Minkowski Spacetime Diagrams Instructions

This is a very simple example that shows the axes for a rest and moving frame. The moving frame has a velocity of 10% c. The grid for the moving frame is also shown.

Note that objects are drawn in the order in which they appear in the script. The grid is "behind" the rest axes, which are behind the moving axes.

// Add Grids

Diagram width: 400, height: 400, origin: (10, 10), scale: 5;

Grid velocity: .1, color: #CCF;

Axes ;
Axes velocity: .1, color: #44C;
      

This diagram shows a single event and then maps the event to the rest and moving axes. The velocity in this case is 80% c.

The example takes advantage of the more advanced forms of coordinates to correctly draw the lines for the moving frame. [20, 20, .8] converts rest coordinate (20, 20) into the equivalent x or t coordinates for the moving frame. (0, [20, 20, .8], .8), for example, become (0, 6.666667, .8). The coordinate (0, 6.666667) is the inverse mapped back to the rest frame.

// Map an event to two different frames

Diagram height: 400, width: 400, origin: (10, 10), scale: 2;

// Map an event to different axes

Axes ;
Axes velocity: .8 color: #008800;

// Lines to rest axes

Line endpoints: (0, 20) (20, 20), dashed: true, color: #CCCCCC;
Line endpoints: (20, 0) (20, 20), dashed: true, color: #CCCCCC;

// Lines to moving axes

Line endpoints: (0, [20, 20, .8], .8) (20, 20), dashed: true, color: #88FF88;
Line endpoints: ([20, 20, .8], 0, .8) (20, 20), dashed: true, color: #88FF88;

// The event

Event origin: (20, 20) color: #CC0000;
      

This diagram is a geometrical representation of length contractions. It highlights that length contraction is actually a "foreshortening" of length in the time axis. The front of the length (measured at a specific moment in the rest frame) is from an earlier time than the rear.

The light blue lines represent the worldlines of each end of the length. The moving frame's t axis is the worldline for the middle of the length. The length, which is exactly 1 unit long in its rest frame, is shown at two points along its time axis.

The dashed lines and the thick green line represent the length as seen in the rest frame.

// Show Length Contraction

Diagram height: 400, width: 400, origin: (25, 25), scale: 25;

// Show the worldlines for the ends of the length

Worldline velocity: .8, origin: (-.5, 0, .8) color: #CCF;
Worldline velocity: .8, origin: (+.5, 0, .8) color: #CCF;

// Highlight the positions of the ends of the length, as seen by the
// rest frame

AngledLine angle: 90, origin: (0, .6, .8, -.3, 0) dashed: true, color: #CCC;
AngledLine angle: 90, origin: (0, .6, .8, +.3, 0) dashed: true, color: #CCC;

// Rest and moving axes

Axes ;
Axes velocity: .8, color: #008;

// Draw a line at rest time t = 1. which is where we will measure the
// length

AngledLine angle:  0, origin: (0, 1), color: #C00;

// This length we will measure is 1 unit long in its rest frame

// Draw the length so that the front of the length is at rest time t = 1

Line endpoints: (-1, .6, .8, +.3, 0) (0, .6, .8, +.3, 0) color: #840, thickness: 5;

// Draw the length so that the rear of the length is at rest time t = 1

Line endpoints: ( 0, .6, .8, -.3, 0) (1, .6, .8, -.3, 0) color: #840, thickness: 5;

// Draw the length seen in the rest frame

Line endpoints: (0, .6, .8, +.3, 0) ( 0, .6, .8, -.3, 0), color: #080, thickness: 5;
      

This is a variant of the famous twin paradox. A ship traveling at 80% c passes Earth heading for a point 4 light years distant. The Earth and ship clocks are synchronized to 0 as the ship passes. When the ship reaches its destination, it passes its clock time to another ship traveling toward Earth also at 80% c. When this ship reaches Earth, its clock says 6 years have passed; the Earth clock says 10 years have passed.

The diagram shows the axes for both ships. The blue worldline traces the outward and inward legs. Lines of simultaneity are shown for Earth (green) and the ships (red). Both the Earth and the ships see each other's clocks running slower, yet less time passes for the blue worldline.

// Twin Paradox

Diagram height: 400, width: 400, scale: 8.5, origin: (10, 10);

// Draw the base grid and axes

Grid ;
Axes ;

// Draw the axes for the two ships

Axes velocity:  .8, color: #FF00FF, positiveOnly: true;
Axes velocity: -.8, color: #FF00FF, origin: (8, 0), positiveOnly: true;

// Draw the lines of simultaneity from S' to S

Line endpoints: (0, 1, .8) (-0.8, 1, .8), dashed: true, color: #CC0000;
Line endpoints: (0, 2, .8) (-1.6, 2, .8), dashed: true, color: #CC0000;
Line endpoints: (0, 3, .8) (-2.4, 3, .8), dashed: true, color: #CC0000;

Line endpoints: (0, 3, -.8, 8, 0) (-2.4, 3, -.8, 8, 0), dashed: true, color: #CC0000;
Line endpoints: (0, 4, -.8, 8, 0) (-1.6, 4, -.8, 8, 0), dashed: true, color: #CC0000;
Line endpoints: (0, 5, -.8, 8, 0) (-0.8, 5, -.8, 8, 0), dashed: true, color: #CC0000;

// Draw the lines of simultaneity from S to S'

Line endpoints: (0, 1) ([ .8, 1, .8], [ .8, 1, .8], .8), dashed: true, color: #00AA00;
Line endpoints: (0, 2) ([1.6, 2, .8], [1.6, 2, .8], .8), dashed: true, color: #00AA00;
Line endpoints: (0, 3) ([2.4, 3, .8], [2.4, 3, .8], .8), dashed: true, color: #00AA00;
Line endpoints: (0, 4) ([3.2, 4, .8], [3.2, 4, .8], .8), dashed: true, color: #00AA00;
Line endpoints: (0, 5) ([4.0, 5, .8], [4.0, 5, .8], .8), dashed: true, color: #00AA00;

Line endpoints: (0, 6) ([-4.8, 6, -.8], [-4.8, 6, -.8], -.8, 8, 0), dashed: true, color: #00AA00;
Line endpoints: (0, 7) ([-5.6, 7, -.8], [-5.6, 7, -.8], -.8, 8, 0), dashed: true, color: #00AA00;
Line endpoints: (0, 8) ([-6.4, 8, -.8], [-6.4, 8, -.8], -.8, 8, 0), dashed: true, color: #00AA00;
Line endpoints: (0, 9) ([-7.2, 9, -.8], [-7.2, 9, -.8], -.8, 8, 0), dashed: true, color: #00AA00;

// Draw the worldline for the trip out (ship 1) and back (ship 2)

Worldline velocity:  0.8,                 from: 0, to:  5. thickness: 5;
Worldline velocity: -0.8, origin: (8, 0), from: 5, to: 10, thickness: 5;
      

This diagram shows an accelerated worldline. The units are light years (x) and years (t). The acceleration is .01 Earth gravities. The greenish axes show the Instantaneous Moving Frame (IMF) at rest time 20.

// An accelerated worldline with an Instantaneous Moving Frame (IMF)

Diagram height: 400, width: 400, origin: (10, 10), scale: 4;

// Draw the base grid and axes

Grid color: #EEE;
Axes ;

// Create an accelerated world line with an IMF at time 20

IMFAxes      g: .01  origin: (10, 0), time: 20, thickness: 1, color: #0BB;
AccWorldline g: .01, origin: (10, 0), from: 0   thickness: 3, color: #C00;