Special Relativity and clocks

Applet by Fu-Kwun Hwang

The special theory of relativity is the result of two postulates:

1. The laws of physics are the same in all inertial reference frames. Extension from the Newtonian principle of relativity: the laws of mechanics are the same for all observers in uniform motion.

2. The speed of light in empty space is the same for all inertial frames. The speed of an object depends on the reference frame, however the speed of light in space is the same. From the above two postulates, our understanding of space and time has to be modified. This java applet invites you to the world of space and time in special relativity.

There are two devices that utilize photons to measure time differences (some kind of clock). A mirror will reflect the photon when it reaches the top or bottom of the device.

First, press the Start button to begin the animation. Both devices are synchronized.

Two Light pulses emitted from the ends (yellow rings) reach the center of the device at the same time. There is no relative motion between the two devices. Now, change the relative velocity from the selections (0.6c or 0.8c), where c is the speed of light in a vacuum. One of the devices starts to move relative to your frame of reference. You can change your frame of reference by moving your mouse button within the area of the blue device or out of it. You can toggle the animation by clicking the right mouse button.

The width of the moving device becomes smaller (from the markers, figure out the contraction factor!) and the photons are not synchronized. Light from the two devices is initiated when the sources touch each other. Two light cones (in yellow) from two ends will reach the center of the device in your frame of reference at the same time. So, those two events are simultaneous in your reference frame.

However, one light cone arrives at the center of the moving device earlier than the other one. So, those two events are not simultaneous in the moving frame.

There is no such thing as absolute simultaneity. Time is relative: it depends on the space-time coordinate system. Try to change your frame of reference by moving your mouse in and out of the moving device (the timing in your frame of reference will reset to 0).

When you look out the window, the scene on your retina - the scene you see - is not all happening at the same moment. The stars in a photo of the night sky were not all there looking as they do at the same moment, even though their light arrived on the film at the same instant. If you saw two stars explode at the same time, one of the events might have happened earlier to an observer from some other galaxy.

The period of the clock in your frame is 1.0s.

The number T at the left shows the period for the photon's motion measured in your rest frame. The period of the moving frame large than 1.0s, so the moving clock runs slower. An observer at rest with the clock sees the pulse moving up and down with speed c.

The picture is very different when viewed from the other frame.

As seen by the observer who is stationary with respect to the moving device, the pulse travels a distance given by ct' (red path in the following figure). In your frame, the pulse travels a longer diagonal path (white path).

The speed of light is the same to all observers in inertial frames.

The pulse seen by you must take a longer time t to traverse the longer distance (ct). During the same interval, the moving device advances a distance (vt). It follows from the Pythagorean Theorem that:

ct2 = vt2 + ct'2
t'2 = t2 (c2 - v2)/c2
t = (1 - (v/c)2) - 1/2 t' = γt'

The time interval seen by the outside observer with respect to whom the clock is moving must be greater than the corresponding time interval seen by the inside observer with respect to whom the clock is stationary.