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In the top answer for this question on the Space Exploration StackExchange, it's calculated that a rocket traveling at constant 1g of acceleration over 100,000 light years would, under Newtonian physics, accomplish the journey in 662 years while reaching a peak velocity of 321 times the speed of light, while when you take relativity into account, it reaches its destination in just 22 years (in its own reference frame), while traveling at just under light speed.

Since as far as the rocket can tell, it's still accelerating just as fast, why does the relativistic rocket reach its destination in 22 years instead of 662 years? I haven't really studied relativity in any formal context, but I would have expected intuitively it to arrive just as quickly, with the time dilation cancelling out the superluminal acceleration - is there something about the time dilation/space compression that isn't symmetrical?

The_Sympathizer
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nick012000
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  • That's just the way those hyperbolic functions work. It would be a remarkable coincidence if the time dilation cancelled out like you intuitively expect, (although there are some coincidental agreements between Newtonian physics and relativity). It's hard to explain without going through the relevant mathematics. How's your calculus? – PM 2Ring Jun 17 '19 at 03:11
  • FWIW, the formulae for constant acceleration in relativity are given (without derivation) on the relativistic rocket page. Those formulae can be derived using fairly basic calculus from the usual equations for time dilation, etc. One very useful equation is the velocity addition formula of special relativity. – PM 2Ring Jun 17 '19 at 03:15
  • @PM2Ring My calculus is good enough to teach high school maths, though I don't really remember anything from the course on university-level maths I took when I was trying to get an engineering degree about 15 years ago. – nick012000 Jun 17 '19 at 03:50
  • Ok. Towards the end of this answer I show how to derive the formula for velocity in special relativity given constant acceleration. You can derive other quantities in a similar fashion. – PM 2Ring Jun 17 '19 at 03:54
  • @PM2Ring I'm not certain how that math relates to this question, since the mass of the rocket is also changing, and you're talking about photons acting as propellent? Assume that the method of the rocket's propulsion doesn't matter for this question - it just has a magic reactionless drive or something. – nick012000 Jun 17 '19 at 04:03
  • The derivation of $v=c\tanh\left(\frac{aT}{c}\right)$ ignores mass. It just assumes that $a$ is constant in the ship frame, that is, occupants of the ship feel a constant g force due to the ship's acceleration. – PM 2Ring Jun 17 '19 at 04:54
  • At very high speeds gamma increases at almost constant rate in the launchpad frame, because gamma is proportional to kinetic energy which increases at constant rate if speed is constant - so in the rocket frame the gamma of the milestones that move by increases even faster - so distance between two consecutive stones shrinks rapidly - so the rate at which the milestnes fly past increases rapidly. – stuffu Jun 17 '19 at 05:31
  • Things can happen in thought experiments which can't happen in real life. In Newton's day,relativistic mass increase was never taken into account. As speed approaches a significant fraction of c,more and more of the spaceship's energy would be used up in creating this relativistic mass,which is why not even a cosmic ray or accelerated particle can reach speed c. The mass increase would have to be provided by the fuel,but there is no fuel capable of doing it. Man- carrying relativistic spaceships must always remain a pipe dream or the stuff of thought experiments. – Michael Walsby Jun 17 '19 at 09:16
  • By the way, I think the question is not about time dilation, but about length contraction, and the answer is that in the relativistic case, looking forward from the ship's cabin, you see a shorter distance ahead because of length contraction. You divide that with the time passing on your own clock onboard the ship, and you will see that the trip takes shorter time (compared to the Newtonian). – Árpád Szendrei Jun 17 '19 at 11:09
  • The question that this question is linked to as a duplicate does not answer my question. I'm not asking "what is time dilation", I'm asking "why does a relativistic rocket experience less time in transit than a Newtonian rocket experiencing the same acceleration and course, even though it's moving slower." An answer would need to talk about time dilation, but it will also need to talk about how that differs from/is related to Newtonian physics. I don't want a general answer about relativity, I want an answer about this specific case. – nick012000 Jun 17 '19 at 12:22
  • @MichaelWalsby IIRC there were some calculations for nuclear-pulse rockets that showed that they could potentially reach relativistic speeds, though not the ludicrous relativistic speeds of the rocket in this question. – nick012000 Jun 17 '19 at 12:26
  • I think I know what kind of rockets you're talking about,but none have ever been built & in my opinion never will be. The drawbacks include protection of the spaceship and its occupants from the blast & radiation of the nuclear pulses. It is one of those concepts which sound superficially plausible,like restoring the atmosphere of Mars to what it used to be billions of years ago,but it's impractical. – Michael Walsby Jun 17 '19 at 12:45
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    @MichaelWalsby The main reason none of them ever got built wasn't technological (we could have built them in the 60s), but political: the Nuclear Test Ban Treaty banned the civilian use of nuclear explosives. – nick012000 Jun 18 '19 at 02:31
  • @nick012000 John Rennie won't know you've posted comments unless you ping him with the @ syntax. You can ask a new question if the linked question is inadequate, but you need to clearly explain how your new question isn't a dupe. But I'm not sure what info an answer would need in order to satisfy you if the material mentioned & linked here already isn't sufficient. – PM 2Ring Jun 18 '19 at 14:43
  • @JohnRennie The question that this question is linked to as a duplicate does not answer my question. I'm not asking "what is time dilation", I'm asking "why does a relativistic rocket experience less time in transit than a Newtonian rocket experiencing the same acceleration and course, even though it's moving slower." An answer would need to talk about time dilation, but it will also need to talk about how that differs from/is related to Newtonian physics. I don't want a general answer about relativity, I want an answer about this specific case. – nick012000 Jun 19 '19 at 10:07
  • @PM2Ring An actual answer to my actual question would be nice. I don't want a general "this is what relativity is" answer or just a bunch of math thrown at me with minimal explanation, I want an answer about why relativity gives this counter-intuitive result in this specific case. – nick012000 Jun 19 '19 at 10:08

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