Theory: block universe, length contraction and time dilation

The geometry drawn by the relativity simulations, with the same notation as the code in js/physics.js. Sections 1–8 apply to length-contraction and time-dilation. Section 9 covers the clocks, present from time-dilation onwards. Section 10 covers the accelerating body of accelerated-motion, which has a different physics.js. Usage, parameters and checks are in block-universe-simulations-guide.en.md; Italian version: block-universe-relativity-theory.it.md.

1. Block universe in 2+1 dimensions

In the block universe, spacetime is a geometric object given once and for all: past, present and future exist in the same way. A body does not "move", it occupies a region of spacetime, its world tube. Motion is the tilt of the tube with respect to the time axis.

Four dimensions cannot be drawn, so the simulator uses two space dimensions (x, y) and one time dimension (t, vertical). Motion is along x; y gives the bodies thickness and shows that the transverse directions do not change.

2. Coordinates and Lorentz transformations

Natural units with c = 1: the light cone has a 45° slope in the x–t plane.

• Frame of the body at rest ("laboratory"): coordinates (x, y, t).
• Frame of the body moving at speed β = v/c along x: coordinates (x′, y′, t′), written x1, t1 in the code.
• γ = 1 / √(1 − β²) (function gamma).

Transformations (functions lorentzToMoving and lorentzToLab):

x′ = γ (x − β t)        t′ = γ (t − β x)
x  = γ (x′ + β t′)      t  = γ (t′ + β x′)
y′ = y

The origin of the moving frame is the laboratory event (x₀, 0), i.e. the parameter x0 of all functions.

3. Building the world tubes

Body at rest (stationaryTube): it occupies x ∈ [xs, xs + L₀], y ∈ [−W/2, W/2] at every t. The tube is a vertical prism.

Moving body (movingTube): in its own rest frame it occupies x′ ∈ [0, L₀]. The side faces are the world lines of the two ends, x′ = 0 and x′ = L₀. The code uses no contraction formula: for each end x′ and each laboratory instant t it solves

t = γ (t′ + β x′)   ⇒   t′ = t/γ − β x′

and transforms the event (x′, t′) into the laboratory (worldlineAtLabTime). The result is a straight line of slope β in the x–t plane, so the tube is tilted by atan(β) with respect to the t axis.

4. Why the t = constant section is L₀/γ long

The "now" plane of the body at rest is t = constant. Cutting the tilted tube with this plane (sliceAtLabTime) gives the ends

x(x′=0)  = x₀ + β t
x(x′=L₀) = x₀ + β t + L₀/γ

The second follows by substituting t′ = t/γ − β L₀ into the transformation: x = x₀ + γ (L₀ + β t/γ − β² L₀) = x₀ + β t + γ L₀ (1 − β²) = x₀ + β t + L₀/γ.

The section is L₀/γ long: this is length contraction. Nothing acts on the body; the body at rest cuts the tilted tube with a plane other than the simultaneity plane of the moving body.

5. Why the t′ = constant section is L₀ long

The "now" plane of the moving body is t′ = τ. In the laboratory it is the tilted plane (movingNowPlane)

t = β (x − x₀) + τ/γ

Cutting the moving tube with this plane (movingSliceAtMovingTime), the ends are the images of (x′=0, t′=τ) and (x′=L₀, t′=τ). Their separation (Δx, Δt) = (γL₀, γβL₀) has Minkowski length

√(Δx² − Δt²) = γ L₀ √(1 − β²) = L₀

that is, the proper length: the body measures itself as L₀ long.

6. Reciprocity

The same plane t′ = τ also cuts the vertical tube of the body at rest (stationarySliceAtMovingTime). The ends are (xs, t(xs)) and (xs + L₀, t(xs + L₀)), with Δx = L₀ and Δt = β L₀, hence

√(L₀² − β² L₀²) = L₀/γ

The moving body sees the body at rest contracted by the same factor. Enable Also show the "now" of the moving body to see both sections dimensioned: L₀ (proper) on the moving tube and L₀/γ on the tube at rest.

7. Caveat: screen ruler and Minkowski metric

The drawing is Euclidean, spacetime is not. On the tilted planes a segment with components (Δx, Δt) appears on screen with length √(Δx² + Δt²), but its physical length is √(Δx² − Δt²). For segments on the t′ = constant plane the "Readings" table gives both values (row "Same segments, Euclidean length in the drawing"). On the t = constant plane the two measures coincide because Δt = 0.

For the same reason the tilted tube, measured with the screen ruler perpendicular to its faces, looks "wider" than the vertical one. This is an artefact of the Euclidean projection.

8. How to use the page

Example values with β = 0.6:

With β = 0.8: γ = 1.667 and L₀/γ = 0.600.

• Side x–t view: the Minkowski diagram. The horizontal section of the tilted tube is shorter than that of the vertical tube.
• Top x–y view: only the two sections on the t = constant plane, with the same width W and lengths L₀ and L₀/γ.
• 3D view: drag to rotate, wheel to zoom, right button to pan.
• Light cone: the world lines of the moving body stay inside the cone (slope β < 1).

The t′ plane slider goes from 0 to t_max/γ, so it depends on β: these are the values for which the rear edge of the moving body stays within the drawn window. The tubes are infinite; the drawing extends them as far as needed to contain the tilted sections.

9. Time dilation: clocks and ticks (time-dilation)

Time dilation is geometry, like contraction: contraction is the Minkowski length across the tube, dilation the Minkowski length along it. The proper time of a body between two events on its world line is

Δτ = √(Δt² − Δx²)

For the body at rest Δx = 0 and Δτ = Δt. For the moving body, between t = 0 and t = T, Δx = βT and Δτ = T √(1 − β²) = T/γ.

Clock ticks

Each body carries a clock synchronized in its own frame. On each tube the simulator draws one section per unit of proper time:

• on the tube at rest (stationaryTicks), the horizontal planes t = 0, 1, 2, 3, 4;
• on the moving tube (movingTicks), the planes t′ = 0, 1, 2…, i.e. the tilted sections of movingSliceAtMovingTime with integer τ. The rear edge (x′ = 0) of tick t′ = n falls in the laboratory at t = γn.

For the same block height the tube at rest contains t_max ticks and the moving one t_max/γ: with β = 0.8 and t_max = 4 that is 4 versus 2.4 (row "Proper time along the whole block" of the table). Only ticks entirely inside the drawn tube are drawn, so with β = 0.8 t′ = 0 and t′ = 1 are visible. The tilted tick looks longer on screen but is worth a single unit (see section 7).

Clock readings on the sections

On the plane t = t_now the body at rest shows t_now at both ends. The clocks of the moving body (movingClocksAtLabTime) show t′ = γ (t − β (x − x₀)). Substituting the positions of the ends of the section (section 4):

rear  (x′ = 0):   t′ = t_now / γ
front (x′ = L₀):  t′ = t_now / γ − β L₀

The rear shows the dilation (t′/t = 1/γ). The difference βL₀ between rear and front is the relativity of simultaneity: events simultaneous for the body at rest are not simultaneous for the moving body, so time is not the same even within a single rectangle.

On the plane t′ = τ the opposite happens (reciprocity): the moving body shows τ at both ends, and the clocks of the body at rest show the times t of the ends of the section of stationarySliceAtMovingTime,

t = β (x − x₀) + τ / γ      for x = xs and x = xs + L₀

A clock at rest at x = x₀ would show exactly τ/γ: each body sees the other's clock slowed by the same factor. Values with β = 0.8, L₀ = 1, t_now = 2 and τ = 1.2:

In the top view the ticks are hidden: all t values project onto the same plane and the sections would overlap.

10. Accelerating body: hyperbolic motion (accelerated-motion)

In accelerated-motion the moving body starts at rest at x₀, accelerates, brakes and stops at x₀ + X. The parameters are no longer the speed but the proper acceleration a of the rear and the distance X. All trip quantities come from trip({a, X, L0, x0}).

Hyperbolic motion and Born rigidity

A point with constant proper acceleration α traces in the x–t plane a hyperbola of "radius" ρ = 1/α centred on an event (x_c, 0):

x(t) = x_c + √(ρ² + t²)        β(t) = t / √(ρ² + t²)        τ(t) = ρ · asinh(t/ρ)

If every point of an extended body had the same acceleration, the body would stay L₀ long in the laboratory and so stretch in its own frame. The body is instead rigid in Born's sense: the point at proper distance ξ from the rear follows the hyperbola with the same centre and radius ρ_R + ξ, where

ρ_R = 1/a            (rear)
ρ_F = 1/a + L₀       (front, proper acceleration 1/ρ_F < a)
ρ_C = 1/a + L₀/2     (midpoint)
x_c = x₀ − 1/a

(worldlineAtLabTime, speedAtLabTime, properTimeAtLabTime). All the hyperbolas share the same centre, so every line through (x_c, 0) is a simultaneity line for all points of the body. The "now" plane of the body is the line through the centre with slope tanh η, where η is the rapidity (movingNowPlane). On this plane the section of the tube has Minkowski length exactly L₀ (movingSliceAtMovingTime), with no contraction formula imposed.

Push and braking

The trip is symmetric: braking is the push reflected through the midpoint event Q = (x₀ + L₀/2 + X/2, T/2). Requiring the midpoint of the body to arrive halfway gives the maximum rapidity, and from it the duration and maximum speed:

sinh η_max = √(ρ_C X + X²/4) / ρ_C
T          = 2 ρ_C sinh η_max           (duration in the laboratory)
β_max      = tanh η_max                 (halfway)

During braking the roles swap: the rear decelerates with 1/ρ_F and the front with a, because the reflection maps the hyperbola of radius ρ_R + ξ onto the one of radius ρ_F − ξ, centred on (2x_Q − x_c, T). The point ξ changes phase at t = (ρ_R + ξ) sinh η_max, which differs from point to point: in the laboratory the front starts braking after the rear.

Proper time and clocks

During the push the clock of point ξ shows τ = (ρ_R + ξ) η, so on the simultaneity plane of the body the front is ahead of the rear by a factor ρ_F/ρ_R. During braking the factors are reversed, and on arrival the two clocks show the same value, equal for every point of the body:

τ_tot = (ρ_R + ρ_F) η_max = 2 ρ_C η_max

The page's τ slider is the proper time of the rear. Before departure and after arrival the planes are horizontal and τ grows like t. The ticks (movingTicks) are the sections with integer τ: more and more tilted during the push, less and less during braking.

On the t = constant plane the section of the curved tube (sliceAtLabTime) is not L₀/γ, because rear and front have different speeds at the same laboratory instant. The "Readings" table shows the measured length next to the two L₀/γ values for comparison.

Example values

With a = 1, X = 3, L₀ = 1 (x₀ = −1):

The Node check is in block-universe-simulations-guide.en.md.