Special Relativity · Kinematics §2 & §4

Simultaneity is relative;
moving bodies contract.

How does relative motion affect the synchronization of clocks, the coordinate measurement of moving rods, and the shape of moving spheres?

Read §2 (On the Relativity of Lengths and Times) of Einstein’s 1905 paper →

SR-03 · An executable laboratory

Rod measurement, simultaneity, and causal order laboratory

Rod measurement and simultaneity, host calculation

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In §2 of Zur Elektrodynamik bewegter Körper (1905), Einstein demonstrates that two events simultaneous from the perspective of one reference frame are not simultaneous when viewed from another in relative motion. In §4, he derives the physical meaning of moving rigid bodies, proving that measuring the length of a moving rod requires taking coordinate positions of its endpoints at one single instant of the measuring frame, contracting its measured length to L = L₀/γ, while a sphere of radius R is measured as an ellipsoid with axes (R/γ, R, R).

Discovery Mode: Predict Before Calculating

At v = 0.6c, two events that mark the ends of a moving 10-ls rod are measured simultaneously in platform frame K (dt = 0, dx = 8 ls). What is their time separation dt' in the rod's rest frame k?

Front end earlier (dt' < 0)

The front end measurement occurs earlier than the rear end measurement in the rod frame.Relativity of simultaneity: dt' = -gamma * v * dx / c^2 = -6.0 s.

Simultaneous in both frames (dt' = 0)

Events simultaneous in one frame must be simultaneous in all frames.Classical absolute time assumption.

Front end later (dt' > 0)

The front end measurement occurs later than the rear end measurement in k.Wrong sign for Lorentz coordinate boost.

Presets:

Spatial Rod Strip Projection

Rest: frame k | Measuring: frame K | v = 0.60c

Coordinate measurement: positions of both ends taken at one single time of the measuring frame. (Coordinate geometry, not what an optical camera sees).

Interval: Δx=10.00 ls, cΔt=0.00 s [Simultaneous: L = 10.00 ls]

Spacetime Event Diagram

Pair: platform-simultaneous | Rest: k | Measuring: K | γ = 1.25 | L₀ = 10.0 ls

Spacetime coordinates: light lines at 45°. Boosted axes $x'$ and $ct'$ tilt toward the light cone by angle $\theta = \arctan(v/c)$.

Moving Sphere Measured as an Ellipsoid (§4)

v = 0.60c | Axes: (0.80, 1.00, 1.00) ls

A sphere of radius R at rest in k, when measured from K at one instant of K, has axes R/γ, R, R = R√(1 - v²/c²), R, R.

Spacetime Event Coordinates & Invariant Interval

Frame	Δt (s)	Δx (ls)	Simultaneity	s² = Δx² - c²Δt² (ls²)	Causal Order
K (Platform)	0.000	10.000	Simultaneous	100.000	Spacelike
k (Moving)	-7.500	12.500	Ordered (-)	100.000	Spacelike

Accepted Laboratory Telemetry Snapshot

Quantity ID	Status	Value / Result	Unit	Owner ID
spatialSeparationK	value	10.0000	ls	sr03-rod-simultaneity-v1
temporalSeparationK	value	0.0000	s	sr03-rod-simultaneity-v1
spatialSeparationKPrime	value	12.5000	ls	sr03-rod-simultaneity-v1
temporalSeparationKPrime	value	-7.5000	s	sr03-rod-simultaneity-v1
simultaneityK	value	0.0000	1	sr03-rod-simultaneity-v1
simultaneityKPrime	value	-1.0000	1	sr03-rod-simultaneity-v1
measuredLength	value	10.0000	ls	sr03-rod-simultaneity-v1
spacetimeIntervalSquared	value	100.0000	ls^2	sr03-rod-simultaneity-v1
causalOrder	value	1.0000	1	sr03-rod-simultaneity-v1
gammaFactor	value	1.2500	1	sr03-rod-simultaneity-v1
ellipsoidAxisLongitudinal	value	0.8000	ls	sr03-rod-simultaneity-v1
ellipsoidAxisTransverseY	value	1.0000	ls	sr03-rod-simultaneity-v1
ellipsoidAxisTransverseZ	value	1.0000	ls	sr03-rod-simultaneity-v1

Limits of this Kinematic Reference Model (Not Modeled)

This reference owner implements exact special-relativistic coordinate transformations, coordinate length measurements, and invariant spacetime intervals between inertial reference frames. The following physical regimes require general relativity, dynamical stress mechanics, or optical ray tracing and are explicitly not modeled:

Optical camera image appearance (Terrell-Penrose rotation and light-travel-time distortion), which differs from coordinate measurement at a single instant.
Accelerating reference frames, Rindler horizons, or Thomas precession.
Internal stress, elasticity, Born rigidity breakdown, or relativistic wave propagation during rod acceleration.
Gravitational time dilation or spacetime curvature (General Relativity).
Quantum uncertainty or field fluctuations at Planck-scale event intervals.
Superluminal observers (|v| ≥ c) or tachyonic coordinate frames.

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The Physical Context

§2: The Relativity of Simultaneity

In §2 of Zur Elektrodynamik bewegter Körper, Einstein investigates a rigid rod of length

r_{AB}

moving with velocity

v

relative to a stationary frame

K

. Clocks mounted at the two ends

A

and

B

are synchronized by light signals emitted from

A

at time

t_A

, reflected at

B

at time

t_B

, and returning to

A

t'_A

From the perspective of stationary observers in

K

, light travels forward with speed

c - v

relative to the rod, and backward with speed

c + v

t_B - t_A = \frac{r_{AB}}{c - v}, \qquad t'_A - t_B = \frac{r_{AB}}{c + v}

Because

t_B - t_A \neq t'_A - t_B

, observers in

K

judge the moving clocks to be desynchronized:

“We see that we cannot attach any absolute meaning to the concept of simultaneity, but that two events which, viewed from a system of coordinates, are simultaneous, can no longer be viewed as simultaneous events when viewed from a system which is in motion relative to that system.”

§4: Physical Meaning of Moving Rods and Spheres

In §4, Einstein uses the Lorentz transformation to determine the coordinate dimensions of moving bodies measured simultaneously in the observer’s frame:

x' = \gamma (x - v t) = \frac{x - v t}{\sqrt{1 - v^2/c^2}}

Taking positions of both ends at one time of the stationary frame (

\Delta t = 0

) yields a measured coordinate length:

\Delta x = \Delta x' \sqrt{1 - \frac{v^2}{c^2}} = \frac{L_0}{\gamma}

Similarly, a rigid sphere of radius

R

at rest in

k

whose surface satisfies

(\xi - \xi_0)^2 + \eta^2 + \zeta^2 = R^2

, when measured at

t = 0

from the stationary system, is an ellipsoid of revolution with semi-axes:

a = R \sqrt{1 - \frac{v^2}{c^2}}, \qquad b = R, \qquad c = R

Coordinate Measurement Versus Visual Appearance

Einstein’s length contraction describes coordinate measurement (positions recorded simultaneously in the measuring frame by a network of synchronized clocks). It is not what a human eye or camera sees. An optical image collects light rays that arrive at the shutter at the same instant, which left different parts of the moving object at different past times (Terrell-Penrose effect, 1959), causing a moving sphere to appear visually rotated rather than flattened.

Invariant Spacetime Intervals and Causal Order

Between any two events

E_1

and

E_2

, the squared spacetime interval is strictly invariant under all Lorentz transformations:

s^2 = \Delta x^2 + \Delta y^2 + \Delta z^2 - c^2 \Delta t^2 = \Delta x'^2 + \Delta y'^2 + \Delta z'^2 - c^2 \Delta t'^2

When

s^2 < 0

(timelike) or

s^2 = 0

(lightlike), a subluminal or light signal can causally connect the events, and their chronological order is invariant across all inertial frames. When

s^2 > 0

(spacelike), no signal can connect them, and observers in different states of relative motion disagree on which event occurred first.

Read §2 of the Special Relativity Paper →Read §4 (Physical Meaning of Moving Bodies) →