This sim draws two events, A and B, on a spacetime diagram and lets a second observer, S′, move past at any speed below c. It tests whether the time order of A and B depends on the observer, and what a faster-than-light reply would do to cause and effect.

What's on screen

• A (blue dot) sends the signal, from (x, ct) = (0.00, 0.00).
• B (rose dot) receives the signal, at (3.21, 4.70). Both dots drag.
• Light cone (dashed yellow): 45° lines through A.
• A → B arrow and tag: the signal speed |Δx/Δct|, green, yellow or red by interval class.
• S′ grid and axes (teal): tilt as v/c changes. The grey grid is S.
• Tags top-left: v/c and the time order each frame sees. The Readings panel holds the numbers.
• Return signal on: a dashed pink arrow from B to the amber triangle E₃, and a ⚠ LOOP tag when E₃ falls before A.

Press ⌖ on a row to see where that control sits in the sim.

Fixed: c = 1 on both axes (ct and x share units), S′ moves along x only, and the reply always heads for A's worldline.

Frame S describes the pair by Δct = ct_B − ct_A and Δx = x_B − x_A. With time measured as ct, light runs at 45°. The interval of the pair is:

• The same in every frame. Frame S′ computes Δct′² − Δx′² for the same two events and gets the same number. The Readings panel shows both.
• It sorts the signal by speed. A signal that carries A's influence to B has average speed |Δx/Δct| in S. Δs² < 0 means that is below c, so B lies inside A's light cone. Δs² > 0 means it is above c, so B lies outside.
• On the cone, Δs² = 0 and the signal moves at c.

Frame S′ moves along x at v = βc relative to S, with γ = 1/√(1 − β²). For the same two events, S′ measures the time separation:

• γ is positive, so the sign of Δct′ is the sign of Δct − βΔx.
• Inside the cone, |Δx| < |Δct|. Since |β| < 1, |βΔx| stays below |Δct|, so Δct′ keeps the sign of Δct. Every frame agrees which event comes first.
• Outside the cone, |Δx| > |Δct|. Now βΔx can exceed Δct while |β| is still below 1, so Δct′ can change sign. S′ then sees B before A.

The tags on the canvas and the Readings panel print the order each frame sees.

Setting Δct′ = 0 gives the boost at which A and B become simultaneous in S′: β_flip = Δct/Δx. A boost of the same sign and larger size reverses the order.

• Farther apart, easier. β_flip falls as Δx grows at fixed Δct. For Δct = 2.00 it is 0.40 at Δx = 5.00 and 0.29 at Δx = 7.00.
• Direction matters. β must have the sign of β_flip. A boost the other way only makes Δct′ larger.
• Inside the cone there is none. There |Δct/Δx| exceeds 1, which no boost reaches.

Turn on Return signal. B answers with a reply that moves at v₂ in S′, aimed at A's worldline x = x_A. It arrives at event E₃, which the sim transforms back to S:

• A and E₃ share a worldline. A is at rest in S, so E₃ sits straight above or below A, and every frame agrees which comes first. If ct₃ < ct_A, the reply arrives before A sent the message: a loop.
• The loop needs more boost than the flip. For B to the right of A, β_loop is above β_flip, and a faster reply lowers it toward β_flip.
• Example, B = (5.00, 2.00), v₂ = 3.00 c: β_loop = 11/17 ≈ 0.65. At β = 0.60, ct₃ = +0.33. At β = 0.80, ct₃ = −1.18.
• For a pair inside the cone, ct₃ stays after A at every v/c and v₂ the sim allows.

The argument assumes that if a faster-than-light signal is allowed, it is allowed in every frame.

For a particle with mass, p = γmv and E = γmc² give v = pc²/E. Combined with E² = p²c² + m²c⁴, a speed above c forces:

• A tachyon needs a negative m². v > c means pc > E, so m²c⁴ = E² − p²c² is negative and m is not a real number.
• Speeding up an ordinary particle does not get there. E = γmc² grows without limit as v → c, so no finite energy reaches c, let alone passes it.

The sim draws signals, not particles, so it shows what any faster-than-light signal would do to cause and effect, not a tachyon itself.

• Accelerators. A 6.5 TeV proton in the LHC has γ ≈ 6900 and moves within about one part in 10⁸ of c. Added energy raises γ, not the speed.
• The OPERA neutrinos. In 2011 that experiment reported neutrinos arriving about 60 ns earlier than light would. The cause was traced to timing equipment, a loose fibre-optic connection and a clock fault, and the result was withdrawn. The loop argument is part of why the claim needed such a check.
• Cherenkov light. In water, light travels at about 0.75 c. A particle faster than that makes a blue glow. It is still slower than c, so no light cone is crossed.

Key points

• Δs² is the same in every frame, and its sign says whether B is inside, on or outside A's light cone.
• Δct′ = γ(Δct − βΔx) can change sign for |β| < 1 only when |Δx| > |Δct|, at β_flip = Δct/Δx.
• A reply at v₂ in S′ arrives before the message once β passes β_loop, which is above β_flip.
• A particle faster than c would need m² < 0.

Common mix-ups

• "A moving observer can reorder any two events." Only pairs outside each other's light cones. Inside, every observer agrees.
• "Anything faster than c breaks causality." Only a signal that carries influence does. A laser spot sweeping across a distant screen can move faster than c, because each position of the spot does not cause the next.
• "This sim shows tachyons cannot exist." It shows what a usable faster-than-light signal would do to cause and effect. It tests no particle.

Things to try on your own

• Drag B to (7.00, 2.00) and find where Δct′ changes sign on the v/c slider. Compare it with Δct/Δx.
• Load the outside pair, set v/c = 0.60, turn on Return signal, and raise v₂ until ct₃ drops below zero.
• Drag B well inside the cone, sweep v/c across its whole range, and watch Δct′.