Simulation
Acceleration
Reference Frame
Display Options
Ship Status
Ship A:
Position: 0.00
Velocity: 0.00c
γ: 1.00
Ship B:
Position: 5.00
Velocity: 0.00c
γ: 1.00
String:
Coordinate distance: 5.00
Proper distance: 5.00
String status: Intact
Watch two spaceships accelerate identically while a string between them snaps—see how Born rigidity works in special relativity
Ship A:
Position: 0.00
Velocity: 0.00c
γ: 1.00
Ship B:
Position: 5.00
Velocity: 0.00c
γ: 1.00
String:
Coordinate distance: 5.00
Proper distance: 5.00
String status: Intact
In 1959, E. Dewan and M. Beran proposed a paradox in special relativity, later championed by John Bell. Two spaceships (A and B) start at rest, separated by distance L, connected by a string. At t=0, both ships begin accelerating with identical proper acceleration profiles. What happens to the string?
Since both ships undergo identical acceleration programs, shouldn't they maintain constant separation? The ships start together and accelerate the same way, so the distance between them should stay constant, and the string shouldn't break.
The string does break, and here's why: the key insight is the relativity of simultaneity. When we say both ships "start accelerating simultaneously," we mean simultaneous in the initial rest frame. But as the ships accelerate, their notion of simultaneity changes.
In the initial rest frame, the ships maintain constant coordinate separation L. However, the string doesn't care about coordinate distances—it cares about proper distance (the distance measured in the instantaneous rest frame of the ships). As the ships accelerate, the proper distance between them increases, stretching and eventually breaking the string.
When a spaceship undergoes constant proper acceleration a, its worldline in spacetime is a hyperbola. The trajectory is given by:
Both ships follow this same hyperbolic trajectory, offset by their initial positions (x₀=0 for ship A, x₀=L for ship B).
A "Born rigid" object is one that maintains constant proper distances between all its parts throughout acceleration. Bell's spaceships are not Born rigid—if they were, ship B would need to accelerate slightly differently than ship A.
For Born rigid motion, rear points must accelerate harder than front points. The proper acceleration of ship B would need to be:
Since both ships use the same proper acceleration, the object (the two-ship-plus-string system) is not rigid, and internal stresses develop.
In this frame, both ships maintain constant coordinate separation L. Their worldlines are parallel hyperbolae. The string appears to maintain constant length in coordinates, but its proper length increases due to Lorentz contraction effects. When the proper length exceeds the string's breaking strength, it snaps.
In the instantaneous rest frame of ship A, ship B appears to be accelerating away. This is because "simultaneous" events in the comoving frame correspond to events at different proper times for the two ships. Ship B, being farther forward, experiences "later" proper time when viewed simultaneously in ship A's frame, so it has already achieved higher velocity—thus it's moving away.
Watch the String Break: Start the simulation and observe both reference frames. In the rest frame, the ships maintain constant separation, yet the string still breaks. In the comoving frame, ship B visibly accelerates away from ship A.
Vary the Acceleration: Try different values of proper acceleration. Higher acceleration causes the string to break sooner (at lower velocity). Can you see why?
Compare Reference Frames: Switch between rest and comoving frames. Notice how ship B's position changes depending on your frame. This is the relativity of simultaneity in action—what counts as "now" for ship B depends on your reference frame.
Examine the Worldlines: The worldlines are hyperbolae in the rest frame. Notice they remain parallel—constant coordinate separation. Yet the proper distance (measured along a simultaneity line in the ships' frame) grows.
When this paradox was first proposed, it caused significant controversy even among physicists. The resolution highlights a subtle but crucial aspect of special relativity: simultaneity is relative, and what seems "obvious" from Newtonian intuition (identical accelerations → constant separation) fails in special relativity.
John Bell used this example to emphasize that special relativity is not just about coordinate transformations—it has real physical consequences. The string really breaks. This is not a coordinate artifact or an optical illusion. The tension in the string is a proper, frame-independent physical quantity, and it genuinely increases until the string snaps.