Time dilation and approaching speed of light

So i read this short bit on Time Dilation which left me with a question. It states that when approaching speed of light at 0.999999 then for each day on board of a vessel about 2 earth years pass. When we speed up to 0.99999999999999 of speed of light this single day will amount to about 20000 earth years.

When looking at this from a SF writing point of view, does this mean that as long as we speak of approaching the speed of light basically anything within those boundaries goes? I'm doing a short story where picking the ship/time earth/time ratio is important and I'd like it to be at least somewhat scientifically accurate.

Apart from that, it makes me wonder how the difference between those two very close approximations can differ so much in effect? I understand speed of light represents a very large distance/time and as such even so far after the comma differences might be great, but still this feels a bit unintuitive. Is the source more or less correct?

• Yes, the source looks accurate. What sort of answer are you looking for? Physics SE is normally used to give more indepth answers. – Lio Elbammalf Feb 1 '17 at 7:47
• Well instinctively these two approximations are really close, so how do they result in such huge differences? – hoppa Feb 1 '17 at 7:52
• Consider the equation given, $t' = \frac{t}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$, specifically the $1 - \frac{v^{2}}{c^{2}}$ bit. As $\frac{v}{c}$ tends to 1 this denominator tends to 0 and so the result will tend to infinity. Look at the plot provided on the link you gave us, you can see how much of a difference you get as the ratio approaches 1. – Lio Elbammalf Feb 1 '17 at 8:10
• So I realise (by now) that this sub-exchange deals mostly with the mathematics of physics but this triggers a philosophical question for me. Obviously the second approximation of speed of light (SOL) makes that whatever body travels at that speed is quicker than the first approximation of SOL. Being quicker in distance travelled per unit of time you'd not expect the ratio of days passed outside an on/in the body to grow right? Or is this because the actual duration of the flight as seen from the point of leaving remains constant and only the time on the body is relative? – hoppa Feb 1 '17 at 11:54
• So basically it is not 1 day inside compared to a relative period of time outside. It is a fixed amount of time outside compared to a relative amount of time inside a moving body? – hoppa Feb 1 '17 at 11:56