03. Two rockets and a thread

Two identical rockets float at rest, one ahead of the other, joined by a taut, fragile thread. At $t=0$ in your frame both fire identical engines and follow identical velocity profiles, so in your frame their separation never changes. Does the thread break?

Solution

Yes.

In your frame the separation is constant by construction — but the thread is now moving, so its unstressed length is Lorentz-contracted. And contraction here is not a description but a dynamical fact: the thread’s internal forces obey Lorentz-covariant dynamics, so a moving thread left to itself settles into a lab-frame length $L_0/\gamma$. Pinned instead across a gap held at $L_0$, it cannot settle; the mismatch is real strain, growing as $\gamma - 1$, and a fragile thread parts almost at once ($0.2\%$ strain — enough for most materials — arrives by $v \approx 0.06c$). In the rockets’ instantaneous rest frames the same story reads differently: there the separation grows, to $\gamma L_0$ — by relativity of simultaneity the leading ship “started earlier,” and in the momentarily comoving frame it is already pulling ahead while the trailer still lags — and the thread is stretched outright by exactly the factor it is overstretched in yours.

To keep the thread relaxed, the ships would have to maintain constant proper separation — Born-rigid motion — and that requires the trailing ship to accelerate harder than the leader: the two must ride different Rindler hyperbolae $x^2 - c^2t^2 = x_0^2$, each with constant proper acceleration $a = c^2/x_0$, where $x_0$ is the distance to the common pivot event of the family — the trailing ship, sitting nearer it, pushes harder. The required difference is absurdly small instant by instant (one part in $10^{13}$ for ships a kilometer apart at $1g$), yet a year of ignoring it accumulates order-unity strain: the paradox’s bite is cumulative, not instantaneous. Identical accelerations and rigid formation are incompatible demands; Bell’s spaceships are the cleanest demonstration that length contraction has dynamical teeth.

Deeper in the notebook: 03. Bell’s Spaceship Paradox and Born Rigidity · 01. Uniform Acceleration; the Rindler Wedge and Horizon