Time Dilation: Traveling to a Distant Star in 4.5yr

[versine]

If there is a spaceship traveling at 0.999c, the time to reach a star 100 lyr away would be approx 100 yr (assuming no accel and decel). But on the spaceship, It would be 100 yr * sqrt(1-0.999^2) = 4.5yr.

Why do we take 100 yr as the time seen on Earth and not the time on the spaceship?

 

[hutchphd]

Because presumably we are on Earth? Your question is not clear. The times are specific and different.

 

[Drakkith]

Because most people's perspective is primarily rooted on Earth, so when we talk about distances, speeds, and time we usually only specifically state the reference frame when it is different from Earth's.

 

[Sagittarius A-Star]

If there is a spaceship traveling at 0.999c, the time to reach a star 100 lyr away would be approx 100 yr (assuming no accel and decel). But on the spaceship, It would be 100 yr * sqrt(1-0.999^2) = 4.5yr.

Why do we take 100 yr as the time seen on Earth and not the time on the spaceship?

Because the (length contracted) distance from Earth to the star in the frame of the spaceship is only
$100 lyr * \sqrt{1-0.999^2} = 4.5 lyr$.

 

[Nugatory]

Why do we take 100 yr as the time seen on Earth and not the time on the spaceship?

Because that’s how much time a clock at rest on Earth (strictly speaking, at rest relative to the spaceship before it started on the journey) would count between the departure event and the arrival event.
There is a subtlety here: someone back on Earth doesn’t see the spaceship arrive at the destination at time 100; they see the arrival event happen after their clock has counted off 200 years (the light took 100 years to reach their eyes). Only after they subtract the light travel time from 200 do they conclude that the spaceship arrived at the same time that their clock had counted off 100 years.