216 Time
An absolute parameter that is used to order events and mark how long motion lasts in classical physics, where an absolute parameter is a single shared clock that does not depend on the observer.
Absolute time. Time flows as an independent backdrop that is the same everywhere and is not changed by matter or by the motion of observers. This principle is used to compute velocity and acceleration without transforming the time variable between frames.
The Galilean time transformation is
\[ t' = t \]
where
- \(t\) is the time of an event in one inertial frame.
- \(t'\) is the time of the same event in another inertial frame.
Universal time. All observers share one clock, so a duration between two events is the same in every inertial frame. This principle is used to write the equations of motion of many bodies with one common time.
The invariance of a time interval is
\[ \Delta t = \Delta t' \]
where
- \(\Delta t\) is the duration in one inertial frame.
- \(\Delta t'\) is the duration of the same pair of events in another inertial frame.
Homogeneity of time. The laws of physics do not depend on when an experiment is done. Homogeneity of time is that uniformity under a shift of the clock. This principle is used to guarantee that a closed system keeps its energy.
Independence of the Lagrangian from explicit time is
\[ \dfrac{\partial L}{\partial t} = 0 \]
where
- \(L\) is the Lagrangian.
- \(t\) is time.
Absolute simultaneity. If two events share the same time for one observer, they share the same time for every observer. Simultaneity is the sharing of one time coordinate. This principle is used to give a universal meaning to the word now in classical physics.
Absolute simultaneity is
\[ t_{A} = t_{B} \iff t'_{A} = t'_{B} \]
where
- \(t_{A}\) and \(t_{B}\) are the times of two events in one frame.
- \(t'_{A}\) and \(t'_{B}\) are the times of those events in another frame.
The separation of time from space. Time is kept separate from the three dimensions of space and enters the laws as the independent variable of motion. This principle is used to write a classical path as a function of one time parameter.
Note: Also called Newtonian time. Homogeneity of time is also called time-translation symmetry.
216.1 References
- Lovett, S. T. Differential Geometry of Manifolds. CRC Press, 2020. §7.2 — classical mechanics treats time as absolute and independent of any frame.
- Shankar, R. Fundamentals of Physics I. Yale University Press, 2019. §12.1, §13.1 — one time for all observers, and time-translation invariance.
- Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — only one kind of time in classical physics, measured the same independent of motion.