Tachyon Causality Violation on a Minkowski Diagram

Source

Lecture: 18 · Faster than Light
Simulation: Simulation 1 — Tachyon causality violation (guided inquiry), reusing the L9 spacetime-interval Minkowski-diagram interface.
Concept: A faster-than-light signal between two spacelike-separated events can have its time-ordering reversed in a boosted frame, enabling a round-trip causal loop that violates causality.
Serves: Explain why the speed of light is an absolute speed limit · Analyse causality using lightcone structure · Connect superluminal propagation, acausality and instabilities.
Inquiry: Using spacetime intervals and frame dependence: can a signal travelling faster than light ever create a causal paradox? Build a scenario that demonstrates this or rules it out.
Misconception: "Faster-than-light signals can still respect causality as long as they aren't infinitely fast" — probed by prediction option B with a targeted rebuttal.

Pedagogy principles applied

ID	Principle	How it shows up here
A1	One concept	Every element serves the single claim: FTL signal ⇒ frame-dependent time order ⇒ causal loop.
A2	Elements serve the concept	Only Δs², Δt, signal speeds, and E₃ arrival are shown — each used by a step.
A3/A4	Equations are tools	Lorentz boost & interval formulae live in The Physics revision sheet, opened on demand.
B1	Real field terms	worldline, light cone, spacetime interval, boost, frames S / S′.
B3	Symbols defined on first use	Δs² sign rule shown; β ≡ v_boost/c; v₁, v₂ labelled as signal speeds.
C1	Numbers are used	Δs² in Step 1, Δt in Step 2, E₃ arrival vs E₁ in Step 3.
C2	Frame + units	"v₁ = 1.25c (S)", "cΔt = +4.00 ls (this frame)", "Δs² = +9.00 (ls)²".
D1	Visual dominates	The Minkowski diagram fills the hero; readouts are compact.
D3	Shape + label	E₁/E₂ circles, E₃ triangle, worldlines carry speed labels.
D5	No clipping	Events clamp to the diagram; off-window markers fade; lines clip cleanly.
D6	Incremental complexity	Guide steps add highlight layers only.
D7	No lockstep gating	All sliders, dragging, return toggle and prediction live from load.
D8	Labels on scene, prose in panels	Canvas shows names/speeds; explanations sit in the aside.
F1	Name before reference	E₁, E₂, E₃ labelled on the diagram before prompts mention them.
F2	Conceptual prediction	"Can the reply beat the original?" — yes/no/conditional, no arithmetic.
F9–F14	Phase-matched prompts	Observe (Step 1) → relate (Step 2) → explain (Step 3).
G2	Invariants labelled	Δs² row marked "(invariant)" and stays steady under boost.
G3	Frame switch re-renders	Boost recomputes all coordinates & worldlines, relabels axes S→S′.
G4	Rest frame is stationary	The observer worldline (x=0 in S) is drawn explicitly in every frame.
G5	Prompts bound to live state	Prompts reference "the displayed Δs²" / "the time-order row".
G7	Operational coordinates	v = Δx/Δt; simultaneity = Δt′ = 0; loop = E₃ at ct < ct₁.
H1	Structural misconception	No standing wrong-idea card — it is a prediction option only.
H2	Targeted rebuttal	Option B reveals the frame-dependent-ordering rebuttal when chosen.
J2	Faithful numbers	Δs², cΔt, E₃ computed from actual coordinates, no rescaling.
J3	Correct model	Lorentz boost, interval, and return intersection use the stated SR formulae.
J5	Stable axes	Fixed ±6 light-second window; no per-frame rescale.

Tachyon Causality Violation

Teach me

Speed

Guided Inquiry

1 · Alice, Bella, and a reply

Let’s meet the cast. Alice sits at x = 0. At E₁ she sends a faster-than-light signal to Bella, who receives it at E₂ and at once sends a faster-than-light reply back; the reply meets Alice's worldline at E₃. The pair E₁E₂ carries the interval Δs² = −(cΔt)² + (Δx)², negative for timelike separation and positive for spacelike, and a boost v_boost re-expresses every event in the coordinates of an observer moving past Alice.

2 · Predict — a universal "before"

Let’s start with the old picture. Newtonian physics gives every observer the same clock. On that picture, do you think an observer moving past Alice could ever see the reply reach her at E₃ before she sent the signal at E₁?

3 · Predict — where the order turns over

Now let’s bring in relativity. In Alice’s frame the outgoing leg spans Δx = +5.00 ls and cΔt = +4.00 ls, and cΔt′ = γ(cΔt − βΔx). At which boost do you think cΔt′ passes through zero?

4 · Predict — closing the round trip

Now let’s close the round trip. Return v_boost to 0. E₂ stays at x = 5.00 ls, ct = 4.00 ls, and the reply reaches Alice’s worldline at E₃. Which change do you think drives E₃ below ct = 0 — before Alice sent?

5 · Predict — which separations can turn over

One last experiment. Drag E₂ inside the dashed 45° lines — near x = 2 ls, ct = 4 ls — and sweep v_boost across its full range. So which pairs do you think a boost can re-order?

6 · The antitelephone

Here’s what it all adds up to. A round trip that lands E₃ before E₁ is a tachyonic antitelephone: each leg is spacelike, so its own order is frame-dependent, but E₁ and E₃ both sit on Alice's worldline at x = 0 — a timelike pair whose order every frame agrees on. c is an absolute limit because superluminal signalling turns that frame-independent verdict into "reply first".

Listen

Controls

▾

Boost v_boost

+0.00 c

Return speed v₂

3.00 c

Live readouts

▾

Active frameS

Δs² (invariant)—

Separation—

Time order—

cΔt (this frame)—

Δx (this frame)—

Signal v₁—

Return v₂—

E₃ arrival (frame S)—

Causality—

Δs² = −(cΔt)² + (Δx)²: <0 timelike · >0 spacelike · =0 lightlike. ls = light-seconds.

Lorentz boost (display in frame S′)

Every event, worldline and light cone is re-computed under this boost — not just relabelled.

Lorentz factor

β is the velocity of frame S′ relative to S, in units of c.

Spacetime interval (invariant)

<0 timelike · >0 spacelike · =0 lightlike. Identical in every frame.

Time-order reversal

Δt′ flips sign when v_boost > c²Δt/Δx — only possible for spacelike pairs (|Δx| > c|Δt|).

Return-signal arrival

x₂ = E₂'s position, sent back toward x = 0. Causal loop when ct₃ < ct₁. E₁ & E₃ both lie on x = 0, so this ordering is frame-invariant.

The Physics

Tachyon causality violation · what this sim shows, the physics behind it, and every control

At a glance

Alice sends a signal to Bella, and Bella replies at once. The sim draws that round trip on a spacetime diagram and lets you view it from a moving frame, so you can test what faster-than-light signalling would do to cause and effect.

What you'll learn

How to read a spacetime diagram: events, worldlines and a light cone.
What the interval Δs² says about a pair of events.
When the time order of two events depends on the frame, and when it does not.
How the speed of a reply decides when it reaches Alice.

Frame S. Alice sits at x = 0, so her worldline is the ct axis. E₂ lies outside the dashed light cone of E₁.

What's on screen

Alice's worldline (dotted cyan, x = 0) with E₁ (blue dot) where she sends. Drag E₁ along the line.
E₂ (orange dot) where Bella receives and replies. Drag it anywhere within ±6 ls.
Light cone of E₁ (dashed 45° lines): where a light signal from E₁ could reach.
Outgoing leg E₁ → E₂ (rose), and the dashed return leg to E₃ (purple triangle).
Badges (top left): the interval Δs² with its type, and the time order of E₁ and E₂ in the frame shown.

See itShow the diagram in Alice's own frame, at reset. Watch for: the badge Δs² = +9.00 (ls)² · spacelike, and the leg labelled v₁ = 1.25c.

The controls

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

Control	What it changes	Range
Boost v_boost	The speed of the frame S′ you view the diagram from, relative to Alice's frame S. Every event, worldline and cone is recomputed in S′.	−0.99 c to +0.99 c opens at 0
↩ Return signal	Shows or hides Bella's reply and the event E₃ where it meets Alice.	on / off opens ON
Return speed v₂	How fast the reply travels, measured in frame S. It decides where E₃ lands.	1.01 c to 5 c opens at 3 c
E₁ and E₂ (drag)	The two events. E₂ moves anywhere in the window; E₁ moves only along Alice's worldline.	±6 ls opens E₁ (0, 0), E₂ (5, 4)
Live readouts	The frame shown, Δs², separation type, time order, cΔt, Δx, the speeds v₁ and v₂, the E₃ arrival and the causality verdict.	panel
⏸ Pause / ▶ Play	Pauses or restarts the animation of the signals along their legs.	top bar
Speed	How fast that animation runs. The physics is unchanged.	0.25×–4× opens at 1×
↻ Reset	Puts the events and sliders back to their opening values and restarts the mode you are in.	top bar

Units: c = 1 ls per second, so ct and x are both in light-seconds (ls). The sim draws Alice's frame S, or the boosted frame S′.

The interval sorts pairs of events

Two events are separated by Δx in space and by cΔt in time, both in ls. Their interval combines the two:

Timelike pairs are negative, spacelike pairs positive.

The dashed 45° lines are the light cone of E₁: pairs with Δs² = 0, joined by a light signal.
Inside the cone Δs² < 0 (timelike). Outside it Δs² > 0 (spacelike).
A frame S′ moving at v_boost sees every event at new coordinates:

The sim redraws everything through this transform.

Key idea. Δs² has the same value in every frame. cΔt and Δx do not: a boost changes how the interval is split between time and space.

See itView the diagram from a frame moving at +0.50 c. Watch for: cΔt and Δx changing while Δs² stays at +9.00 (ls)².

A boost can reverse the order of two events

Take the outgoing leg, E₁ to E₂. In S′ its time gap is:

γ is always positive, so the sign of cΔt′ is the sign of cΔt − βΔx.
It passes through zero at β_flip = cΔt/Δx. At reset, Δx = +5 ls and cΔt = +4 ls, so β_flip = +0.80. There the two events are simultaneous in S′.
Below β_flip, E₁ comes before E₂ in S′, as in S. Above it, E₂ comes before E₁.
Δs² stays at +9.00 (ls)² throughout.

See itBoost just past β_flip, to +0.82 c. Watch for: the badge changing to ↓ E₂ before E₁, with Δs² unchanged.

Only spacelike pairs can be reversed

No frame moves at c, so a flip needs |β_flip| = |cΔt/Δx| < 1. That means |Δx| > c|Δt|, which is exactly Δs² > 0.

E₂ at (x, ct), in ls	Δs² (ls)²	β_flip = cΔt/Δx
(5, 4) · spacelike	+9	+0.80, reachable
(4, 4) · on the cone	0	+1.00, not reachable
(2, 4) · timelike	−12	+2.00, not reachable

A pair on or inside the cone keeps its order in every frame the slider can reach.
A signal between such a pair would not need to be faster than c.

See itMove E₂ inside the cone, to (2, 4), and sweep the boost across its full range. Watch for: Δs² = −12.00 (ls)² and the time order staying ↑ E₁ before E₂ at every boost.

The reply's speed sets when it arrives

Bella's reply runs from E₂ back to Alice's worldline at speed v₂ (in units of c, in frame S). It covers |x₂| ls, so E₃ lands at:

v_boost v₂

These sliders move the sim too.

reply speed v₂ (reset geometry)	E₃ at ct₃ = 4 − 5/v₂
3.00 c	+2.33 ls, after E₁
1.25 c	0.00 ls, level with E₁
1.20 c	−0.17 ls, before E₁
1.05 c	−0.76 ls, before E₁

The slower the reply, the lower E₃ lands. It falls below E₁ when v₂ < v₁, the outgoing leg's speed in frame S (E₂ above E₁). That is the causal loop.

See itLower the reply speed to 1.20 c. Watch for: E₃ arrival going negative, and the flashing ⚠ CAUSAL LOOP banner on the diagram.

Every frame agrees which of E₁ and E₃ came first

E₁ and E₃ are both on Alice's worldline, so Δx = 0 and they are a timelike pair.
Each leg is spacelike, so the order of its two ends depends on the frame. The order of E₁ and E₃ does not.
So if E₃ lands below E₁ in frame S, it is below E₁ in every frame.

The same events at v_boost = 0 and +0.85 c, with v₂ = 3 c. The order of E₁ and E₂ turns over, but E₃ stays above E₁ in both.

Key idea. With a loop, the reply arrives before the message was sent, and no choice of frame removes that. This is the case against signals faster than c.

See itBoost to +0.85 c with the reply at 1.05 c. Watch for: E₂ before E₁ in this frame, while E₃ stays below E₁ and the loop banner stays on.

Where it matters

OPERA, 2011. A neutrino experiment reported arrival times about 60 ns shorter than light's. Later checks found a faulty fibre-optic connection and a clock problem, and the claim was withdrawn in 2012.
Apparent speeds above c. A laser spot swept across the Moon can move faster than c, and so can a wave's phase velocity. Neither carries information from a cause to an effect, so the loop argument does not apply.
Field theory. Relativistic quantum field theories are built so that events at spacelike separation cannot influence each other. The frame-dependent order of such pairs is then harmless.

Go deeper: why a tachyon would need m² < 0

The four-momentum relation, read backwards.

For a particle with mass, E = γmc² grows without limit as v → c, so no finite energy reaches c.
A hypothetical particle with v > c would need an imaginary rest mass. It is called a tachyon, and none has been observed.
A "tachyonic" mass term in field theory means something else: an unstable vacuum, as in the Higgs mechanism, not a faster-than-light messenger.

Summary

In one line. A signal faster than c joins events whose order depends on the frame, and two such signals can bring the reply back before the message, a result every frame agrees on.

Key points

Δs² = −(cΔt)² + (Δx)² sorts pairs of events and has the same value in every frame.
A boost can reverse the order of two events only if Δs² > 0.
The reply lands at ct₃ = ct₂ − |x₂|/v₂, below E₁ when v₂ < v₁.
E₁ and E₃ share a worldline, so every frame agrees on their order.

Common mix-ups

"Faster than light is fine, as long as it isn't infinitely fast." Any v > c gives a spacelike pair, and at reset any reply slower than the 1.25 c outgoing leg closes the loop.
"The boost causes the loop." E₂ and v₂, in frame S, fix where E₃ lands. The boost only redraws the events.

Things to try on your own

Drag E₂ to x = 6 ls, ct = 4 ls, then find the flip boost and the reply speed at which the loop closes.
With v₂ = 1.20 c, sweep the boost from −0.99 c to +0.99 c. Which readouts change, and which do not?