Time Dilation: Moving Rocket 0.5c vs Accelerating Rocket 0.5c

[pavelbure9]

A non-moving observer is looking at two different rockets in space. One rocket is moving at a steady velocity of 0.5c, and the other rocket is currently moving in 0.5c but has steadily accelerated from 0.3c and will continue accelerating until it gets to 0.7c.

Assuming that there is no gravity, currently (when both rockets are moving at a velocity of 0.5c), which rocket's time will the non-moving observer think is running slower?

 

[Orodruin]

Does the time dilation formula include the acceleration anywhere?

 

[pavelbure9]

Does the time dilation formula include the acceleration anywhere?

I was just wondering if I had to take General relativity into thought because there is an acceleration

 

[Ibix]

No. Special relativity can handle acceleration just fine. And, as Orodruin implies, this question is straightforward.

Studying accelerating reference frames (not accelerating objects) was, I gather, one of the things that led Einstein to general relativity. Also, many special relativity courses avoid accelerating objects because you have to use fairly complicated calculus almost immediately. Taken together, this can lead to the impression that you can't do acceleration in special relativity. But that is wrong - as long as you're willing to do the maths, you can.

 

[Histspec]

A non-moving observer is looking at two different rockets in space. One rocket is moving at a steady velocity of 0.5c, and the other rocket is currently moving in 0.5c but has steadily accelerated from 0.3c and will continue accelerating until it gets to 0.7c.

Assuming that there is no gravity, currently (when both rockets are moving at a velocity of 0.5c), which rocket's time will the non-moving observer think is running slower?

The transformation formulas for acceleration in special relativity are well known:
https://en.wikipedia.org/wiki/Acceleration_(special_relativity)

From that, one obtains the formulas for proper acceleration (i.e. the acceleration that a comoving observer feels):
https://en.wikipedia.org/wiki/Acceleration_(special_relativity)#Proper_acceleration

by which one finally obtains the formulas for constant proper acceleration, also known as hyperbolic motion, see
http://math.ucr.edu/home/baez/physics/Relativity/SR/Rocket/rocket.html
https://en.wikipedia.org/wiki/Hyperbolic_motion_(relativity)
https://en.wikipedia.org/wiki/Acceleration_(special_relativity)#Curved_world_lines

Using those formulas, and plugging in some numbers, you should be able to compute the time of both rockets.