Source

Twin Paradox Animated Spacetime Diagram

Adapted from

Twin Paradox Simulation

Left clock: Earth clock, with one full rotation corresponding to one grid spacing on the time axis of the diagram.

Middle clock: ship clock (as seen by ship), time-dilated in the twin's frame of reference.

Right clock: seen by Earth of ship clock through a telescope.

The right clock (unsteady rate!) 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 middle clock.


Distance ly Velocity c
As seen by the Earth, the Relativistic Doppler effect will alter the "rates" of signals coming to the Earth:
outbound = sig/Earth year, inbound = sig/Earth year
 Black dot: Earthbound twin.

Green dot: simultaneous (according to Earth) position of the traveling twin.

Red dot: what the Earthbound twin actually sees. (Intersection of 45° light-path & green trail.)

The red dot lags the other green dot, 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 (Doppler shift) and rapidly catches up to the green dot.

To the left of each clock is a time bar, indicating the total number of rotations of the clock, i.e., the age. Eg: with distance = 1 cyear, and velocity = 0.8 c ... pause when Earth clock = 1.25 years .. note that the ship twin is "at" the planet, but the Earth has not seen the signal yet. In those 2.5 years, observe:

  1. The Earth Clock moves at a constant rate (arrow sweeps around) (as seen by the EARTH observer).
  2. 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).
  3. The Ship Clock moves at a constant rate — but is some fraction of the Earth clock (as seen by the SHIP observer).
  4. 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!).
  5. 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 bottom).
  6. 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.
  7. We could think of the trailing dot as being "where was the ship physically, when it sent the signal I just received".
  8. 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!
  9. 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!)
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