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Chapter 3: Spacetime Diagrams

Spacetime diagrams (Minkowski diagrams) are powerful tools for visualizing special relativity. They let us see time dilation, length contraction, and the relativity of simultaneity as geometric relationships, making the abstract concrete.

Basic Setup

Axes

• Vertical axis: ct (time × c, so both axes have length units)
• Horizontal axis: x (space)
• Each point = an event in spacetime

Light Lines

• Light travels at 45° (slope = 1 since x = ct)
• All observers agree light lines are at 45°
• Light cone = future and past light rays from origin

Worldlines

• Stationary object: vertical line (moving through time only)
• Moving object: tilted line (slope < 1 since v < c)
• Must always be inside the light cone (no FTL travel)

A Moving Observer's Axes

For an observer moving at velocity v, the ct' and x' axes are tiltedtoward the light line.

Key Features

• The ct' axis is the worldline of the moving observer
• The x' axis (simultaneity surface) is tilted the same angle from horizontal
• Both axes tilt toward the light line as v → c
• The axes meet the light line at v = c (impossible)

The angle θ from the vertical satisfies tan(θ) = v/c. Lines of constant t' are tilted; events that are simultaneous in S' are NOT simultaneous in S!

Visualizing Relativistic Effects

Time Dilation

Equal proper time intervals on the moving clock correspond to longer coordinate time intervals in the rest frame. The moving clock's tick marks are spread out along the time axis.

Length Contraction

The ends of a moving rod trace out two worldlines. "Length" means the spatial separation at the same time—but "same time" differs between frames! The rod appears shorter in the frame where it moves.

Relativity of Simultaneity

Lines of constant t are horizontal; lines of constant t' are tilted. Events on a horizontal line are simultaneous in S but NOT in S'—and vice versa.

Light Cones and Causality

Future Light Cone

Upper 45° cone. Events here CAN be influenced by the origin. Timelike separated.

Past Light Cone

Lower 45° cone. Events here COULD HAVE influenced the origin. Timelike separated.

Elsewhere

Outside both cones. No causal connection possible. Spacelike separated.

Interactive Simulations

ASCII Spacetime Diagram with Multiple Worldlines

Python

script.py66 lines

import numpy as np

print("=" * 62)
print("SPACETIME DIAGRAM: Multiple Worldlines")
print("=" * 62)

W = 61   # width
H = 31   # height (time axis)
cx = W // 2

grid = [[' ' for _ in range(W)] for _ in range(H)]

# Axes
for i in range(H):
    grid[i][cx] = '|'
for j in range(W):
    grid[0][j] = '-'
grid[0][cx] = '+'

# Light cone from origin
for i in range(1, H):
    if cx + i < W: grid[i][cx + i] = '/'
    if cx - i >= 0: grid[i][cx - i] = '\\'

# Worldlines
worldlines = [
    (0.0, 'A', "at rest"),
    (0.3, 'B', "v=0.3c"),
    (0.6, 'C', "v=0.6c"),
    (-0.4, 'D', "v=-0.4c"),
]

for v, ch, _ in worldlines:
    for i in range(1, H):
        x_pos = cx + int(i * v)
        if 0 <= x_pos < W and grid[i][x_pos] == ' ':
            grid[i][x_pos] = ch

# Simultaneity lines for boosted frame (v=0.6c)
# In boosted frame, lines of constant t' have slope v/c in original diagram
v_boost = 0.6
for x_off in [-15, -5, 5, 15]:
    for dx in range(-20, 21):
        x_pos = cx + x_off + dx
        t_pos = int(abs(dx) * v_boost) + 5
        if 0 <= x_pos < W and 0 <= t_pos < H and grid[t_pos][x_pos] == ' ':
            grid[t_pos][x_pos] = '.'

print("  ct ^")
for i in range(H - 1, -1, -1):
    print("     " + ''.join(grid[i]))
print("     " + " " * cx + "+---> x")

print("\nLegend:")
print("  / \\ = light cone (45 degrees)")
print("  A = object at rest")
print("  B = v = 0.3c    C = v = 0.6c")
print("  D = v = -0.4c")
print("  . = simultaneity lines for v=0.6c frame")

print("\nSPACETIME REGIONS:")
print("  Inside light cone: TIMELIKE (causally connected)")
print("  On light cone:     LIGHTLIKE (null)")
print("  Outside light cone: SPACELIKE (causally disconnected)")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Fortran: Spacetime Diagram Event Coordinates

Fortran

program.f9065 lines

program spacetime_diagrams
  implicit none
  double precision :: t, x, tp, xp, beta, gamma_val, s2
  integer :: i, j
  double precision :: velocities(5)

velocities = (/0.0d0, 0.3d0, 0.5d0, 0.7d0, 0.9d0/)

print '(A)', repeat('=', 70)
  print '(A)', 'SPACETIME DIAGRAM: WORLDLINE COORDINATES'
  print '(A)', repeat('=', 70)

! Generate worldline points for different velocities
  print '(A)', 'Worldline points (ct, x) for particles starting at origin:'
  print '(A6, A10, A10, A12)', 'v/c', 'ct', 'x', 'Proper time'
  print '(A)', repeat('-', 40)

do i = 1, 5
    beta = velocities(i)
    if (beta < 1.0d-15) then
      gamma_val = 1.0d0
    else
      gamma_val = 1.0d0 / sqrt(1.0d0 - beta**2)
    end if

print '(/,A,F4.1,A)', '  Particle at v = ', beta, 'c:'
    do j = 0, 10
      t = dble(j)
      x = beta * t
      ! Proper time: tau = t / gamma
      print '(F6.1, F10.3, F10.3, F12.4)', &
        beta, t, x, t / gamma_val
    end do
  end do

! Light cone classification
  print '(/,A)', repeat('=', 70)
  print '(A)', 'EVENT CLASSIFICATION FROM ORIGIN'
  print '(A)', repeat('=', 70)
  print '(A10, A10, A12, A16)', 'ct', 'x', 's^2', 'Region'
  print '(A)', repeat('-', 50)

block
    double precision :: test_t(8), test_x(8)
    character(len=16) :: region
    test_t = (/5.0d0, 3.0d0, 4.0d0, 2.0d0, 1.0d0, 7.0d0, 0.0d0, 3.0d0/)
    test_x = (/2.0d0, 3.0d0, 4.0d0, 5.0d0, 0.0d0, 1.0d0, 6.0d0, 3.0d0/)

do i = 1, 8
      s2 = -test_t(i)**2 + test_x(i)**2
      if (abs(s2) < 1.0d-10) then
        region = 'LIGHTLIKE'
      else if (s2 < 0.0d0) then
        region = 'TIMELIKE'
      else
        region = 'SPACELIKE'
      end if
      print '(F10.1, F10.1, F12.2, A16)', &
        test_t(i), test_x(i), s2, region
    end do
  end block

end program spacetime_diagrams

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

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