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This question already has an answer here:

I am trying to wrap my mind around special relativity and I have these questions:

If I am moving at 70% of the speed of light (relative to the ground) to the right, and my friend moves at 70% of the speed of light (relative to the ground) to the left, are we moving apart at faster than the speed of light?

Does this mean that from my perspective, he is moving away from me at faster than the speed of light?

How about from the perspective of a third party not moving (relative to the ground). Do they see us moving apart at faster than light?

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marked as duplicate by ACuriousMind Sep 9 '17 at 19:06

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  • $\begingroup$ Obviously, you are moving apart faster than the speed of light in the frame of a third party on the ground. But you cannot be moving apart faster than the speed of light in any frame in which either of you is stationary. $\endgroup$ – WillO Sep 9 '17 at 18:23
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"From my perspective": no, he is seen moving away at less than the speed of light.

"from the perspective of a third party on the ground": Every physical interaction involves 2 parties making an exchange. Therefore, whilst a 3rd party could choose to add .7c and .7c to arrive at a figure of 1.4c, this figure is most definitely not the relative velocity of you and your friend, and will not be relevant to interactions between you and them.

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Focusing on the 1st question (which is a great place to start if you're trying to wrap your head around SR): You can't add velocities, but you can add "rapidities", which is (with $c=1$):

$ w = \tanh^{-1}{v} $

so you see your friend moving at:

$ v = \tanh{w} = \tanh{(w_1 + w_2)} = \tanh(2\tanh^{-1}{0.7}) = 0.93..$

Of course, now you have to understand why this formula works.

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Picture an extremely obtuse angle expanding at the speed of light from the point of the observer. Redshift .7/light speed as seen from observer with relation to each "traveller".

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