A.5.4 Space-time diagrams (HL only) Notes

Graphical Representation of Time and Space in Different Frames

1. 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).

Axes and Scale Interpretation

1. The horizontal axis ($x$) measures the position of events in space, marked in kilometers.
2. 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.

The Worldlines

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

Events on the Diagram

1. Event Z is positioned at some spatial distance from the origin but occurs at an earlier time.
2. Event Y occurs later in time and at a farther position along the $x$-axis.
3. The intersection of $x'$ and $ct'$ represents the transformation of space and time coordinates from the original frame to the moving reference frame.

Implications

1. The slanted axes ($ct'$ and $x'$) indicate that the moving observer perceives space and time differently than the stationary observer.
2. 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:

1. Vertical Worldline ($\theta = 0^\circ$):
   1. When the particle is at rest, its velocity $v = 0$, and $\tan \theta = 0$.
   2. The worldline is vertical, indicating no spatial displacement over time.
2. Slanted Worldline ($0 < \theta < 45^\circ$):
   1. For a moving particle, the worldline slopes away from the time axis.
   2. The slope increases with velocity, reflecting greater spatial displacement per unit time.
   3. The steeper the worldline, the slower the particle.
3. Worldline at 45° ($\theta = 45^\circ$):
   1. A particle moving at the speed of light ($v = c$) has a worldline making a $45^\circ$ angle with the time axis.
   2. 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:

1. As $v$ approaches $c$, $\tan \theta \to 1$, and the worldline approaches $45^\circ$.
2. 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?
   1. Time dilation: From the Earth’s frame, the muons’ lifetime is extended due to their high speed.
   2. Length contraction: From the muons’ frame, the distance to the Earth is contracted, allowing them to reach the surface before decaying.