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Suppose I have no idea about relativity and find an ancient spaceship that takes me to Proxima Centauri in just 1 year more than it takes light to make the trip (so some observer might say it took about 5 years).

Now from my point of view it only took 1 year (if it had been at light speed, the trip would have been instant). My new friends at Proxima Centauri also agree that they saw it take 1 year (the light from Earth when I left reached them just 1 year before they saw me arrive). None of us could perceive anything other than classical Newtonian mechanics happening.

Meanwhile others on Earth who never understood relativity would observe that my trip took 9 years and the photos I send to them upon my arrival show I only aged 1 year. They would be surprised.

Is this correct? Meaning for a naïve space traveler from Earth and like-minded observers at the destination, relativity wouldn't seem to set any speed limit?

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  • $\begingroup$ The Doppler effect would give it away, one way or another. It's not the same in Newtonian mechanics and in relativity. Relativity is inescapable, and even though you can chose to close your eyes, put your hands over your ears and sing "Lalalalalala... I can't hear you.", it doesn't change the fact that an observer who has his eyes open would find distinct differences. $\endgroup$
    – CuriousOne
    Apr 1, 2016 at 0:33
  • $\begingroup$ To clarify, when you say that the new friends "saw it take 1 year" you are referring to the time in their frame since the light from your departure arrived. We have to be careful here because normally when talking about the timing of events in a frame, we use the time as we reckon it along the frame's time coordinate, not the time when light arrives at some point. So, assuming Earth and PC have no relative motion, they would reckon that your journey took 4.6 years, using Schwern's math. $\endgroup$
    – M.M
    Apr 1, 2016 at 3:15
  • $\begingroup$ Just as a comment: you really have to be careful as to what you mean by "see". Does "see" mean to write down an event in spacetime, in whatever physics equation you're working with? Or does "see" mean to actually have a little camera with a photosensitive surface inside it? The common statements would be, "the trip took five years in both Earth's and Proxima Centauri's coordinate systems", "a video camera on Earth looking at the spaceship's launch and landing would give a nine yr video", and "a video camera on Proxima Centauri of the launch and landing would give a 1yr video". [cont.] $\endgroup$
    – user12029
    Apr 1, 2016 at 6:13
  • $\begingroup$ (Schwern's answer has more accurate numbers of course, I'm just ballparking!) The video effects are directly related to redshifting/blueshifting, whereas the coordinate system effects are consequences of Lorentz transformations and time dilation. My word of caution is that it's really the latter you want to focus on in understanding special relativity. $\endgroup$
    – user12029
    Apr 1, 2016 at 6:15

3 Answers 3

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The part where you say the traveler wouldn't see any set speed limit is correct. They could always go faster and get there in less time in their frame of reference.

An observer who sees the traveler moving would never see them move at or faster than the speed of light.

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  • $\begingroup$ Isn't it true that an observer at the destination who doesn't know light has a speed limit, would agree with the traveler about the travel time? If both start a timer when they observe the ship depart, the timers show the same duration when both observe it arrive (assuming it travels in a straight line). $\endgroup$
    – Doofus
    Apr 1, 2016 at 0:52
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Now from my point of view it only took 1 year (if it had been at light speed, the trip would have been instant). My new friends at Proxima Centauri also agree that they saw it take 1 year (the light from Earth when I left reached them just 1 year before they saw me arrive).

No, they'd see you take 0.1 years.

For the ~4.5 year trip to appear to the spaceship inhabitants to take 1 year, time dilation has to be 4.5. That happens at 0.975c. This means your spaceship would arrive about 0.1 years behind the light of you taking off from Earth. The people on Proxima would see your ship take off, seemingly approach faster than the speed of light as the distance between your ship and them decreases, and then land 0.1 years later.

The spaceship inhabitants would think they took one year, the Proxima inhabitants would say they just saw you take off 0.1 years ago, and you'd know something was wrong. Also the people on Proxima would see some crazy optical effects, like your ship would appear to be elongated and a bit distorted (distorted because it's not traveling directly at Proxima but where it will be in 4.5 years) through a combination of Lorenz transformations and optical effects.

If the people on the ship observe Proxima on their approach they will see similar effects, to the observer on the ship they are stationary and Proxima is approaching them at 0.975c! Everything on Proxima will appear to be time dilated and it will appear elongated and distorted. As the ship get near and slows, these effects will diminish.

Here's a great site visualizing the optical effects on observing objects moving near the speed of light.

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  • $\begingroup$ Relativistically traveling objects don't appear "shorter", they appear rotated, if they are traveling at an angle relative to the observer. $\endgroup$
    – CuriousOne
    Apr 1, 2016 at 1:31
  • $\begingroup$ @CuriousOne The ship would be heading nearly straight at Proxima Centauri, though off by a little bit because it's aimed at where Proxima will be. I think I got it wrong anyway, it should appear elongated. $\endgroup$
    – Schwern
    Apr 1, 2016 at 2:00
  • $\begingroup$ Your "elongated" link refers to spheres that are expanding, however that is not the case for the spaceship. In Penrose's 2004 book he says that you can't "see" the Lorentz transformation: even though you would reckon the spaceship to be elongated, the difference in timing between emission of light which reaches your eye at the same time "cancels" this effect and you still see a sphere. But the sphere would be rotated. $\endgroup$
    – M.M
    Apr 1, 2016 at 3:22
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Easy to make simple mistakes in time dilation. M.M. Is right (on the time, the rotation is irrelevant).

The dilation factor is 4.5, it means both your friends back on Earth and your new friends in Proxima saw you take about 4.6 years, whereas you felt or aged only 1 year. Proxima and earth people are in the same coordinate frame (with their x=0 offset by 4.5 ly, but they can do that classical Galilean transformation). They are at rest wrt each other. They measure time the same way. The people in Proxima would know when they saw the light of you taking off from Earth .1 years before you arrived that the light took 4.5 years, and you took 4.6 years. They say 4.6, you say 1, you know there is a relativity effect.

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