Transform events between inertial frames, verify the spacetime interval, and visualize causality, simultaneity, and light cones
Setup
Examples
Mode
Two events enable interval, causality, and temporal-order analysis.
Input frame
β is always the signed velocity of S′ relative to S. When the frame changes, the current numbers are interpreted there; they are not transformed automatically.
β
Signed fraction of c; the origins coincide at x = x′ = 0, t = t′ = 0
0.6
c
-0.950.95
Units
Changing a unit re-expresses the same physical event; it never reinterprets it.
Event A — coordinates in S
light-years
years
ct = 0.4 ly
β = 0.6 · γ = 1.25
Frame SFrame S′Light cone (±45°)grid step: 0.1 ly
Spacetime diagram with β = 0.6. Events: A — S: (0.3, 0.4) ly · S′: (0.075, 0.275) ly
Results
Lorentz factor
1.25
dimensionless
Event A in S′
x′0.075light-years
t′0.275years
ct′0.275light-years
Galilean comparison
x′_G 0.06 light-years
t′_G 0.4 years
|x_L − x_G| 0.015 light-years
|t_L − t_G| 0.125 years
The Galilean transformation (x′ = x − βct, t′ = t) converges to the Lorentz one as |β| becomes small.
An event is a point in spacetime: something that occurs at a definite position and time. In one spatial dimension, it is represented by the coordinates (x, t). The sequence of events occupied by a body forms its worldline.
An inertial frame assigns coordinates using rulers and clocks at rest relative to one another, with the clocks synchronized within that frame. In the standard configuration, S′ moves at constant velocity v = βc along the +x direction of S, and the two origins coincide at t = t′ = 0.
β=cv,−1<β<1
The transformation is passive: it describes the same event with two coordinate systems. It does not represent a force, an acceleration, or a physical change of the event.
2. The Lorentz transformation
The principle of relativity states that the laws of physics have the same form in every inertial frame. The invariance of the speed of light rules out the Galilean transformation when speeds are comparable with c. For the standard configuration, the coordinates are related by:
x′=γ(x−βct),ct′=γ(ct−βx)
γ=1−β21=(1−β)(1+β)1
The Lorentz factor γ has a minimum value of 1 and grows without bound as |β| approaches 1. In the |β| ≪ 1 regime, γ ≈ 1 and the Galilean approximation is recovered. In matrix form, the forward boost is:
(ct′x′)=(γ−γβ−γβγ)(ctx)
The inverse transformation follows by replacing β with −β. Successive boosts form a hyperbolic transformation. With rapidity φ = artanh(β), β = tanh(φ) and γ = cosh(φ); rapidities therefore add even though velocities do not add in the ordinary way.
3. The interval and causal classification
For two events A and B, the spacetime interval is defined—with signature (+, −)—as:
Δs2=(cΔt)2−(Δx)2=(cΔt′)2−(Δx′)2
Its value is invariant: every inertial frame agrees on the sign of Δs², although different frames assign different spatial and temporal separations.
Timelike (Δs² > 0): there is a frame in which both events occur at the same place. The proper time is Δτ = √(Δs²)/c. If B is in the future of A, a signal traveling no faster than c can connect them.
Lightlike (Δs² = 0): this is the path of a light signal in vacuum. The proper time between the events is zero, and no inertial rest frame exists for light.
Spacelike (Δs² < 0): there is a frame in which the events are simultaneous. The proper distance is D = √(−Δs²). No causal signal traveling at or below c can connect them.
4. Light cones, simultaneity, and temporal order
The lines x = ±ct form the light cone of an event. Its interior contains timelike-related events, its boundary contains lightlike events, and its exterior corresponds to spacelike separations.
Each inertial frame defines its own family of simultaneous events. In an (x, ct) diagram, the lines ct′ = constant are parallel to the x′ axis. Their tilt expresses the relativity of simultaneity, not a clock delay or a visual propagation effect.
The causal order of timelike or lightlike events is preserved. Only two spacelike-separated events can appear in a different temporal order in different frames, precisely because neither event can influence the other.
5. Reading a Minkowski diagram
The horizontal axis represents x and the vertical axis ct. Using the same unit on both axes makes light rays appear at 45°.
The ct′ axis is the worldline of the S′ origin: x = βct. The x′ axis contains the events for which t′ = 0: ct = βx.
Lines parallel to x′ indicate constant ct′; lines parallel to ct′ indicate constant x′. Their intersections make it possible to read an event's primed coordinates.
The primed axes have a hyperbolic calibration. Euclidean lengths and angles on the screen must not be interpreted as physical distances or times.
The cone drawn from the origin classifies events relative to that origin. The relation between A and B depends on the B − A interval, which is equivalent to translating the cone so that its vertex is at A.
A segment between two events represents a physical trajectory only if both events belong to the worldline of the same object.
6. Kinematic consequences
Time dilation and length contraction arise from comparing precisely defined pairs of events. Time dilation compares, for two events on the same clock's worldline, the proper time with the coordinate-time interval of a frame in which that clock is moving. Length contraction compares the simultaneous positions of an object's endpoints in the measuring frame.
These relations do not describe a mechanical deformation caused by motion. They express how different inertial frames organize spatial and temporal separations while preserving the same interval.
7. Scope and conventions
The model describes special relativity in flat spacetime with one spatial dimension. It assumes inertial frames, constant relative velocity, coincident origins, and |β| < 1. It does not include acceleration, gravity, or extensions to three spatial dimensions.
The sign of β sets the direction of relative motion. Coordinates may be positive or negative, and primed and unprimed values always identify the same event. The formulas use ct so that time and position are expressed in compatible units.