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I am a student and due to school closures, I am reading ahead in physics. I have been learning about special relativity and I made up a problem for myself. I've drawn a diagram below: enter image description here

The diagram shows Mr Green travelling at $0.5c$ in the $x$-direction with respect to me at the origin and Mr Red is travelling at $0.5 c$ in the $y$-direction with respect to me, where $c$ is the speed of light.

So we can say that:

$$ v_{G/O} = \begin{bmatrix} 0.5c \\ 0\end{bmatrix} $$

Thus, $$ |v_{G/O}|=0.5 c $$

Where $v_{G/O}$ is the velocity of Mr Green with respects to the origin. The same can be said for Mr Red:

$$ v_{R/O} = \begin{bmatrix} 0 \\ 0.5c\end{bmatrix} $$

Thus, $$ |v_{R/O}|=0.5 c $$

where $v_{R/O}$ is the the velocity of Mr. Red with respects to the origin.

So now we could say that the velocity of Mr Green with respect to Mr. Red is:

$$ v_{G/R} = \begin{bmatrix} 0.5c \\ -0.5c\end{bmatrix} $$

So the magnitude of the velocity of Mr Green with respects to Mr Red is:

$$ |v_{G/R}| = \sqrt{0.5^2 +0.5^2} = \frac{\sqrt{2}}{2} $$

Now if we were to look at the time conversions, 1 year for Mr Green would be the same as:

$$ \frac{1}{\sqrt{1-0.5^2}}=1.15 $$

1.15 years for me. The same can be said about Mr Red, 1 year for Mr Red would be the same as 1.15 years for me.

That should mean that 1 year for Mr Green and Mr Red should take the exact same time as one another. However, when we do the maths: $$ \frac{1}{\sqrt{1-(\frac{\sqrt{2}}{2})^2}} = 1.41 $$

This suggests that 1 year for Mr Green is the same as 1.41 years for Mr Red.

Could you tell me what I have done wrong?

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  • $\begingroup$ "That should mean that 1 year for Mr Green and Mr Red should take the exact same time as one another" would you make the same conclusion in 1D case with Mr Green and Mr Red having equal speeds but opposite velocities? Also, check out en.wikipedia.org/wiki/Velocity-addition_formula $\endgroup$ – Umaxo May 19 at 10:17
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Your assumption that

That should mean that 1 year for Mr Green and Mr Red should take the exact same time as one another.

holds true only when you observe the time from your frame of reference . A frame of reference is something like your 'point of view'. More formally, it is a coordinate system with respect to you, from where you make measurements like velocity or time.

Remember time is not absolute. When you say that 1 year for Mr. Green and Mr. Red should be the same, you are looking at things from your reference frame. Once you change the reference frame to, say, Mr. Red's, you are then talking about how Mr. Red measures time passing for Mr. Red. And that should not necessarily be the same as the time observed from your frame of reference.

In your situation, you see 1.15 years pass by for both Mr. Red and Mr. Green, because both are moving with the same speed (but in different directions; Remember in this case the magnitude counts.) But when you look at the situation from Mr. Red's perspective, he sees that he is at rest, and that you are moving with velocity of magnitude $0.5c$, while Mr. Green (according to Mr. Red) is travelling with velocity of magnitude $\frac{\sqrt 2}{2}c$. So while he sees that 1 year for you is 1.15 years for him, but sees that 1 year for Mr. Green is 1.41 years for him.

So, to sum up: the 'contradiction' is caused because of the reference frame. The time dilation you observe in your frame of reference could be different in another frame of difference, just by virtue of how objects seem to be moving in that frame of reference.

By the way, great to see that you made up the problem by yourself. +1 and keep it up.

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