If the bystander measures the distance between the cars as contracted, then two identical experiments do not yield the same result. Or, if the bystander measures the distance between the cars as unchanged from the original distance, then it must be explained how the cars would length-contract in the bystander frame (SR is clear that they would) yet a rope stretched between them would not length-contract. That’s the conundrum, the paradox.

If both cars accelerate identically for the same elapsed time on bystander clocks then they should traverse the same distance in the bystander frame. If the distance between them contracts then the cars do not traverse the same distance.

Here is the relevant equation:

*d* = (sqrt((

*a* *

*t*)^2 + 1) – 1) /

*a*
This equation returns the distance

*d* in the bystander frame that a car traverses for a given acceleration

*a* (acceleration the driver feels, in terms of

*c*) for elapsed time

*t* on bystander clocks. In the paradox

*a* and

*t* are the same for both cars, hence--because

*a* and

*t* are the only inputs to the equation--

*d* should be the same, hence the distance between the cars should be constant in the bystander frame. If the distance between the cars contracts in the bystander frame then the equation is fiction; two experiments involving the same

*a* and

*t* do not yield the same

*d* result.

Let both cars be initially at rest and one light year apart in the bystander frame, then accelerate to so close to

*c* in one second on bystander clocks that the distance between them contracts to one meter in the bystander frame. Now the pursuing car has surpassed the speed of light in the bystander frame, beating its own image that travels toward the pursued car at

*c*, leading to a causality paradox.

So for two important reasons, the distance between the cars in the bystander frame should not change as the cars accelerate.

I may have misread the convoluted web page about Bell’s Spaceship Paradox. The relevant quote may be:

This first picture interprets "two ships with the equal constant accelerations" to mean "constant for the co-moving observers, and equal in the lab-frame". Note that the lab-frame says that the accelerations are not constant, and the co-moving observers say the accelerations are not equal! (More precisely, any particular co-moving observer says this. The phrase "the co-moving observers" does not refer to a single frame of reference, unlike the phrase "the lab-frame".) The lab-frame says the ships maintain a constant distance from each other; the co-moving observers don't agree.

I think this is the resolution to the paradox: the bystanders measure the distance between the accelerating cars as unchanged. A rope between them would break (because the pursuing driver observes that the other car is pulling away--it's a given that the cars have the same constant acceleration so the rope doesn't affect that, but make the rope a thread if you wish the breakage to be more believable) and length-contract along with the cars.