Newtonian and Relativistic Travel

Beginning students often imagine that Lorentz invariance (special relativity) represents an obstacle to reaching distant locations within a human life span. Letting L denote the distance to some far location and T the span of a lifetime devoted to the journey, the student has been told that special relativity implies L/T < c, which he thinks implies that it’s impossible to reach any location at a distance L of more than cT. In contrast (so the student thinks), Newtonian physics placed no limit on the ratio L/T, so there was no limit on the distance that could be traversed in a human life time. Of course, this reasoning overlooks the length contraction between two systems of inertial coordinates. Taking this into account, it’s well known that a person can reach arbitrarily great distances in their life span, both according to Newtonian mechanics and according to special relativity. However, somewhat less well known is the fact that, although travel to great distances is prohibitively expensive (in terms of energy) in either case, reaching a great distance in a certain amount of time would actually take much less energy in the context of special relativity than it would under Newtonian physics.

To quantify this, consider a space capsule of mass m to be launched by an enormous rail gun that accelerates the capsule to the speed at the start of its journey. (By increasing the length of the journey and the rate of acceleration, we can make the acceleration portion of the trip arbitrarily short compared with the overall trip.) If the capsule is to travel to a distance L in time T, then according to Newtonian physics the capsule must be given the speed V = L/T, and the energy that must be imparted to the capsule is

On the other hand, according to special relativity, at a speed v (not to be confused with V) the distance to the desired location in terms of the inertial coordinates of the capsule is reduced due to length contraction such that

Hence we have

The energy we must impart to the capsule to place it in this state of motion is

Substituting from the previous expression for v/c, this gives

The ratio of Newtonian energy to special relativistic energy is therefore

where (remember) V = L/T. In both Newtonian physics and special relativity the value of V can be as large as we like, not limited to less than c. For sufficiently small values of V/c this ratio approaches unity, but for larger values it increases without bound, as shown in the plot below.

For example, if we wish to travel for 20 years and reach a location that is 20,000 light years away, we need V/c = 1000, and this means that according to special relativity the required amount of energy that must be imparted to the capsule is roughly 1000 times the entire energy content (mc²) of the capsule – surely a daunting prospect. But the amount of energy that would be required to accomplish this under Newtonian physics is about 500 times greater, i.e., about half a million times the total amount of energy that would be released if the entire rest mass of the capsule was converted to pure energy.

Of course the kinetic energy imparted to the capsule would not be “lost”. It could (in principle) be re-captured at the destination when braking the capsule, since the destination star is roughly at rest relative to the sun – assuming the capsule’s energy wasn’t dissipated by interactions with interstellar particles, and assuming a means of braking could be devised. This energy might then be used to send the capsule back… but therein lies the circumstance that disappoints the anti-relativity enthusiast:  Upon return, the Newtonian traveler would expect to find his siblings aged just 40 years in his absence, whereas according to special relativity about 40,000 years will have passed on earth. Ironically, when asked why it’s important to leave the earth, many reply that it’s to escape a possible cataclysmic event, such as a colliding asteroid or some other catastrophe that makes the world uninhabitable. Why, then, they would be so concerned about returning is unclear. And in such a case, the time delay might be beneficial.

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