Relativity Lite

Appendix | 101 CONTINUOUS ACCELERATION TO CYGNUS X-1 Suppose we want to travel out to the black hole Cygnus X-1, 7,733 light-years away * in the shortest reasonable time. The rocket would accelerate at 1 g for half the trip, flip over and decelerate at 1 g for the remainder of the trip. At this acceleration, the rocket will reach the halfway point with a time-dilation factor given by (A21) of γ = + + −         = + 1 1 1 0 98715 3866 0 2 2 0 V c A c x x c yr x ( ) . ( .5 0 cyr c −       ) , or γ = 3,992.37 . Equation (A22) gives the rocket’s velocity as it just reaches the halfway point, r = 3,866.5 c yr , as v c = − 1 1 2 γ = 0.99999997 c . Equation (A23b), with V x0 = 0 , gives the time for the half trip at t c A = − γ 2 1 = 3,867.47 yr , where we use the Julian year of 365.25 days as preferred by the IAU. † Both equations (A9c) and (A11d) give τ = 8.70418 yr . In chapter 4, we said that under large acceleration, “figure 1c will stretch out as in figure 1d, so much so that the curved light path will differ little from the straight hypot- enuse. Both high constant speeds and large accelerations (to high speeds) give enormous time dilation.” We can now quantify these two claims in the present case of 1 g accelera- tion. The hypotenuse of the triangle under the hyperbolic curve shown in figure 1d, whose legs are r and c τ (as calculated above using the correct formula for acceleration [A9c] or [A11d]), is c t ' = 3,866.51 c yr , just under 1 light-year shorter than the hyperbola, whose length is c t = 3,867.47 yr , a 0.02 % difference. The intermediate constant velocity that would give such a triangle is r/t = 0.999997 c , indeed a high constant speed, differing in the sixth decimal place from the rocket’s veloc- ity as it just reaches the halfway point. It would give a very big time dilation factor when calculated in the special-relativistic manner: t ' / τ = 444 . But the frame-shifting a rocket undergoes when accelerated at 1 g leads to a time dilation factor nearly ten times larger, γ = 3,992.37 when calculated using the correct (and consistent) formula for acceleration (A21). We see that our intuitive connection between the two cases falls well short of the reality, a divergence that will only grow as we go to even larger accelerations (to much higher speeds). * The 2018 Gaia DR2 catalogue shows a parallax of 0.4218 mas, which gives d = 1/0.0004218 = 2371 pc = 7733 c yr . “DR2 2059383668236814720,” Centre de Données astronomiques de Strasbourg, last modified May 26, 2020, http:// simbad.u-strasbg.fr/simbad/sim-id?mescat.distance=on&Ident=%402905066&Name=DR2 +2059383668236814720&submit=display+all+measurements#lab_meas . Other names one can use for the search include HD 226868 and 1956+350. † G. A. Wilkinson, IAU Style Manual Comm. 5, in IAU Transactions XXB (unpublished pamphlet), 1987, p. S23, https://www.iau.org/static/publications/stylemanual1989.pdf (accessed June 26, 2020); https:// www.iau.org/publications/proceedings_rules/units/ (accessed June 26, 2020).