging of travelers who take different paths.

Time dilation. A moving clock records less proper time than the coordinate time of the frame in which it moves. Time dilation is that slowing of a moving clock relative to synchronized clocks at rest in one frame. This principle is used to relate wristwatch time to laboratory time.

Note: Also called \(\Delta\tau\).

193.1 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. — clock present at both events; \(\Delta\tau=\Delta t\sqrt{1-\dfrac{v^{2}}{c^{2}}}\); time dilation.
  3. Carroll, S. M. Spacetime and Geometry. Cambridge University Press. — proper time as interval length; maximal proper time on a straight path.
  4. Emam, M. H. Covariant Physics. Oxford University Press, 2021. — invariance of proper time.
  5. Shankar, R. Fundamentals of Physics I. Yale University Press, 2019. — wristwatch time and time dilation.
  6. Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. — moving clocks run slow.