Relativity : Twin Paradox

Description

Three clocks are shown - far left = Earth clock (steady rate as seen by Earth) - middle = ship clock (steady rate as seen by ship) - far right = What Earth Sees of Ship Clock (unsteady rate!).

The right side of the animation has 3 moving dots. The black dot corresponds to the Earthbound twin. The leading green dot gives the simultaneous (according to Earth) position of the traveling twin. The trailing  red dot, which will change to blue, indicates what the Earthbound twin actually sees. (The color change represents the Doppler shift as the earthbound twin looks at his brother.)  It lags the other red dot on the outbound leg, as the image of the traveling twin received by the Earthbound twin is late due to the light travel time. On the return leg, this dot turns blue and rapidly catches up to the green dot.

The clock on the left shows proper time for the Earthbound observer. It moves at a constant rate, with one full rotation corresponding to one grid spacing on the time axis of the diagram.
The middle clock corresponds to the leading green dot.  This is the time in the twin's frame of reference and is time-dilated according to special theory of relativity.
The clock on the right shows what the Earthbound observer would actually see through a telescope focused on his traveling twin's clock. This time takes into account both the relativistic Doppler effect and the light travel time. If the earthbound twin were to subtract the light travel time from these readings, he would obtain the values shown in the second clock.  Note that it is slow up until almost the very end, when it speeds up as the blue-shifted signals from the traveling twin begin to arrive back at Earth. 

To the left of each clock is a time bar, indicating the total number of rotations of the clock, i.e., the age. The leftmost bar shows the age of the Earthbound twin, the middle bar gives the simultaneous (according to Earth) age of the traveling twin, and the rightmost bar gives the age of the traveling twin as viewed by the Earthbound twin through his telescope.

Example : with distance = 12 cyears, and velocity = 0.6 c ... pause when Earth clock = 20 years .. note that the ship twin is "at" the planet, but the Earth has not seen the signal yet. In those 20 years,

Important "conceptual connections" with the clocks and the graph on the right:

• The Earth Clock moves at a constant rate (arrow sweeps around) (as seen by the EARTH observer).
• The black dot on the vertical axis of the graph represents the Earth observer (not moving in the distance direction, only moving uniformly upwards in the time direction).
• The Ship Clock moves at a constant rate (arrow sweeps around) - but is some fraction of the Earth clock (as seen by the SHIP observer).
• As the animation starts .. the right-most dot in the graph is the physical ship - it smoothly moves in the distance-time graph out to the planet (notice when that dot gets to the planet, the Earth clock should be half done!).
• The trailing moving dot is the "image" where the ship is in its journey ship, as seen from the Earth .. that is, since there is a delay for the signal to get back to the Earth, when the Earth receives the signal, the ship has moved from that point (the Earth doesn't perceive the Ship getting to the planet as fast as it thought!). Note : let's look at when the ship has actually gotten to the planet, how many signals has the Earth received? (take the Earth clock value at that time, and multiply by the "receeding rate" show in the upper right).
• Why is that "receding rate" less than 1? .. The ship is moving away from the Earth, but sending signals back toward the Earth .. thus, in space, the signals are "stretched out" in relation to each other ... on the way back, the ship sends a signal toward the Earth, and then tries to catch up to it, and sends another signal, etc. .. these signals will "bunch up" relative to each other .. so they will hit in quick succession when they arrive at the Earth.
• We could think of the trailing dot as being "where was the ship physically, when it sent the signal I just received".
• Watch for the trailing dot to "get to the planet" (what that means is that the Earth has received the signal that the Ship sent when it got there!). Where is the actual ship .. very close to the Earth!
• Once the "arrival" signal has been received by the Earth, the signal rate jumps to the "approch" value (because once the "arrival signal" gets to the Earth .. all following signals where sent by a ship that was coming back toward the Earth, trying to catch up to the signals, so they are close together!)

Things to try :

• Set distance = 12 and velocity = 0.6 c (our class example) - watch for