Bell Lab

Two different Bells, two very different shocks.

One section shows why a rope snaps when two ships keep a fixed launch-frame gap while accelerating. The other shows why any local hidden-variable strategy stays inside the CHSH limit while quantum correlations do not.

Bell's spaceship paradox

The ships keep the same distance in the original launch frame, but that does not mean the rope keeps the same proper length. In the ships' momentary comoving frame, the gap grows like $\gamma L$.

Bell / CHSH

A local rule can preassign answers for every setting, but the resulting correlations always satisfy $|S| \le 2$. Quantum mechanics reaches $|S| = 2\sqrt{2}$ for the right angle choices.

Relativity

Bell's spaceship paradox

Rope intact

This model keeps the ships a fixed distance apart in the launch frame. As the shared speed rises, the gap measured in the back ship's instantaneous rest frame becomes larger than the rope's rest length. That is the strain that breaks it.

Elapsed time 0.00 s
Speed 0.00 c
Gamma 1.000
Proper gap 10.00 ft
Rope strain 0.0%
Break point Not reached
Worldlines in launch coordinates
Rope proper length and strain
Launch-frame separation $L$
Lorentz factor $\gamma = 1 / \sqrt{1 - \beta^2}$
Momentary proper gap $L_{proper} = \gamma L$

Quantum foundations

CHSH experiment

Ready to sample

Alice chooses between $a_0 = 0^\circ$ and $a_1 = 90^\circ$. Bob chooses between $b_0 = 45^\circ$ and $b_1 = -45^\circ$. The local model below uses a shared hidden angle $\lambda$ and deterministic local response functions. The quantum model samples singlet-state outcomes with correlation $E(a, b) = -\cos(a - b)$.

Local hidden-variable model

|S| = 0.000
Pair Estimate

Deterministic local rules can shift where the four correlations land, but the CHSH combination remains bounded by $2$.

Quantum singlet model

|S| = 0.000
Pair Estimate

With the angle choices here, the target quantum value is $2\sqrt{2} \approx 2.828$.

Correlation comparison and CHSH bound

What the local simulator is doing

Every trial samples one hidden variable $\lambda$. Alice's answer depends only on her setting and $\lambda$. Bob's answer depends only on his setting and the same $\lambda$. No side can react to the other side's choice during that trial.

What the quantum simulator is doing

It does not preassign all outcomes ahead of time. Instead it samples from the singlet probabilities for the chosen angle pair, producing the exact quantum correlation law in repeated trials.