Bell's spaceship paradox

Bell's spaceship paradox is not about preserving the rest length between objects (as in Born rigidity), but about preserving the distance in an inertial frame relative to which the objects are in motion, for which the Ehrenfest paradox is an example.[26] Historically, Albert Einstein had already recognized in the course of his development of general relativity, that the circumference of a rotating disc is measured to be larger in the corotating frame than the one measured in an inertial frame.[33] Einstein explained in 1916:[34]

"We suppose that the circumference and diameter of a circle have been measured with a standard measuring rod infinitely small compared with the radius, and that we have the quotient of the two results. If this experiment were performed with measuring rods at rest relatively to the Galilean system K′, the quotient would be π. With measuring rods at rest relatively to K, the quotient would be greater than π. This is readily understood if we envisage the whole process of measuring from the "stationary" system K′, and take into consideration that the measuring rods applied to the periphery undergoes a Lorentz contraction, while the ones applied along the radius do not. Hence Euclidean geometry does not apply to K."

As pointed out more precisely by Einstein in 1919, the relation is given[33]

${\displaystyle U=\gamma U_{0}}$,

${\displaystyle U}$ being the circumference in the corotating frame, ${\displaystyle U_{0}}$ in the laboratory frame, ${\displaystyle \gamma }$ is the Lorentz factor ${\displaystyle 1/{\sqrt {1-v^{2}/c^{2}}}}$. Therefore, it's impossible to bring a disc from the state of rest into rotation in a Born rigid manner. Instead, stresses arise during the phase of accelerated rotation, until the disc enters the state of uniform rotation.[33]

Minkowski diagram: Length ${\displaystyle L'}$ between the ships in S′ after acceleration is longer than the previous length ${\displaystyle L'_{old}}$ in S′, and longer than the unchanged length ${\displaystyle L}$ in S. The thin lines are "lines of simultaneity".

Loedel diagram of the same scenario

Similarly, in the case of Bell's spaceship paradox the relation between the initial rest length ${\displaystyle L}$ between the ships (identical to the moving length in S after acceleration) and the new rest length ${\displaystyle L'}$ in S′ after acceleration, is:[3][6][8][16]

${\displaystyle L'=\gamma L}$.

This length increase can be calculated in different ways. For instance, if the acceleration is finished the ships will constantly remain at the same location in the final rest frame S′, so it's only necessary to compute the distance between the x-coordinates transformed from S to S′. If ${\displaystyle x_{A}}$ and ${\displaystyle x_{B}=x_{A}+L}$ are the ships' positions in S, the positions in their new rest frame S′ are:[3]

${\displaystyle {\begin{aligned}x'_{A}&=\gamma \left(x_{A}-vt\right)\\x'_{B}&=\gamma \left(x_{A}+L-vt\right)\\L'&=x'_{B}-x'_{A}\\&=\gamma L\end{aligned}}}$

Another method was shown by Dewan (1963) who demonstrated the importance of relativity of simultaneity.[8] The perspective of frame S′ is described, in which both ships will be at rest after the acceleration is finished. The ships are accelerating simultaneously at ${\displaystyle t_{A}=t_{B}}$ in S (assuming acceleration in infinitesimal small time), though B is accelerating and stopping in S′ before A due to relativity of simultaneity, with the time difference:

${\displaystyle {\begin{aligned}\Delta t'&=t'_{B}-t'_{A}=\gamma \left(t_{B}-{\frac {vx_{B}}{c^{2}}}\right)-\gamma \left(t_{A}-{\frac {vx_{A}}{c^{2}}}\right)\\&={\frac {\gamma vL}{c^{2}}}\end{aligned}}}$

Since the ships are moving with the same velocity in S′ before acceleration, the initial rest length ${\displaystyle L}$ in S is shortened in S′ by ${\displaystyle L'_{old}=L/\gamma }$ due to length contraction. This distance starts to increase after B came to stop, because A is now moving away from B with constant velocity during ${\displaystyle \Delta t'}$ until A stops as well. Dewan arrived at the relation (in different notation):[8]

${\displaystyle {\begin{aligned}L'&=L'_{old}+v\Delta t'={\frac {L}{\gamma }}+{\frac {\gamma v^{2}L}{c^{2}}}\\&=\gamma L\end{aligned}}}$

It was also noted by several authors that the constant length in S and the increased length in S′ is consistent with the length contraction formula ${\displaystyle L=L'/\gamma }$, because the initial rest length ${\displaystyle L}$ is increased by ${\displaystyle \gamma }$ in S′, which is contracted in S by the same factor, so it stays the same in S:[6][14][18]

${\displaystyle L_{contr.}=L'/\gamma =\gamma L/\gamma =L}$

Summarizing: While the rest distance between the ships increases to ${\displaystyle \gamma L}$ in S′, the relativity principle requires that the string (whose physical constitution is unaltered) maintains its rest length ${\displaystyle L}$ in its new rest system S′. Therefore, it breaks in S′ due to the increasing distance between the ships. As explained above, the same is also obtained by only considering the start frame S using length contraction of the string (or the contraction of its moving molecular fields) while the distance between the ships stays the same due to equal acceleration.

The world lines (navy blue curves) of two observers A and B who accelerate in the same direction with the same constant magnitude proper acceleration (hyperbolic motion). At A′ and B′, the observers stop accelerating.

Two observers in Born rigid acceleration, having the same Rindler horizon. They can choose the proper time of one of them as the coordinate time of the Rindler frame.

Two observers having the same proper acceleration (Bell's spaceships). They are not at rest in the same Rindler frame, and therefore have different Rindler horizons

Instead of instantaneous changes of direction, special relativity also allows to describe the more realistic scenario of constant proper acceleration, i.e. the acceleration indicated by a comoving accelerometer. This leads to hyperbolic motion, in which the observer continuously changes momentary inertial frames[35]

${\displaystyle {\begin{aligned}x&={\frac {c^{2}}{\alpha }}\left({\sqrt {1+\left({\frac {\alpha t}{c}}\right)^{2}}}-1\right)={\frac {c^{2}}{\alpha }}\left(\cosh {\frac {\alpha \tau }{c}}-1\right)\\c\tau &={\frac {c^{2}}{\alpha }}\operatorname {asinh} {\frac {\alpha t}{c}},\quad ct={\frac {c^{2}}{\alpha }}\sinh {\frac {\alpha \tau }{c}}\end{aligned}}}$

where ${\displaystyle t}$ is the coordinate time in the external inertial frame, and ${\displaystyle \tau }$ the proper time in the momentary frame, and the momentary velocity is given by

${\displaystyle v={\frac {\alpha t}{\sqrt {1+\left({\frac {\alpha t}{c}}\right)^{2}}}}=c\tanh {\frac {\alpha \tau }{c}}}$

The mathematical treatment of this paradox is similar to the treatment of Born rigid motion. However, rather than ask about the separation of spaceships with the same acceleration in an inertial frame, the problem of Born rigid motion asks, "What acceleration profile is required by the second spaceship so that the distance between the spaceships remains constant in their proper frame?"[36][35][37] In order for the two spaceships, initially at rest in an inertial frame, to maintain a constant proper distance, the lead spaceship must have a lower proper acceleration.[3][37][38]

This Born rigid frame can be described by using Rindler coordinates (Kottler-Møller coordinates)[35][39]

${\displaystyle {\begin{aligned}ct&=\left(x'+{\frac {c^{2}}{\alpha }}\right)\sinh {\frac {\alpha t'}{c}},&y&=y',\\x&=\left(x'+{\frac {c^{2}}{\alpha }}\right)\cosh {\frac {\alpha t'}{c}}-{\frac {c^{2}}{\alpha }},&z&=z'.\end{aligned}}\ (t'=\tau )}$

The condition of Born rigidity requires that the proper acceleration of the spaceships differs by[39]

${\displaystyle \alpha _{2}={\frac {\alpha _{1}}{1+{\frac {\alpha _{1}L'}{c^{2}}}}}}$

and the length ${\displaystyle L'=x_{2}^{\prime }-x_{1}^{\prime }}$ measured in the Rindler frame (or momentary inertial frame) by one of the observers is Lorentz contracted to ${\displaystyle L=x_{2}-x_{1}}$ in the external inertial frame by[39]

${\displaystyle L={\frac {L'}{\cosh {\frac {\alpha t'}{c}}}}=L'{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}$

which is the same result as above. Consequently, in the case of Born rigidity, the constancy of length L' in the momentary frame implies that L in the external frame decreases constantly, the thread doesn't break. However, in the case of Bell's spaceship paradox the condition of Born rigidity is broken, because the constancy of length L in the external frame implies that L' in the momentary frame increases, the thread breaks (in addition, the expression for the distance increase between two observers having the same proper acceleration becomes also more complicated in the momentary frame[17]).