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I have been thinking about this for a while but apparently it is beyond my pop-sci level of understanding:

Imagine a spaceship departing from Earth under constant acceleration of 1g (10 m/s^2) for many years (let's not dwell on how this can be achieved in practice). Since acceleration is absolute, the traveler might do a naïve calculation showing that in less than a year (31.5 million seconds), he has accelerated to more than the speed of light relative to his starting condition.

I understand that I, remaining on Earth, will never observe the spaceship's speed to be greater than the speed of light relative to me. My understanding is that I would observe the spaceship's time to tick ever slower (and its image ever more redshifted), always inching towards but never reaching the speed of light. This reminds me of things falling towards the event horizon of a black hole - I would observe their time to keep slowing down, gradually freezing in time and never reaching the point where the thing crosses the event horizon. In fact, the very formation of the event horizon will be in my infinite future.

My first question is if the point that I will be observing the spaceship forever inching towards is the same point where the traveler would calculate to reach the speed of light, i. e. if I could see a clock ticking on the spaceship, would I only see it going to 29 999 999th second and never see it tick to 30 000 000th second - the point where calculation of the traveler gives 10 m/s^2 times 30 000 000 seconds equals 300 000 000 m/s, the speed of light?

This would imply that the spaceship will at this point seem (to me) almost frozen in time and not moving at all, much like an object falling towards a black hole.

This brings me to the second question - what if this spaceship was not moving away from me, but towards me from a great distance? It seems I should also observe it to never reach the speed of light, and freeze in place some distance away from me, never reaching my position. On the other hand, from the frame of reference of the spaceship, it has no reason to not hit me after enough time. So what gives? My common-reason feeling is that the spaceship should hit me eventually. But how? What is wrong with the reasoning that I will never see it reach the point of "speed of light" (the point where the traveler would calculate he has reached the speed of light given his constant acceleration and elapsed time)?

Thank you in advance for any insight into this "thought experiment".

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My understanding is that I would observe the spaceship's time to tick ever slower (and its image ever more redshifted), always inching towards but never reaching the speed of light.

This is correct. Notice here that what the spaceship is "inching towards" is a speed, not a location. Because you used the colloquial term "inching towards" instead of the more proper term "approaching" you may be confusing yourself.

if the point that I will be observing the spaceship forever inching towards

Here you have mentally changed the "approaching" a speed to "inching towards" a location. The ship is not "inching towards" any location. Any location that it is going towards it is doing so at just under $c$. It is "rushing towards" locations as it gradually "approaches" the speed $c$.

It seems I should also observe it to never reach the speed of light, and freeze in place some distance away from me, never reaching my position.

Here you make the switch in a single sentence. Indeed, it never reaches the speed of light relative to you, but it does approach it. It does not freeze in place at any point, but always approaches you at ever faster speeds which approach $c$.

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  • $\begingroup$ Thank you. What I think was mostly confusing me was that I saw the situation as analogous to an object falling into a black hole - and here it is often described as frozen both in time and space, moving closer but ever slower to the event horizon, never reaching it (as seen by a distant observer). $\endgroup$
    – Tondo PX
    Commented Feb 8 at 14:07
  • $\begingroup$ You are not entirely wrong about that. There are some limited similarities. But within those limited similarities the accelerating observer is similar to the observer that hovers outside the horizon and the non-accelerating observer is similar to the observer that falls through the horizon. $\endgroup$
    – Dale
    Commented Feb 8 at 14:11
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    $\begingroup$ And yeah, I was confusing my observation of ship's clock with my observation of ship's speed... if I see the ship's clock almost not moving, it doesn't mean that I also see the ship almost not moving... $\endgroup$
    – Tondo PX
    Commented Feb 8 at 14:14

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