The direction is hidden in the formula, once we talk of relative velocity, the direction is there.
Nothing is hidden. The minus sign is trumped by the square term. Therefore it matters not whether the moving frame is approaching or receding. It is moving, therefore special relativity engages. An observer in one frame notices that time dilates in the other. Go back to Einstein's example of a person dropping a ball on a train while an observer watches the train go by, and sees the ball fall in the window. To the moving observer, the ball falls straight down. To the person at the station, the ball falls along a slant, which cuts more of local space than it does of the space in the train. Yet it does so in the same amount of time, referred to the station clock. In order for this to be true, time must dilate on the train, according to the person at the station. Yet, when we reverse roles, and let the person at the station drop a ball, the observer on the train sees it traverse a slant, whereas the person on the platform sees it drop vertically - a shorter distance. Now the person in the train has no choice but to infer that time dilates on the platform. To get the point of relativity, allow both observes to drop a ball while observing the other person dropping a ball. Each concludes that the other is undergoing time dilation. Notice direction has nothing to do with this.
You are probably talking about a very specific case of v and -v.
Actually you brought it up, not me.
It is very legitimate to describe the motion (Say, Velocity Vector) of two frames in third frame, in that case direction is crucial to ascertain the relative speed.
No, you are wrong. Introducing a third reference frame only adds a few steps in which you will eliminate the 3rd frame insofar as it is irrelevant. Relativity applies to any TWO frames. Time dilates in the frame that leaves the inertial frame (i.e. moves relative to that frame) regardless of direction. Trying to contrive an exception to the rule by contriving a different problem is not going to work. If A>B and B>C then A>C. B (your third frame) is irrelevant since all that matters here is A>C. And no, direction is irrelevant as I have said before. Consider the following diagram:
Notice that direction of travel has no bearing on the fact that the traveling twin traversed a total distance of
ct regardless of leaving (interestingly marked in blue) or returning (and this is marked in red). All that matters is the magnitude of the slope in each case, not the sign (positive or negativc / receding or approaching). It just doesn't matter; a total of
ct applies regardless of direction of the slope.
A Green Man riding the interstellar bi-cycle can never experience past as there is no time reversal associated with time dilation (relative Motion based).
Yes, when the twin returns a day younger than her sister, she is "living in the past" according to the sedentary twin, whereas the she concludes the sedentary twin (and all sedentary people on Earth) are living in the future. Brian Greene simply came up with a clean way to propound this idea graphically. And notice, he did not need a third reference frame to explain this. Nor did Einstein, nor any of the early discoverers of relativity. (See the so-called Fitzeau Water Experiment).
So you see Brian Greene got it right, as did the early discoverers of modern physics.
