# Time dilation in 2+ dimensions. Relativity problem

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: 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?

• "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 – Umaxo May 19 at 10:17

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.