Question/Doubt about Time Dilation Symmetry in Special Relativity

You write: "From the Spaceship’s point of view, the first signal was sent from Earth when it was 4.8 Spaceship light years away. Thus it would take 4.8 Spaceship years to reach the Spaceship."

Be careful. I, on earth, send out a light signal at the same time (according to me) that the space ship takes off. According to you (on the now-moving spaceship), I sent that signal 10.67 years ago (at time -10.67 by your clock). Also, according to you, I've spent that 10.67 years traveling towards you at speed .8. Therefore, when the light signal was sent, I was not just 4.8 light years away; I was 4.8 plus another (10.67 x .8) light years away --- a total of about 13.33 light years. So the light signal should take a total of 13.33 years to reach you. As it was sent at time -10.67 it should arrive at time 2.67. (I hope I got the arithmetic right!).

The right way to do this, of course, is not to grind through the arithmetic, but to draw the spacetime diagram, which makes everything clear from the start. The black line is the earth, the blue line is the distant planet, the red line is the traveler, and the gold lines are light rays, sent every two years from earth and every two years from the ship. The black numbers are earth times and the red numbers are traveler-times.

This may or may not be helpful.

The diagram above shows the following in the planet frame:

• The solid blue line is Earth, which remains 8 light years distant from the planet at all times. I've drawn dots for t=-8 to t=10.

• The solid green line is the planet, which remains at x=0 at all times. I've drawn dots for t=0 to t=18.

• The solid red line is the ship, which travels from the planet to Earth in 10 years, covering 8 light years, at least in the planet's frame. Note that the dots are drawn every year of planet time, not every year of ship time.

• The dashed blue lines are light-speed signals traveling from Earth to the planet, intersecting the ship as they pass.

• The dashed red lines are light-speed signals traveling from the ship to Earth.

Caveats:

• The x and t scale are not equal on this graph. Because we're dealing with a distance of 8 light years and a timeframe of 26 years, I couldn't find a good way to create a graph with x and t scaled equally.

• I chose x as horizontal and t as vertical. Some people do it the other way.

Things To Notice:

• When the ship leaves the planet, it sees Earth's signal from t = -8. In other words, it sees the Earth as it looked at t = -8. This isn't the time it assigns Earth, it's just what Earth light it's currently receiving.

• By the time the ship arrives, it sees Earth at t = 10. Again, this isn't the time it assigns Earth, just what it actually sees.

• Thus, considering only the light actually reaching the ship, 18 years passes on Earth during the ship's 10 year journey. Notice that this has nothing to do with dilation or contraction, it's pretty much just the Doppler Effect.

• In your example, Earth only starts sending signals at t=0. In this case, the ship receives the first signal at about 4.44 years (still in the planet's frame), a little before it's halfway through it's journey.

• Since the ship still receives the last signal at t=10, we note the ship receives 11 signals during its 10 year journey, but the first signal doesn't arrive until t = 4.4 (Earth time, not ship time), so it really receives the 11 signals in 5.56 years (one signal every 0.505 years or so).

• The first signal the ship sends doesn't arrive until Earth time t = 8. This makes sense, since, in this frame, the Earth is 8 light years away.

• The last signal the ship sends arrives at t = 10 (just as the ship is landing). Thus, the Earth sees 10 signals arriving in the space of two years, one signal every 0.2 years.

Now, let's do a Minkowski transform into the ship's frame.

Just for fun, we'll animate the change as we move from the planet's frame to the ship's frame (for the purposes of the animation, I've removed the labels giving Earth's time).

Focusing on the final image:

While this diagram is important, let's zoom in a bit:

Notice what we see here:

• The ship now remains at x=0 throughout its journey. In other words, we are now looking at things from the ship's point of view.

• When the ship leaves, it still sees light from Earth at t = -8 Earth time.

• When the ship arrives, it still sees light from Earth at t = 10 Earth time.

• During its journey, the ship still sees 11 signals (if we ignore those where Earth t < 0).

• The first signal the ship sends still arrives at Earth time t = 8, and the last signal still arrives at Earth time t = 10.

In other words, in terms of what the ship and Earth actually see, nothing has changed.

The only difference is that we can now talk about ship time and ship distance, instead of planet time and planet distance.

Note that if we look at t = 0, we see the Earth at x = 4.8 (the Lorentz contraction) and at Earth time t = 6.4.

In some sense, however, this is artificial. The light from Earth at t = 6.4 will still reach the ship at the same point it is journey as it did before.

Further, the Earth that's 4.8 light years away is not Earth at t = 0, but Earth at t = 6.4.

Ultimately, regardless of the coordinate system, an observer in a given frame will see the same event at the same time and same distance.

The Lorentz contraction and simultaneity are artificial constructs that insures this happens.