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Humans live for about 80 years. A star, however, lives from 1 million to trillions of years, depending on its mass. If humans want to have a detailed study of the life of a $1 M_\odot$ star, how fast would they have to travel, and would it be viable?

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We want it so that $1 \text{ year at the speed of x% of c} = \dfrac{10^{10}}{80} \text{ stationary years}$. The Lorentz factor (in terms of c) is $\gamma=\dfrac{1}{\sqrt{1-v^2}}$. Solving for $v$ with $\gamma = \dfrac{10^{10}}{80},$ we get $v=0.999999999999999968c = 299792457.999... \text{m}\cdot\text{s}^{-1}$. This is definitely impossible, as the energy requirements are astronomically large: $$5572282108577709241 \text{J}\cdot c^{-2} \cdot 125000000 =696535263572213655125000000 \text{J} = 7.75 \cdot 10^9 \text{ kg of mass converted to energy}$$

This is definitely unrealistic, and is a probably impossible feat to complete. It would be better to study more and more stars instead of exploiting time dilation to view a single star for billions of years.

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    $\begingroup$ How do you travel that fast and stay anywhere near the star you want to study? Or if not staying near it, how do you study it given the redshift the speed implies. $\endgroup$
    – ProfRob
    Apr 21 at 17:53
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    $\begingroup$ While the answer is right, the energy demands do not rule it out (one can in principle imagining an advanced civilization wasting massive resources on it). A better implausibility argument is that the blueshift of the CMB turns it into gamma rays, and they will brake the craft even if they do not fry it. $\endgroup$ Apr 21 at 19:44

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