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If a space ship travels at near the speed of light towards a star, wouldn't the light from the star doppler shift into gamma rays (or worse?) and become lethal to life and destructive to space ship structure?

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Doppler shift will not actually make the spaceship emit Gamma rays. It's just a matter of reference frames. –  Yashbhatt Jul 20 at 15:17
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I think that going that fast you should be more worried about the atoms of hydrogen or helium and the grains of dust that you meet than about gamma rays. –  Francesco Montesano Jul 21 at 10:15

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The equation for Doppler shifting towards the blue spectrum (i.e. when you are traveling towards it) is:

$\lambda_b = \lambda_c/(c + v_b)$

where $\lambda_b$ is the shifted wavelength, $\lambda_c$ is the original wavelength, $c$ is the speed of light and $v_b$ is your velocity towards the source.

BUT! When you get closer to the speed of light you need to go relativistic! When moving closer to the speed of light you also need to take other effects into account like time dilation. So we need to move to an equation that looks more like this:

$v_{ob} = v_{source}\times\sqrt{(1+B)/(1-B)}$

Where $v_{ob}$ is the observed wavelength, $v_{source}$ is the emitted source radiation and $B$ is your velocity/speed of light.

So lets say that someone is shining some UV light on us at about $4\times10^{-7}~\rm{m}$ and we want to see at what velocity this light is shifted into gamma i.e. about $1\times10^{-11}~\rm{m}$. We can re-arange that formula to:

$B = [(\frac{v_o}{v_s})^2-1]/[\frac{v_o}{v_s}+1]$

where $B$ will be a decimal of the speed of light. $(v_o/v_s)^2$ is ridiculously small, we are talking magnitudes of $10^{-10}$.

Then when you add or subtract 1, you really dont move that far from 1. Just under 1/ just over 1 you get the answer that You have to be going ridiculously close to the speed of light for this to happen. Like so ridiculously close MATLAB wont calculate the difference and just gives me the value of 1, i.e. the speed of light.

IN SUMMARY:

Yes but you have to be going so close to the speed of light it would be physically impossible.

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