Distance Between Bell’s Rockets

 

John Bell gave a famous illustration of the effects of special relativity in terms of two rockets, both at rest at time t in terms of a standard system of inertial coordinates S, and both undergoing constant proper acceleration “a” in the positive x direction. This is illustrated in the figure below.

 

 

We have depicted the case where the rockets have been accelerating continuously, but they could just as well have been continually at rest until time t, and then begin their accelerations. Let L denote the separation between the two rockets in terms of S at time t. We will suppose each rocket has an ideal clock that is set to zero at t = 0, and thereafter shows the elapsed proper time. As discussed in “Accelerated Travels”, at some proper time τ1 for the trailing rocket, it has the S coordinates given (in terms of geometric units, so “a” has units of inverse distance) by

 

 

Let S′ denote the standard inertial coordinate system in terms of which the trailing rocket is momentarily at rest when at this event, and at that value of t′ let the leading rocket be located at S coordinates given as a function of its proper time τ2 by

 

 

We also know that the velocity of the trailing rocket at the cited event is v = tanh(aτ1), and the skew of simultaneity gives v = (t2 – t1)(x2 – x1). Therefore we have

 

 

Clearing fractions gives

 

 

Thus we have

 

 

A plot of τ2 versus τ1 for various values of L, given that a = 1, is shown below.

 

We note that when aL = 1 we have τ2 = 2τ1. This is the case when L = 1/a, meaning that the initial distance between the rockets equals the distance of the trailing rocket from its fulcrum event, i.e., the center of its hyperbola.

 

The spatial distance between the rockets in terms of S′ for a given τ1 is

 

 

where τ2 here is understood to be the function τ2(τ1) given previously. Making that substitution and simplifying, we get the result

 

 

A plot of L′ versus τ1 is shown below.

 

 

In the special case when aL = 1 this reduces to

 

 

and therefore in this case we have

 

 

Return to MathPages Main Menu