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Theme A — Space, Time and Motion

A.5.10 Space-Time Diagrams — IB Physics HL Revision Notes & Topical Questions

5 min read 8 parts Lesson 62 of 63 1 quiz

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A.5.10 Space-Time Diagrams — IB PHYSICS HL notes: Interpret and draw space-time diagrams with world lines and tilted axes for a second frame.Spacetime (Minkowski) diagrams show motion in space and time.They help to visualise time dilation, length contraction and simultaneity.Four…

  • Four dimensions cannot be drawn
  • Object at rest
  • Constant velocity
  • Light or a photon
  • Or gradient c/v
  • Measure θ with a protractor if required

Part 1 of 6Space-time diagrams

1
Spacetime (Minkowski) diagrams show motion in space and time.
2
They help to visualise time dilation, length contraction and simultaneity.
3Four dimensions cannot be drawn
  • 3D space collapses into one spatial dimension, with time as another.
4
Plot x on the horizontal axis and ct on the vertical axis.
  • Both axes have dimensions of length, so values compare easily.

Part 2 of 6World lines

World line
World line a line on a spacetime diagram showing an object's motion.
Gradient
Gradient ct/x = c/v.
  • Steeper gradient means slower motion; shallower gradient means faster.
Object at rest
  • Object at rest a vertical line, infinite gradient, moving only in time.
Constant velocity
  • Constant velocity a straight line steeper than 1.
Light or a photon
  • Light or a photon gradient 1, at 45°.
Nothing can have gradient less than 1.
  • Accelerating objects have curved world lines whose gradient is always greater than 1.
  • A line that dips below gradient 1 anywhere is impossible motion.
A world line need not start at the origin.

Part 3 of 6Velocity from a world line

1tan θ
Δx/(cΔt) = v/c, where θ is the angle between the world line and the ct axis.
2Or gradient c/v
  • Or gradient c/v e.g. ct = 10 and x = 8 give v = 0.8c.
3Measure θ with a protractor if required
  • Measure θ with a protractor if required v = c tan θ.
4
Questions involve constant velocity only.

Part 4 of 6Two frames on one diagram

Two frames on one diagram
A second frame S' moving at v has axes ct' and x' tilted towards the light line.
  • The ct' axis is the world line of the origin of S'.
  • The x' axis tilts the same angle on the other side so that light has gradient 1 in both frames.
Scales on the axes differ
  • Scales on the axes differ they are set by lines of constant space-time interval.
Event coordinates in S' are read by drawing lines parallel to the x' axis and the ct' axis.
Clocks in both frames read zero when the origins coincide.
Never label c'
  • Never label c' c is the same in all frames.

Part 5 of 6Using the diagram for relativistic effects

Simultaneity
  • Simultaneity events on a line parallel to x are simultaneous in S, but not in S'.
Time dilation
  • Time dilation two flashes at x = 0 in S are further apart in time in S'.
  • The first flash also lies on the negative x' side.
Length contraction
  • Length contraction measure the rod's ends at the same time in the observer's frame.
  • The right-hand end meets the x' axis at less than the proper length.
Exam questions may use ly for ct and x.

Part 6 of 6More from the specification

1Worldline trajectories of decaying muons
  • Worldline trajectories of decaying muons plotting muon trajectories on Minkowski spacetime diagrams illustrates time dilation and path length contraction.
2Inertial Reference Frames and Relative Motion
  • An inertial frame of reference is a coordinate system moving at constant velocity where Newton's laws hold; observers in different frames measure identical laws of physics but different relative speeds and coordinates.
    • (such as a stationary person on Earth versus a traveller inside a moving train)
3Spacetime Diagrams and Worldlines
  • On a Minkowski spacetime diagram, the vertical axis represents ct and the horizontal axis represents spatial distance x;
    • a stationary object forms a vertical worldline,
  • while an observer moving at constant speed along a given direction produces a tilted worldline with gradient c/v.
4Simultaneity and the Constancy of Light Speed
  • Because the speed of light c is invariant in all directions across all reference frames, events that occur simultaneously for an Earth-bound observer do not occur simultaneously for a moving traveller.
5Inertial reference frames
  • Inertial reference frames coordinate frames moving at constant velocity where stationary observers measure physical events without acceleration.
6Worldlines on spacetime diagrams
  • Worldlines on spacetime diagrams the vertical worldline represents an object at rest, while a tilted worldline shows motion at speed.
7Relative speed
  • Relative speed the speed of a traveller observed from Earth differs depending on relative motion between coordinate frames.
8Reference frame
  • Coordinate system used by an observer.
9Velocity addition
  • U = u' + v.
10Galilean relativity
  • Newton's laws take the same form in all inertial frames.
11Stationary
  • Not moving.
12Distance
  • Total length of path travelled, a scalar.

Key terms

TermMeaning
Space-time diagramPlot of ct against x showing events and world lines.
World linePath of an object through space-time.
Light lineWorld line of light with gradient 1.
EventA point on a space-time diagram.
ct'-x' axesTilted axes of a moving frame S'.
Minkowski diagramAnother name for a space-time diagram.

Avoid these mistakes

1
Plotting t instead of ct on the vertical axis.
2
Drawing a world line shallower than the light line (gradient below 1).
3
Confusing gradient c/v with v/c.
4
Drawing ct' and x' axes at right angles.
5
Labelling the speed of light in S' as c'.

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