Minkowski Metric Explorer

Source

Lecture: 10 — Lorentz transformations
Simulation: 1 — Minkowski space & metric (guided-inquiry); a (t, x) plane with a Local / Global zoom toggle.
Concept: The spacetime interval Δs² = −Δt² + Δx² (c = 1) defines a non-Euclidean "distance" that can be negative, zero, or positive — classifying event pairs as timelike, lightlike, or spacelike.
Serves: Describe Minkowski spacetime and its metric structure; identify invariant quantities under Lorentz transformations (the interval is the central invariant).
Inquiry: Place P₂ at spacelike / timelike / lightlike separations from P₁, read Δt and Δx off the grid, compute Δs², and watch the sign flip as the pair crosses the light cone.
Misconception: "A squared distance is always ≥ 0" — the Euclidean intuition. The Minkowski minus sign on the time term makes Δs² < 0 for timelike pairs.

Pedagogy principles applied

ID	Principle	How it shows up here
A1	Single concept	Everything serves "Δs² can be −, 0, or +".
A3/A4	Formal tools gated	Metric matrix & index form appear only in the Local view (driven by the zoom toggle); the full revision sheet opens only on request.
B3	Define symbols	Δt, Δx, Δs², dt, dx, ds², η defined in the Symbols panel; finite (Δ) vs infinitesimal (d) distinguished.
B4	Descriptive headers	"Spacetime diagram (t, x)" / "Local metric structure", not aspirational names.
C1	Numbers used	Δt, Δx, Δs² are read off, typed, and compared in the Investigate steps.
C2	Units + frame	Readouts carry units (light-seconds) and the classification (timelike/spacelike/lightlike).
D1	Primary content	The (t, x) diagram fills the hero; inputs/panels are secondary.
D3	Identify entities	P₁ is a cyan circle, P₂ a magenta diamond, each labelled.
D4	Plot fits	Canvas sized to fit without an inner scrollbar.
D5	No clipping	All labels are bounds-clamped with backing pills, even for thin near-lightlike triangles.
D6	Incremental complexity	Global view is the entry point; Local view adds the formal metric layer via the always-available toggle.
D7	Playable from load	On load: points draggable, toggle live, inputs accept text, prediction answerable — nothing gated.
D8	Labels not narration	Canvas carries labels (axes, points, Δt, Δx, Δs²); prose lives in panels.
F2	Conceptual prediction	The prediction asks about the sign behaviour of Δs², not a numeric value.
H1/H2	Structural misconception	The Euclidean "always ≥ 0" idea is prediction option (a); its rebuttal + a dashed Euclidean overlay appear only if selected.
J1	Consistent notation	Signature (−, +), η with subscript μν, Δ for finite and d for infinitesimal throughout.
J2	Faithful computation	A negative interval is shown as a negative number (e.g. −12), coloured by type.
J3	Correct metric	Δs² = −Δt² + Δx²; exact 0 on the 45° line.
J5	Stable axes	Fixed −5…+5 range; no rescale while dragging.

Minkowski Metric Explorer

Teach me

Speed

spacelike (Δs²>0) timelike (Δs²<0) lightlike (Δs²=0)

Guided Inquiry

1 · Predict the sign of Δs²

Let’s start with two events. P₁ and P₂ sit on a (t, x) plane, with c = 1 so both axes carry light-seconds, and their separation is Δs² = −Δt² + Δx².

Now imagine P₂ moving from far out in space (Δx > Δt) to inside the dashed light cone (Δt > Δx). In Euclidean geometry a squared distance is never negative. So what do you think Δs² does along the way?

2 · A zero interval

Here’s a special case. P₂ now sits exactly on a dashed 45° line through P₁: Δt and Δx are equal, and neither is zero, so Δs² = 0. What do you think that zero says about this pair?

3 · Inside the cone

Now let’s step inside. P₂ has moved inside the dashed cone, where Δt is the larger of the two gaps and Δs² is negative. What do you think can connect this pair of events?

4 · What sets the length

One last experiment. A displacement of fixed coordinate length is turned through α, away from the t-axis, and the grey dashed step keeps the same length at every α. What do you think the coloured interval ds does as α passes 45°?

5 · The Minkowski metric

So here’s the big idea. The Minkowski metric η turns the two gaps into the interval, Δs² = −Δt² + Δx²: the time gap enters with a minus sign where the Euclidean rule d² = Δx² + Δy² has a plus.

	Euclidean	Minkowski
sign	never negative	+, 0 or −
zero when	the points coincide	light connects the events

Every inertial observer computes the same Δs², so all agree which of the three cases a pair falls into. The curves of constant Δs² are hyperbolae, not circles, and a boost only slides an event along its hyperbola, never off it. When you’re ready, press Finish and explore freely.

Listen

Controls

▾

View

Δt = 3.0 ls Δx = 4.0 ls Δs² = 7.0 ls² · spacelike

Read off the grid & compute: Δt Δx

true (from grid): hidden — read it off, then Compute

Symbols & notation

▾

Δt, Δx — finite coordinate differences between P₁ and P₂ (light-seconds).

Δs² — finite squared interval = −Δt² + Δx² (ls²). The invariant interval: same in every inertial frame.

dt, dx — infinitesimal displacements (Local view). Symbols, not numbers.

ds² — infinitesimal squared interval = −dt² + dx².

η_μν — Minkowski metric tensor, diag(−1, +1) in the (t, x) basis.

Signature (−, +), c = 1 ⇒ Δs² < 0 timelike, = 0 lightlike, > 0 spacelike.

c = 1 — time & space in the same units (light-seconds, "ls"). Spacetime interval: Δs² = −Δt² + Δx² (in ls²).

Spacetime interval (finite)

Metric tensor (infinitesimal form)

Classification

Proper time (timelike pairs)

Worked check points

Invariance

Δs² is a Lorentz-invariant scalar — all inertial observers compute the same value even though they disagree on Δt and Δx separately. (The next simulation demonstrates this under a Lorentz boost.)

The Physics

Minkowski space & the metric · what this sim shows, the physics behind it, and every control

At a glance

Two events, P₁ and P₂, sit on a space–time diagram. This sim computes one number for the pair, the interval Δs², and shows how that number relates to the light cone and to the length of a small step.

What you'll learn

How the interval Δs² is built from the time gap Δt and the space gap Δx.
How the sign of Δs² relates to the light cone.
What √|Δs²| measures, and for which pairs it can be measured.
How the metric sets the length of a step, and what happens to Δs² when the frame changes.

Two events and the gaps between them. P₂ opens at Δt = 3, Δx = 4.

What's on screen

P₁ (cyan circle): the first event. Drag it or P₂ on the grid.
P₂ (magenta diamond): the second event.
Dashed light cone through P₁, with photon dots running along it.
Shaded triangle and thick line: the two gaps Δt and Δx, and the line joining P₁ to P₂.
Readouts and Compute box (sidebar): Δt, Δx, Δs², and the arithmetic.

See itPut P₁ at the origin and P₂ at Δt = 3, Δx = 4. Watch for: the magenta diamond up and to the right of the cyan circle, with the shaded triangle showing the two gaps.

The controls

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

Control	What it changes	Range
P₁ and P₂ markers	Drag either marker on the grid (Global view). Δt and Δx are the differences between them.	±5 ls steps of 0.5 · P₂ opens at Δt 3, Δx 4
View	Global: the two events and the interval between them. Local metric: one small step and the length the metric gives it.	Global / Local opens in Global
Readouts	Δt and Δx, and Δs² with its type. Δs² reads “predict first” until the inquiry's first prediction is made; the Compute box shows it either way. In Local view: α and the sign of ds².	panel
Read off & compute	Type the Δt and Δx you read off the grid. Compute Δs² shows the arithmetic and the true values for comparison. Global view only.	two number boxes
Rotate displacement direction	The angle α of the small step, measured from the t-axis. Local view only.	0°–90° opens at 30°
Symbols & notation	A collapsed panel that defines Δt, Δx, Δs², dt, dx and ds².	panel
▶ Play / ⏸ Pause	Starts or freezes the photon dots. The intervals do not depend on it.	top bar
Speed	How fast the photon dots move on screen.	0.25×–4×
↻ Reset	Restarts the mode you are in from its beginning.	top bar

Fixed: c = 1, so times and lengths are both in light-seconds (ls) and Δs² is in ls². The signature is (−, +).

The interval is a difference of squares

Δt and Δx are the time gap and the space gap between the two events, measured in one inertial frame: the frame of the grid. The interval combines them with opposite signs:

The interval Δs². With c = 1 it is −Δt² + Δx².

The same pair, P₂ at Δt = 3, Δx = 4. The Euclidean distance² adds the squares and is never negative. Δs² gives the time term a minus sign and can be negative.

Signature (−, +): timelike pairs are negative, lightlike pairs zero, spacelike pairs positive.

Key idea. Δs² is not a distance squared. Δt² counts as negative and Δx² as positive, so Δs² can be negative, zero or positive.

See itRead Δt = 3 and Δx = 4 off the grid and let the Compute box do the arithmetic. Watch for: the Compute line starting with −9.0, the time term, and ending in 7.0 ls².

The light cone sorts the pairs

The dashed lines through P₁ are the light cone: the points where |Δt| = |Δx|, which at c = 1 is exactly the path of light. Comparing |Δt| with |Δx| gives the sign.

The three cases.

Timelike (inside the cone): a signal slower than light can link the events.
Lightlike (on the cone): only light can link them.
Spacelike (outside): no signal can link them.

Examples: Δs² = −12 inside the cone, 0 on it, +7 outside. The sign of Δt separates future from past.

Δs² = 0 does not mean the same event. At Δt = 3, Δx = 3 the events are 3 ls apart and Δs² is still 0.

See itKeep Δt = 3 and slide P₂ from Δx = 5 in to Δx = 0. Watch for: the Compute line changing from positive to 0.0 at Δx = 3, then to negative.

See itPut P₂ on the cone at Δt = 3, Δx = 3. Watch for: P₂ sitting on a dashed line, with the photon dots running along that line.

What the interval measures

The square root of |Δs²|, by type.

Timelike: √(−Δs²) is the proper time (c = 1), what a single clock present at both events reads. That clock moves at speed |Δx/Δt| < 1.
Spacelike: √Δs² is the proper distance, the separation measured in a frame where the two events are simultaneous. No clock or signal links them.
Lightlike: Δs² = 0, so both roots are 0. Only a light signal links the events.

Δt Δx

These sliders move the sim too.

See itPut P₂ inside the cone at Δt = 4, Δx = 2. Watch for: the Compute line giving Δs² = −12.0 ls², so a clock between the events reads √12 ≈ 3.46 s.

The metric sets the length of a step

Local metric view takes one step (dt, dx) = (cos α, sin α), so its coordinate length √(dt² + dx²) is 1 for every α. The metric turns the step into an interval:

The metric η, with signature (−, +).

As α turns from the t-axis (0°) to the x-axis (90°), ds² goes from −1 through 0 at α = 45°, the light-cone direction, to +1. The coloured bar has length √|cos 2α|, so it is shortest at 45° and returns to 1 at 90°.

Key idea. The grid does not set the length of a step. The metric does.

See itTurn the step from α = 0° to 90°. Watch for: the coloured bar shrinking to a point at α = 45° (ds → 0), then growing again.

Why every observer agrees on Δs²

A second inertial frame S′, moving at speed v along x, assigns the same two events different gaps Δt′ and Δx′ (Lorentz transformation, c = 1):

Both gaps change; the difference of their squares does not.

So every inertial observer finds the same Δs², and the same type: timelike, lightlike or spacelike cannot depend on the frame. A rotation of the diagram would preserve Δt² + Δx² (circles). A boost preserves −Δt² + Δx² (hyperbolas).

P₂ at Δx = 4, Δt = 3, boosted by v = 0.5 c. It moves along the hyperbola, not the circle.

The sim has no moving frame. The boost is done on paper, and the sim shows the two resulting pairs.

See itBoost the pair Δt = 5, Δx = 3 by v = 0.6 c (γ = 1.25) on paper, to Δt′ = 4, Δx′ = 0. Watch for: the Compute line giving 16.0 ls² for both pairs.

Where it matters

Cause and effect. Only timelike or lightlike pairs can be linked by a signal. Since Δs² is the same in every frame, all observers agree which pairs those are.
Time order. For a timelike pair, every observer agrees which event comes first. For a spacelike pair, different frames can disagree, and the pair cannot be causally linked anyway.
Muons. Creation and decay of a muon form a timelike pair. √(−Δs²) is the muon's own mean lifetime, 2.2 μs. In the Earth frame at 0.99 c (γ ≈ 7.1) the time between the events is about 16 μs.

Go deeper: the same structure for energy and momentum

The energy–momentum pair (E, p) transforms like (t, x), and −E² + p² = −m² (c = 1) is its invariant, with the same signature (−, +).
A particle with m > 0 is timelike, a photon with m = 0 is lightlike.

Summary

In one line. Δs² = −Δt² + Δx² (c = 1) is the same in every inertial frame, and its sign sorts every pair of events as timelike, lightlike or spacelike.

Key points

√(−Δs²) is a proper time (timelike), √Δs² a proper distance (spacelike).
The metric, not the grid, sets the length of a step.
A change of inertial frame changes Δt and Δx but not Δs².

Common mix-ups

"A squared quantity can't be negative." Δs² is not a square of a real length. It is a difference of two squares.
"Δs² = 0 means the same event." It means the events lie on each other's light cone, which can be several ls apart.
"The interval is the length of the thick line on screen." The line is drawn on the Euclidean grid. Δs² is computed from Δt and Δx by the metric.

Things to try on your own

Read Δt and Δx off the grid, type them into the Compute box, and compare with the true values.
In Local view, find the angle other than 30° where the bar has the same length as at 30°.
Place P₂ at Δt = 5, Δx = 3, then at Δt = 4, Δx = 0, and compare Δs².