Elapsed Time

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60. The standing wave exists in a microwave resonator if the length of the resonator cavity is equal to multiple half-wavelengths of microwave. The stationary interference of standing wave will travel in another inertial reference frame. The vibrating pattern of the standing wave is conserved.

The existence of nodes in all reference frames requires the wavelength of the microwave to be conserved in all inertial reference frames. The angular frequency of microwave is different in every reference frame. Hence, the apparent velocity of the microwave depends on the choice of reference frame while the elapsed time remains invariant in all reference frames.

63. A parity symmetry can be established between any pair of identical clocks. The rest frames of each clock can also form a parity symmetry.

The time of each rest frame is independent of the direction of the motion of each clock. The elapsed time is identical for both rest frames.

Consequently, the elapsed time is conserved in all reference frames. Conservation of elapsed time is a property of parity symmetry.

30. The parity symmetry in physics shows that the motions in two different reference frames are related to each other. By comparing the displacement and the velocity from two reference frames, the elapsed time can be shown to be conserved in both reference frames. For two frames in relative inertial motion, the elapsed time is conserved in all inertial reference frames. If both frames are in circular motion, then the elapsed time is conserved in all local reference frames on the same circle. If both frames are free to move in any direction at any speed, then the elapsed time is conserved in all non-inertial reference frames. Two simultaneous events are simultaneous in all reference frames.

28. The application of symmetry to physics leads to conservation law and conserved quantity. For inertial reference frames, the reflection symmetry generates not only conservation but also transformation. Under reflection symmetry, the elapsed time is conserved in all inertial reference frames. The displacement in space is also conserved in all inertial reference frames. From the conservation of the elapsed time and the displacement, the coordinate transformation between inertial reference frame is derived. Based on the coordinate transformation, both the time transformation and the velocity transformation are also derived. The derivation shows that all three transformations are dependent exclusively on the relative motion between inertial reference frames.

26. The relative reflection symmetry exists for an isolated system of two stationary persons. The first person sees the second person in a distance away. The second person sees the first person in the same distance away but in the opposite direction. Such symmetry also exists for two mobile persons. Both persons see each other moving at the same speed but in opposite direction in their own rest frames. From the definition of velocity, the time in the rest frame of the first person can be compared to the time in the rest frame of the second person. The result shows that the time in the rest frame of the first person differs from the time in the rest frame of the second person by a constant. Two simultaneous events in one inertial reference frame are also simultaneous in another inertial reference frame.

21. In the projectile motion under vertical gravity, the horizontal speed remains constant while the vertical speed increases. The elapsed time to travel over a fixed length in the horizontal direction remains constant regardless of the vertical speed. Such elapsed time is independent of any reference frame in vertical motion. Therefore, the elapsed time is independent of the reference frame. Consequently, time is independent of the reference frame.

19. Time in a reference frame can be represented by a rotating ring with constant angular velocity. The rotational motion of this ring is not affected by any acceleration parallel to its axis of rotation. The rotation period remains constant as the ring accelerates along the axis of rotation. The rotation period in the rest frame of the ring is always constant. Therefore, the rotation period is independent of reference frame. The rotation period represents the elapsed time in a reference frame. As a result, the elapsed time is also independent of reference frame. Consequently, time is independent of reference frame.

4. Two identical stopwatches moving at the same speed will elapse the same time after moving the same distance. If both stopwatches were started at the same time, there will be no time difference between these two stopwatches after both stopwatches have elapsed the same time. Both stopwatches will continue to show no time difference under identical acceleration. Therefore, both stopwatches show identical time in an accelerating reference frame if both stopwatches were restarted at the same time in a stationary reference frame. Consequently, a physical system that exhibits Translational Symmetry in its motion demonstrates that two simultaneous events in one reference frame should be simultaneous in another reference frame.

2. Two identical stopwatches moving at the same speed will elapse the same time after moving the same distance. Start one stopwatch later than the other stopwatch. The time difference between these two stopwatches will remain constant after both stopwatches have elapsed the same time. Such time difference will remain constant while both stopwatches are under identical acceleration. Therefore, the elapsed time in an accelerating reference frame is identical to the elapsed time in a stationary reference frame. Consequently, a physical system that exhibits Reflection Symmetry in its motion demonstrates that the time of a moving clock is independent of the relative motion between the clock and its observer.