Spaceship moves away from the Earth

[Pritamstar]

Spaceship moves away from the Earth at a constant speed of 300 m/s. How long would it take for a clock on the Earth to differ from a clock in the spaceship by 1 second.

 

[A Lazy Shisno]

Interesting question.

Recall that

$$t=\frac{\tau}{\sqrt{1-\left(\frac{v}{c}\right)^2}}$$

Where $t$ is the time measured on Earth, $\tau$ is the proper time measured on the spaceship, $v$ is the velocity of the spaceship, and $c$ is the speed of light.

What we want to find is $\tau$ when

$$\tau+1=\frac{\tau}{\sqrt{1-\left(\frac{300}{299\,729\,458}\right)^2}}$$

which gives

$$\Rightarrow\tau+1=1.000000000000500692525630328937507832865359772...\tau$$

which means that for every second that passes on the spaceship, ~1.0000000000005 passes on Earth. Making $1.0000000000005...=\gamma$ for ease,

$$\tau+1=\gamma\tau$$

$$\frac{\tau+1}{\tau}=\gamma$$

$$\Rightarrow\frac{\tau}{\tau}+\frac{1}{\tau}=\gamma$$

$$\Rightarrow 1+\frac{1}{\tau}=\gamma$$

$$\Rightarrow \tau=\frac{1}{\gamma-1}$$

$$\therefore \tau=1.9972337289059503848273053279826226...\times 10^{12}$$

Therefore, it would take $~1.997234\times 10^{12}$ seconds, or around $63\,332$ years(!), for a clock on the Earth to differ from a clock in the spaceship by 1 second.

 

[Ibix]

Spaceship moves away from the Earth at a constant speed of 300 m/s. How long would it take for a clock on the Earth to differ from a clock in the spaceship by 1 second.

There is no non-arbitrary answer to this question (@A Lazy Shisno is making assumptions). There is no unique way to compare clocks that aren't in the same place, so there is a range of possible approaches to answering the question.

Conventionally, you would adopt the Einstein synchronisation convention. But you didn't say how long it takes for who. An observer in Earth's rest frame? An alien passing by at half the speed of light? Somebody doing 300m/s relative to the Earth in the opposite direction? All will have different answers.

 

[A Lazy Shisno]

There is no non-arbitrary answer to this question (@A Lazy Shisno is making assumptions).

That may be so, but both you and I know what answer the OP was looking for ;)

 

[Orodruin]

That may be so, but both you and I know what answer the OP was looking for ;)

On the contrary, I believe that the OP was making the false (and very common among laypeople) assumption that there somehow is a unique and objective time difference. As such, the OP might be looking for an answer that does not exist and Ibix is certainly correct in pointing this out rather than perpetuating the OP's misconceptions about relativity.

 

[Ibix]

That may be so, but both you and I know what answer the OP was looking for ;)

Failing to be aware of the assumptions you made is how people end up with the notion that relativity is inconsistent. The thread is marked for university-level answers and anyone not developing a reflex of worrying about this kind of detail is going to start struggling with relativity at that level.

 

[berkeman]

Spaceship moves away from the Earth at a constant speed of 300 m/s. How long would it take for a clock on the Earth to differ from a clock in the spaceship by 1 second.

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