This interactive tutorial visualizes the Lorentz Transformation in a Minkowski spacetime diagram. The goal is to show how space and time "tilt" and "stretch" as velocity v approaches the speed of light c. The simulation uses an interactive coordinate system where the vertical axis is time (t) and the horizontal axis is space (x).

The simulation provides: (1) Stationary Frame (S) — a static blue grid with axes t and x; (2) Moving Frame (S') — an orange grid that deforms in real time as you change the velocity slider (hyperbolic skew); (3) Light Cone — dashed 45° lines (x = ±ct) that remain invariant in both frames; (4) Velocity slider — set β = v/c from −0.95 to 0.95; (5) Lorentz factor (γ) — displayed in real time; (6) Pulse — sends a photon along the light cone; (7) Split view — side‑by‑side Time Dilation (stopwatches with t vs τ) and Length Contraction (reference bar L₀ on the x‑axis, ghost bar, and contracted orange rod); (8) Sync Clocks & Reset — resets time to zero and aligns both stopwatch hands to 12 o'clock.

Key insight: The transformation is not a simple rotation but a hyperbolic rotation (skew). As v increases, the S' axes close in on the light cone like a pair of scissors; nothing can exceed c. Length contraction and time dilation are visible in the spacing of the S' grid lines along the x- and t-axes.

The Lorentz coordinate system as the moving observer's POV: In physics we call S' the Primed Frame. It represents the point of view (POV) of an observer moving relative to you. Three ideas make this concrete:

• Your map vs. their map: If you are at rest, your axes (S) form a "square" grid. Someone flying past you uses a different "map" — the skewed orange grid (S'). The same event has coordinates (t, x) in your frame and (t′, x′) in theirs.
• A different "now": The moving observer's t′-axis (where they think t′ = 0) is tilted. So their definition of simultaneous is not yours: events on a horizontal line in S lie on a tilted line in S′, and vice versa. This is relativity of simultaneity.
• A unified fabric: The transformation is a linear matrix operation on the vector (t, x). That shows space and time are not separate; they are components of one spacetime vector. When an observer moves, they effectively "rotate" their perspective into the time direction.

NOTE: Units are chosen so that c = 1 (e.g. time in seconds, space in light-seconds). The canvas vertical axis is time (up = future); screen Y is inverted so that "up" in the diagram is drawn upward.

Mathematical Foundation

1. Lorentz transformation

For a frame S' moving with velocity v along the x-axis of frame S, the coordinates transform as:

t' = γ (t − vx/c²),   x' = γ (x − vt)

where the Lorentz factor is γ = 1/√(1 − v²/c²). With c = 1: γ = 1/√(1 − v²).

2. Invariant light cone

The lines x = ct and x = −ct (45° in the diagram) are the worldlines of light rays. They have the same form in all inertial frames, so the speed of light c is invariant.

3. Grid lines in S'

In the diagram (with c = 1): lines of constant x' are x = vt + (x'/γ); lines of constant t' are t = vx + (t'/γ). So the S' grid appears as a skewed set of lines that "shear" toward the light cone as |v| increases.

 

Usage Example

Follow these steps to explore the Lorentz Transformation:

1. Initial view: You see a Minkowski diagram with a blue grid (S) (stationary frame), an orange grid (S') (moving frame), and dashed 45° light cone lines. Vertical axis = time (t), horizontal = space (x).
2. Velocity slider: Move Velocity (β = v/c) from 0 toward ±0.95. Watch the orange S' grid skew: the t' and x' axes tilt toward the light cone. This is the hyperbolic shear; as |v| → c, the axes "scissor" onto the 45° lines.
3. Lorentz factor (γ): Read the displayed γ. It grows as |v| increases (e.g. v = 0.6 → γ ≈ 1.25; v = 0.95 → γ ≈ 3.2). Length contraction and time dilation are reflected in the S' grid spacing.
4. Pulse (photon): Click Pulse (photon). Two dots move outward from the origin along the two 45° lines, showing that light travels at c in both directions. They stay on the light cone regardless of the velocity slider, illustrating that c is invariant.
5. Play (time animation): Click Play to run coordinate time. A blue clock (stationary) and an orange clock (moving) advance; filled arcs and hands show elapsed time. A horizontal dashed line is "now" in S. Use Step Fwd / Step Bwd to scrub time. Sync Clocks & Reset sets t = 0 and both stopwatch hands to 12 o'clock.
6. Show Length Contraction (single view): With Play on, check Show Length Contraction to display a moving rod of proper length L₀ = 2. In S it appears contracted (L = L₀/γ). Vertical dashed lines to the x-axis show the measurement. Rod color shifts with velocity (Doppler).
7. Split view (Time | Space): Select Split (Time | Space) for two panels. Left: Time Dilation — blue and orange stopwatches with filled arcs; blue shows coordinate time t, orange shows proper time τ (τ < t when v > 0). Right: Length Contraction — a blue reference bar L₀ (Rest) on the x-axis, a faint ghost bar (full L₀ moving with the rod), and an orange contracted rod L = L₀/γ with vertical projections. Play and velocity affect both panels.
8. Labels: t, x mark the S axes; t', x' are drawn along the S' axes and rotate with the skew.
9. Twin Paradox: Check Twin Paradox to show the Stay-at-Home twin (blue vertical worldline) and the Traveling twin (pink kinked worldline). Ages (proper time) are shown: Earth twin = coordinate time; Traveler = Earth time / γ. Drag the pink kink to change the turnaround point; velocity and total trip time update so the traveler always returns to Earth. The bent worldline is shorter in spacetime than the straight one.
10. Simplify View (Tutorial): Check Simplify View to reduce clutter: the grid is dimmed and only the axes stand out. Move the mouse over the diagram to see (1) Simultaneity lines — a blue horizontal "now" in S and an orange tilted "now" in S′ through the cursor; (2) an Event dot with blue dashed projections to the S axes (x, t) and orange projections to the S′ axes (x′, t′); (3) a floating HUD with S and S′ coordinates, ratio u = x/t and u′ = x′/t′, and (in SI mode) Simultaneity shift in nanoseconds. This shows how "now" and length depend on the frame and illustrates length contraction.
11. Matrix Λ and units: Below the canvas, the Lorentz matrix Λ is shown for [t′; x′] = Λ [t; x]. Use Natural (c=1) to see symmetric-like coefficients (γ, −vγ). Use SI (m, s) to see real units: the top-right term (−vγ/c²) becomes very small (scientific notation), showing why simultaneity effects are tiny at everyday speeds. In SI mode with Simplify View, the HUD shows Simultaneity shift: [X] ns — the time offset in the moving frame due to position x.

Tip: Set v ≈ 0.6 first to see a clear skew, then try v ≈ 0.9 to see the S' grid squeeze toward the light cone. Use Pulse to confirm that photons always lie on the 45° lines.

Parameters

Short descriptions of each parameter:

• Velocity (β = v/c): Relative speed of frame S' with respect to S, in units of c. Range: −0.95 to 0.95. Positive = S' moves in the +x direction. Controls the skew of the S' grid and the value of γ.
• Lorentz factor (γ): γ = 1/√(1 − v²/c²). Displayed in real time. Determines time dilation (τ = t/γ) and length contraction (L = L₀/γ). γ = 1 when v = 0; γ → ∞ as |v| → c.
• Reference length (L₀): Proper length of the rod = 2 units (c = 1). In split view, the blue reference bar on the x-axis shows L₀; the orange rod shows contracted length L = L₀/γ.

Controls and Visualizations

• Velocity (β): Slider adjusts v/c. Value and γ update live; S' grid deforms.
• Pulse: Emits a photon along both light-cone directions (invariant c).
• Play / Stop: Runs or pauses coordinate time; stopwatches and rod advance in sync.
• Step Fwd / Step Bwd: Advance or rewind time by one step (pauses Play).
• Sync Clocks & Reset: Sets t = 0 and pulse off; stopwatch hands return to 12 o'clock.
• View: Single / Split: Single = one diagram with optional overlays. Split = left panel Time Dilation (stopwatches t vs τ), right panel Length Contraction (L₀ reference bar, ghost bar, orange contracted rod).
• Show Length Contraction / Show Time Dilation: Toggle moving rod and time-dilation labels + invariant box in single view.
• Twin Paradox / Simplify (Tutorial): Overlays for worldlines and interactive HUD.
• Canvas: Minkowski diagram. Blue = S, orange = S'. Dashed 45° = light cone. Legend (single view) explains S, S', light cone, twin.