Relativity Illuminated, Addendum IV

Addendum IV
Twin Paradox Without Accelerations

The famed "Twin Paradox" is not a contradiction in the Theory of Relativity. A space traveler can leave Earth, travel at relativistic speeds, and return to Earth younger than his homebound twin.

The issue that perplexes most folks is that since Relativity precludes any notion of absolute motion – ie. it's all relative – then how come the space traveling twin seems to benefit from all the time dilation, as if it were he who is truly in motion. But there is no contradiction: at any juncture, the Earth can be thought of as the frame in motion instead, and the astronaut twin simply sitting stock still in space. It's not the astronaut's motion that is absolute, it's his modification of course that is inarguable. It's that course modification that allows him to benefit from all the relativistic clock dissynchronicity, time dilation being almost inconsequential to the outcome.

BACKGROUND

According to Relativity's famed Lorentz Transformation, two separated clocks affixed to an elongated frame, that are perfectly synchronized according to a native of that frame, are not synchronized to an observer for whom the aforementioned frame is in relative motion along the direction that defines the separation of those clocks. Instead, to such an observer, the front-most clock has a reading earlier than the rear-most clock (yet they both advance at the same time-dilated rate). The amount by which the clocks are out of sync is equal to the time it would take light to travel the two clocks' uncontracted separation distance, multiplied by the frame's velocity expressed as a fraction of lightspeed.

TWIN PARADOX, NO ACCELERATIONS

The standard version of the twin paradox has an astronaut twin – call her Stella – accelerating away from her Earthbound twin sister, Terra. Stella rockets through the cosmos at relativistic speeds, then later turns around and flies home, eventually landing back on Earth.

But in order to eliminate pesky accelerations, we'll say that the astronaut twin, Stella, was already in motion as she passes over Earthbound twin Terra. At the pass-by, their clocks each start at zero. Then, instead of later turning around, Stella eventually passes another astronaut, Alf, moving equally fast in the opposite direction, who adopts Stella's clock reading and continues back toward Earth. When Alf arrives at Earth, he passes over homebound Terra without slowing, at which time their clock readings are compared. Thus there are no accelerations and the scenario can be examined with the simplest (SR) arithmetic, like what's been shown in diagrams thus far.

In order to make things clearer, we'll say that there exists a space buoy, unmoving with respect to Earth, exactly one light-year distant from Earth, and the buoy has a clock on it that is synchronized with Earth's clock. It is at that space buoy that Stella will pass by Alf. We'll say that Stella, and later Alf, each have a velocity of .75c relative to Earth, which makes the relativistic gamma factor compute out to 1.5, almost exactly. At that gamma factor, a moving clock will tick off only two thirds as much time as the observer's clock does; and moving collinear distances will be like foreshortened.

TERRA'S RECKONING: It's Stella and Alf that move

According to Earthbound Terra's point of view, Stella, traveling at .75C, will reach the distant buoy in 1.33 years. Terra predicts that Stella will witness the buoy clock to display that precise reading upon her arrival there. And Alf's return leg will take the same again, another 1.33 years; so a total of 2.67 years should elapse on Terra's clock. But, deduces Terra, relativistic time-dilation must be ascribed to the traveling clocks (of Stella-cum-Alf), so instead of 2.67 years, Alf's clock is expected to read only 1.78 years upon his arrival.

STELLA-CUM-ALF'S RECKONING: No, it's Terra that travels

According to Stella's viewpoint, stock still in space, the "real estate" consisting of Earth-to-buoy is a moving frame, and so is length-contracted to 0.67 light-year. And since it has a velocity of .75C with respect to her, the distant buoy should arrive when her clock reads 0.89 year. Upon its arrival, Stella figures that Terra's clock back on Earth, being time-dilated from motion, must read only 0.59 year. But the buoy clock reads 1.33 years. However, this is as expected due to the clock dissynchronicity: the buoy clock has a reading 0.75 year later than Earth's, according to Stella's perspective.

When Alf takes over Stella's clock reading of 0.89 year, to him the Earth-to-buoy frame is moving in the opposite direction. Alf notices that the buoy clock reads 1.33 years and computes, per relativistic dissynchronicity of moving separated clocks, that Earth's clock must be at 2.09 years. Under Relativity, Alf cannot adopt Stella's evaluation of Earth's then-current clock reading because differently moving observers do not reckon distant clock readings the same. But Alf can and does take at face value the reading on the buoy clock as a valid indicator, since it is right then and there local to him and Stella both.

The remainder is simplicity: Alf's clock advances another 0.89 year, just as Stella's clock had, as he sits stock still in space awaiting the arrival of the speeding Earth. Terra's Earthbound clock is presumed time-dilated, and so advances only another 0.59 year, to add to the 2.09 reading that Alf had reckoned for it when he was at the buoy. So when Earth gets to Alf, Terra's clock will read 2.67 years, while Stella-cum-Alf's time is 0.89+0.89, or 1.78 years.

CONCLUSION

By either reckoning, Terra's Earthbound clock aged the greater, and by the same amount... so there is no disparity. And more importantly, it corroborates the core concept of Relativity, that any observer can rightly claim his vantage to be stock still: it's the other guy who's moving.