So I read that if an astronaut travels at the speed of light and goes to a nearby star system , when he or she returns, he or she will have aged 10 years but 1000 years will have gone by on Earth. Travel at the speed of light is not achievable ,so I was wondering if the spacecraft was traveling at an achievable speed of 10% the speed of light , how much will the astronaut have aged and how much time has elapsed here on Earth upon the return of the astronaut ?
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1$\begingroup$ Relevant: en.wikipedia.org/wiki/Twin_paradox#Specific_example $\endgroup$– tmwilson26Commented Dec 9, 2015 at 20:15
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$\begingroup$ As you suggest, nobody can travel at the speed of light, that requires infinite energy for a body with mass (that is if we assume, as all the evidence suggests, that relativity is correct). The time dilation function is here: en.wikipedia.org/wiki/Time_dilation#Overview_of_formulae $\endgroup$– adrianmcmenaminCommented Dec 9, 2015 at 21:01
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1$\begingroup$ @userLTK Carl Sagan and friends designed "flyspeck", a small rocket that could realistically achieve 40% the speed of light within our solar system. However 1) as the name suggests, it would be very small, just enough to contain a camera/controller (and not a human being), 2) I can't find a single resource to confirm that flyspeck ever existed, even theoretically. $\endgroup$– user21Commented Dec 10, 2015 at 0:42
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1$\begingroup$ This is basic special relativity. You can find the calculation all over the internet. The source you read that said "if an astronaut travels at the speed of light" is nonsense. Only massless particles can do so, and suffer infinite time dilation. No time passed in a photon's frame of reference. The effect of moving at 0.1c is small. $\endgroup$– ProfRobCommented Dec 10, 2015 at 7:54
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6$\begingroup$ I'm voting to close this question as off-topic because it would fit better with physics or space exploration, instead of astronomy. $\endgroup$– James KCommented Dec 14, 2015 at 19:26
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You're actually describing the twin paradox. That's a little beyond Special Relativity, since the astronaut accelerates, and deaccelerates. Since Special Relativity treats only inertial frames of reference, accurate treatment would require some extension towards General Relativity. But you may consider two astronauts, one travelling with constant velocity towards Earth, the other one with the same (by amount) velocity away from the Earth, and then apply the formula for time dilation of Special Relativity to both astronauts, as an approximation, and to avoid the twin paradox. Then you get the almost ubiquitous factor $\sqrt{1-\beta^2}$ of Special Relativity, with $b=v/c$ the ratio of the relative velocity to the speed of light. For $\beta=0.1$, hence 10% the speed of light, we get a factor of
$\sqrt{1-\beta^2}=\sqrt{1-0.1^2}=\sqrt{1-0.01}=\sqrt{0.99}\approx 0.9949874371$.
Divide 10 years by 0.9949874371, and you get 10 years plus 18.4 days on Earth.
For $\beta=1$ this division would fail, implying a break-down of the formulas of Special Relativity. That's why travelling with the speed of light doesn't work for astronauts respecting Special Relativity.
