Relativistic Generalization of the Incentive Trap of Interstellar Travel with Application to Breakthrough Starshot

Abstract: As new concepts of sending interstellar spacecraft to the nearest stars are now being investigated by various research teams, crucial questions about the timing of such a vast financial and labor investment arise. If humanity could build high-speed interstellar lightsails and reach the alpha Centauri system 20 yr after launch, would it be better to wait a few years, then take advantage of further technology improvements to increase the speed, and arrive earlier despite waiting? The risk of being overtaken by a future, faster probe has been described earlier as the incentive trap. Based on 211 yr of historical data, we find that the speed growth of human-made vehicles, from steam-driven locomotives to Voyager 1, is much faster than previously believed, about 4.72 % annually or a doubling every 15 yr. We derive the mathematical framework to calculate the minimum of the wait time (t) plus travel time (tau(t)) and extend two exponential growth law models into the relativistic regime. We show that the minimum of t+tau(t) disappears for nearby targets. There is no use of waiting for speed improvements once we can reach an object within about 20 yr of travel, irrespective of the actual speed. In terms of speed, the t+tau(t) minimum for a travel to alpha Centauri will occur once 19.6 % the speed of light (c) become available, in agreement with the 20 % c proposed by the Breakthrough Starshot Initiative. If interstellar travel at 20 % c can be obtained within 45 yr from today and if the kinetic energy could be increased at a rate consistent with the historical record, then humans can reach the ten most nearby stars within 100 yr from today.