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According to time dilation formula $$ \Delta t_{observer} = \frac{\Delta t_{proper}}{\sqrt{1-(v/c)^2}} $$

I have basically two questions:

(1) Is this formula applicable only for 1D motion, i.e. what if the observer is aligned along the Y axis and the moving entity moves along X axis. Let the speed be 0.1c so there is a considerable amount of time dilation but since the relative motion is now 2D, what effect does this have on the time dilation formula.

(2) After answering the first question, I would like to know that if in a similar case (i.e. observer along y axis and entity moves in x axis), if the velocity of the entity approaches the speed of light, what happens to the $\Delta t_{observer}$ ? Does it tend to infinity?

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Yes, the formula applies regardless of motion direction. Length contraction, instead, occurs only in the direction of motion. And yes, the dilation gets larger the closer you get to $c$. Regarding the question in your comment below, both, the traveler and the observer will see each other's clocks running slower. Now you are talking about acceleration here, in such a case you have the twin's paradox. The resolution of the paradox is that the traveler, which is the one that accelerated, remains younger, in the sense that when they meet his clock will show less time has passed. The reason is that when he decelerates he sees the clock of the non accelerating observer speeding up.

Another note, you can never reach the speed of light, so the amounts will always be finite, even if as large as you want.

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  • $\begingroup$ Referring to the second question. Thought the dilation obviously gets larger as we get closer to c. But, say that the traveler travels at near light speed from say point A (x= -150000000 meters) to point B ( x=150000000 meters) and stops at point B, the traveler would record a time of 1 sec. But at the same time the observer would see that the whole while the traveler's clock didn't move forward in time but still he reached in a finite amount of time to point B. $\endgroup$ – Mike Victor Oct 27 '18 at 13:33

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