Due to relativity, if you are onboard a ship going at 99.999% the speed of light, it seems to you that you travel multiple light years per year, even though from an outside perspective you take just over a year per light year. So, at what point as a fraction of C does it seem to you onboard a relativistic spacecraft like you are moving at one-light year per internal year?
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1$\begingroup$ All velocities are relative, so you can't really go 99.999% the speed of light, you can only do that in reference to other objects. However, I understand what you're askng. You can google to find the relativistic formulas, but I'm almost sure the answer is sqrt(2)/2 times the speed of light. $\endgroup$– user7073Commented Mar 2, 2020 at 17:48
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$\begingroup$ I don't think there is a good definition of "seem like", as there are no studies (AFAIK) on how humans perceive time/space dilation effects. $\endgroup$– MefiticoCommented Mar 2, 2020 at 20:07
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$\begingroup$ @Mefitico it does not have to be a human, I'm just talking about what the internal clocks would show $\endgroup$– qazwsxCommented Mar 2, 2020 at 20:08
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1$\begingroup$ @qazwsx : Clocks themselves (i.e. with no external reference) are not aware of time dilation, they just keep doing their thing. If they were able to notice relativistic effects, they'd be able to compute their speed relative to an inertial frame. $\endgroup$– MefiticoCommented Mar 2, 2020 at 20:26
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$\begingroup$ The concept of rapidity is relevant to your question: en.wikipedia.org/wiki/Rapidity $\endgroup$– Christopher James HuffCommented Mar 2, 2020 at 23:12
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2 Answers
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Clarifying the question (I hope): your spaceship leaves point A, accelerates up to some relativistic speed relative to A, coasts for a while, then decelerates to rest at some point B. A and B are at rest relative to each other, with a distance of say 1 light year (we could send a light signal from A to B to A in two years), and the ship accelerates quickly enough that we can ignore the acceleration and deceleration phases and pretend that the ship was traveling at a constant speed (relative to both A and B) basically 100% of the time. How fast should that speed be so that the subjective time experienced by someone on the ship is equal to the light-time from A to B? We know that the answer is larger than 0 (at non-relativistic speeds it takes more than 1 year to go 1 light-year) and less than $c$ (at the limit of $c$ it takes zero subjective time to go any distance).
Let's use natural units for speed, i.e. $v = 0.5$ means one-half lightspeed. Then time dilation is $\gamma = \sqrt{1 - v^2}$. We want to know when our "subjective speed", which is our speed divided by our time dilation factor, equals 1: $\frac{v}{\sqrt{1 - v^2}} = 1$. A little re-arranging turns this into $v^2 = 1 - v^2$ and then to $v = \frac{1}{\sqrt 2}$. So there you are: under some simplifying assumptions, a 1-light-year trip takes 1 subjective year at approximately 70.7% times the speed of light.
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This never happens due to length contraction. As the person moving, you will think you are traveling less distance, and an outside observer will observe you taking more time. These effects lead to both observers measuring the same speed.
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$\begingroup$ I meant external distance- if I'm going to a star 4 light years away, how fast do I need to go there and age 4 years $\endgroup$– qazwsxCommented Mar 2, 2020 at 22:42