Accelerating spaceship paradox

[yeknod71]

Hi,

Please consider:

At time zero a spaceship takes off from Earth and keeps traveling under constant acceleration.

From Earth's perspective, the spaceship's speed keeps increasing but never reaches c. Also from Earth's perspective, the clock on the spaceship keeps slowing down and asymtotes a certain time value (let's call tMax).

However, from spaceship's perspective, their clock has not slowed down and passes tMax. They can take a photograph of their clock showing greater values than tMax, and return to earth. It can be arranged that in the background of this photograph there is evidence that it was taken before the spaceship turned around. How will observers on Earth explain this photograph?

Thanks in advance.

 

[Janus]

Hi,

Please consider:

At time zero a spaceship takes off from Earth and keeps traveling under constant acceleration.

From Earth's perspective, the spaceship's speed keeps increasing but never reaches c. Also from Earth's perspective, the clock on the spaceship keeps slowing down and asymtotes a certain time value (let's call tMax).

No such value as tMax exists. From Earth, the ship clock does slow down as it approaches c, but so does the rate at which the ship approaches c. There is no maximum value that the ship clock can read as seen from Earth.

However, from spaceship's perspective, their clock has not slowed down and passes tMax. They can take a photograph of their clock showing greater values than tMax, and return to earth. It can be arranged that in the background of this photograph there is evidence that it was taken before the spaceship turned around. How will observers on Earth explain this photograph?

Thanks in advance.

 

[yeknod71]

No such value as tMax exists. From Earth, the ship clock does slow down as it approaches c, but so does the rate at which the ship approaches c. There is no maximum value that the ship clock can read as seen from Earth.

Thanks, so that was my misunderstanding. If not too much trouble, what would be the formula to calculate ship clock value as seen from Earth, as a function of Earth clock value as seen on Earth, assuming a constant acceleration?

 

[Janus]

That would be:

$$t = \frac{c}{a}\sinh^{-1} \left( \frac{aT}{c} \right )$$

Where t is the shiptime
T is the time on Earth
and a is the acceleration.
sinh^-1 is the inverse hyperbolic sin.