This post contains Minkowski diagrams of flat spacetime with light cones to illustrate the causal structure, as well as graphical interpretations of Lorentz transformations (“boosts”), and more. Some figures were inspired by Very special relativity – An illustrated guide by Sander Bais. For more related figures made with TikZ, please see Relativity category. An excellent and animated explanation of Lorentz transformation is given by this YouTube video by ScienceClic English.

Table of content:

• World lines
• Causal structure of Minkowski spacetime
• Coordinate transformations
• Simultaneity
• Time dilation
• Length contraction
• Paradoxes
• Hyperbolae
• Analogy to Euclidean rotation
• Lorentz factor
• Full code

Empty spacetime diagram.

Several spacetime worldlines with different velocities.

Two observers at rest (w.r.t. to each other) communicating by reflecting a light signal.

Derivation of the relativity of simultaneity with the thought experiment of two moving observers (with the red and green worldlines) communicating with a light signal. Assuming the speed of light is c in all inertial frame of references (second postulate of special relativity), the dashed lines represent the plane of simultaneity. All events on a given dashed line happen at the same time from the point of view of the moving observers.

Future and past lightcone at a 45 degrees angle to illustrate the causal structure of Minkowski space.

Different types of spacetime vectors and separation: timelike (s² > 0), spacelike (s² = 0) and lightlike or null (s² < 0).

Overlapping light cones of two observers separated in spacetime to illustrate the causal structure.

Simple 2D rotation in the (Euclidean) xy plane.

Galilean transformation for a moving frame. Clocks in both frames are equal (the grid lines at constant time ct and ct’ coincide).

Lorentz transformation, or “boost”. Notice that the angle θ is given by the slope β = v/c = tanθ.

Inverse Lorentz transformation, or equivalently, if the boosted frame is moving to the left (in the negative x direction).

Events A and B are simultaneous in the rest frame S, but in the boosted frame S’, B happens before A.

Events A and B are simultaneous in the boosted frame S’, but in the rest frame S, A happens before B.

Relativity of simultaneity. Two observers experience events at in different order: In frame S, A happens before B, while in frame S’, B happens before A.

Time dilation in the boosted frame S’ for events at a fixed point of space in frame S. The time separation between A and B is longer in S’ than in S; Δt’ > Δt. The derivation is given in these lecture notes.

Time dilation for events at a fixed point of space in the boosted frame S’. The time separation between A and B is longer in S than in S’; Δt > Δt’.

Length contraction of a rod at rest in frame S. The rod is shorter in the boosted frame S’, because it is moving relative to this frame. The derivation is given in these lecture notes.

Length contraction of a rod moving in frame S. The rod is shorter in the frame S’, because it is moving relative to this frame.

Ladder paradox: Due to length-contraction a long ladder moving at relativistic speed is short enough to fit in a barn with both doors simultaneously closed. In the moving frame S’, however, the barn is contracted and cannot fit the ladder. The solution to the paradox is that the doors are never simultaneously closed in S’ due to the relativity of simultaneity.

From the perspective of the observer in their rest frame S’, the barn is moving, and the doors are not closed simultaneously:

Twin paradox explained with planes of simultaneity.

Hyperbolae in Minkowski spacetime to derive the Lorentz transformation in terms of hyperbolic functions sinh and cosh in analogy to rotation. The red point is boosted along the red hyperbola, such that its spacetime seperation from the origin remains constant.

Equations for Lorentz transformation in terms of hyperbolic functions:

Equations for Lorentz transformation in terms of the γ factor and β = v/c:

Lorentz transformation matrix with hyperbolic functions sinh and cosh in analogy to the rotation matrix. A boost is a hyperbolic rotation with rapidity 𝜑, where 𝛽 = tanh 𝜑.

Different hyperbolae in Minkowski spacetime. All points on the same hyperbola have an equal spacetime separation s from the origin (t,x) = (0,0). A boost (i.e. a hyperbolic rotation) moves a point along such a hyperbola, and therefore preserves the spacetime distance (i.e. spacetime distance between two points are invariant under Lorentz transformations).

Notice that spacelike separation squared are negative, s² < 0, so this distance will be imaginary, while timeline separations will be real.

Both boosts and rotations preserve the spacetime distance given by the metric ds² = cdt² – dx² – dy² – dz². They are part of the so-called Lorentz group.
The transformation matrix for boosts in terms of hyperbolic functions looks very similar to the rotation matrix.

The 2D rotation matrix in the xy plane can be expressed as follows:

Spacelike slice at t = 0. All points on the same circle have the same (Euclidean) distance r to the origin. Euclidean rotations move points along the circle, and hence preserve this spacetime distance.

Spacelike slice to illustrate different types of separations.

Plot of the Lorentz factor as a function of velocity v.

Plot of the Lorentz factor as a function of velocity β = v/c (i.e. in units of c).

Edit and compile if you like:

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% Author: Izaak Neutelings (October 2021)
% Inspiration
%   "Very special relativity - An illustrated guide", Sander Bais (2007)
%   http://people.uncw.edu/hermanr/GR/Minkowski/Minkowski.pdf
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\usetikzlibrary{calc} % for adding up coordinates
\usetikzlibrary{decorations.markings,decorations.pathmorphing}
\usetikzlibrary{angles,quotes} % for pic (angle labels)
\usetikzlibrary{arrows.meta} % for arrow size
\usepackage{xfp} % higher precision (16 digits?)
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\tikzset{>=latex} % for LaTeX arrow head
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\colorlet{mylightred}{red!85!black!12}
\colorlet{myfieldred}{mydarkred!5} % for S' background
\colorlet{myredhighlight}{myred!20} % highlights simultaneity in ladder paradox
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\colorlet{mypurplehighlight}{mydarkpurple!20} % highlights simultaneity in ladder paradox
\tikzstyle{world line}=[myblue!40,line width=0.3]
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