Relativity Lite

ATrip to Alpha Centauri | 29 CHAPTER 3 A Trip to Alpha Centauri Let us take a trip to Alpha Centauri and leave our twin behind. This is a G-type star that is 4.37 light-years from Earth’s Sun, * so if we travel at v = 0.6 c , then the one-way Earth coordinate time will be t d v c yr c yr = = = 4 37 6 7 283 . . . , or a round-trip Earth coordinate time of 14.56 years . (Note that we have written the distance unit light-year as c year , since this allows us to manipulate the units correctly; note also that c/c = 1 just as 2/2 = 1 or pig/pig = 1 , and multiplying an expression by one gives us that expression back.) Now you and your twin agree to communicate using light pulses the whole time. Your twin will send 10 pulses of light over the 14.56 years , or one pulse every 1.456 years . Since you are trying to outrun the pulses (though failing) on the outbound trip, you will see fewer pulses, at longer intervals, than your twin sends. On the return trip, the pulses you see will pile up faster than they are sent. We wish to actually see what happens on this trip, so I will show screenshots from a 1993 Macintosh OS 9 application called RelLab, which allows us to program in the motion of our flying saucer between these stars as well as the propagation of the light pulses we use to signal with the twin left on Earth. The initial setup is shown in figure 1. In the upper left-hand corner, one sees “Frame: Earth,” which indicates that this is the frame of reference for our twin left on Earth. That means that what we are seeing is what is transpiring in the Earth coordinate system, courtesy of an omnipotent observer (defined as a collection of ob- servers all at rest with respect to the Earth and with watches synchronized). No individual observer at rest with respect to the Earth, such as our twin, “sees” this omnipotent view, but she can construct it from all such observers if each of them sent her messages detailing the arrival times of the pulses of light at their locality. I have added RelLab to an application called WPMacApp, which may be downloaded * Due to the extreme brightness of Alpha Centauri, the new Gaia space telescope has, as of 2019, not given a parallax measurement of its distance. So we must rely on other instruments. P. Kervella, F. Mignard, A. Mérand, and F. Thévenin, A&A 594 , A107 (2016), used the Very Large Telescope (VLT) and the New Technology Telescope (NTT) to find the parallax. Their result was 747.17 ± 0.61 milli-arc-seconds (mas), giving a distance in parsecs (from the phrase “parallax arc-­ seconds,” the means by which distances are found) and light-years of d = 1/(0.74717 ± 0.00061)" = (1.3384 ± 0.0011 pc) × 3.2616 c yr/pc = 4.3653 ± 0.0036 c yr . We will round this up to 4.37 and hope that their error bars will survive the test of time. Actually, an error of even a few percentage points would not alter this story in any significant way.