A.5.4 Space-time diagrams (HL only)

Graphical Representation of Time and Space in Different Frames

1. Imagine you’re watching a train pass by while standing on a platform.
2. You see the train moving, but a passenger inside sees themselves as stationary.
3. How do we reconcile these different perspectives?

This is where spacetime diagrams come into play.

Spacetime Diagrams: Visualizing Events

A spacetime diagram is a graphical tool that helps us understand how events are perceived in different reference frames.

The Axes

1. Time Axis ($ct$): Represents time, scaled by the speed of light ($c$) to ensure both axes have the same units (meters).
2. Space Axis ($x$): Represents position in space.

Worldlines: Tracing Paths in Spacetime

1. Object at Rest: Its worldline is vertical, as its position doesn’t change over time.
2. Object in Motion: Its worldline is slanted, with the slope indicating its velocity.

Light Cones: Defining Causality

Light cones illustrate the boundaries of causality:

1. Future Light Cone: Events that can be influenced by a signal from the origin.
2. Past Light Cone: Events that could have influenced the origin.
3. Outside the Light Cone: Events that are causally disconnected from the origin.

Space-Time Diagram Example: Explanation

This spacetime diagram represents the motion of objects in different reference frames, with the horizontal axis ($x$) representing position in kilometers and the vertical axis ($ct$) representing time in kilometers of light-travel distance (since $ct$ is used to maintain consistent units between space and time).

1. Axes and Scale Interpretation

• The horizontal axis ($x$) measures the position of events in space, marked in kilometers.
• The vertical axis ($ct$) represents the time coordinate, scaled by the speed of light ($c$), so each unit corresponds to the distance light travels in a given time interval.

2. The Worldlines

• The vertical worldline labeled X represents an object at rest.
  • Since it does not move in space, its position remains fixed at $x=0$, and time progresses upwards.
• The diagonal worldline labeled $x^{\prime }$ represents a moving object, with the slope of the line related to its velocity, following the relationship $\tan \theta =v/c$.
  • Since it is slanted, it indicates that the object is moving through space as time progresses.
• The steeper diagonal worldline labeled $c{t}^{\prime }$ represents the time axis for a moving reference frame, suggesting an observer moving with velocity $v$ relative to the original reference frame.

3. Events on the Diagram

• Event Z is positioned at some spatial distance from the origin but occurs at an earlier time.
• Event Y occurs later in time and at a farther position along the $x$-axis.
• The intersection of $x^{\prime }$ and $c{t}^{\prime }$ represents the transformation of space and time coordinates from the original frame to the moving reference frame.

4. Implications

• The slanted axes ($c{t}^{\prime }$ and $x^{\prime }$) indicate that the moving observer perceives space and time differently than the stationary observer.
• This visualization helps to analyze relativistic effects such as time dilation and length contraction, as well as the relativity of simultaneity.

Relating the Angle of the Worldline to a Particle’s Speed

1. In a spacetime diagram, the angle between the worldline of a moving particle and the time axis provides a visual representation of the particle’s velocity.
2. This angle, denoted as $\theta$, is related to the particle’s speed, $v$, and the speed of light, $c$, by the relationship:

$$\tan \theta =\frac{v}{c}$$

Understanding the Relationship:

• Vertical Worldline ($\theta =0^{\circ }$):
  • When the particle is at rest, its velocity $v=0$, and $\tan \theta =0$.
  • The worldline is vertical, indicating no spatial displacement over time.
• Slanted Worldline ($0<\theta <{45}^{\circ }$):
  • For a moving particle, the worldline slopes away from the time axis.
  • The slope increases with velocity, reflecting greater spatial displacement per unit time.
  • The steeper the worldline, the slower the particle.
• Worldline at 45° ($\theta ={45}^{\circ }$):
  • A particle moving at the speed of light ($v=c$) has a worldline making a ${45}^{\circ }$ angle with the time axis.
  • This represents the maximum possible speed in relativity, as no object with mass can exceed this limit.

Significance of $\tan \theta =\frac{v}{c}$:

This relationship connects the particle’s motion to its representation on a spacetime diagram:

• As $v$ approaches $c$, $\tan \theta \to 1$, and the worldline approaches ${45}^{\circ }$.
• This visualization helps understand relativistic motion intuitively by linking geometric properties to physical quantities.

Muon Decay: Evidence for Time Dilation and Length Contraction

1. Muons are unstable particles with a short lifetime of about $2.2\times {10}^{-6}$ s in their rest frame.
2. However, muons created in the upper atmosphere reach the Earth’s surface, even though they should decay long before doing so.
3. How is this possible?
   • Time Dilation: From the Earth’s frame, the muons’ lifetime is extended due to their high speed.
   • Length Contraction: From the muons’ frame, the distance to the Earth is contracted, allowing them to reach the surface before decaying.

Reflection and Self-Assessment