Graphical Representation of Time and Space in Different Frames
- You’re watching a train pass by while standing on a platform.
- You see the train moving, but a passenger inside sees themselves as stationary.
- 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
- Time Axis ($ct$): Represents time, scaled by the speed of light ($c$) to ensure both axes have the same units (meters).
- Space Axis ($x$): Represents position in space.
Plotting $ct$ instead of $t$ ensures the axes have the same units, making it easier to interpret worldlines and light cones.
Worldlines: Tracing Paths in Spacetime
Worldline
A worldline is the path an object traces in spacetime.
- Object at Rest: Its worldline is vertical, as its position doesn’t change over time.
- Object in Motion: Its worldline is slanted, with the slope indicating its velocity.
- A photon’s worldline forms a 45° angle with the axes because it travels at the speed of light.
- This angle is a universal feature of light in spacetime diagrams.
Light Cones: Defining Causality
Light cones illustrate the boundaries of causality:
- Future Light Cone: Events that can be influenced by a signal from the origin.
- Past Light Cone: Events that could have influenced the origin.
- Outside the Light Cone: Events that are causally disconnected from the origin.
- The light cone’s shape reflects the fact that nothing can travel faster than light.
- This constraint defines the limits of cause and effect in spacetime.
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
- 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.
In spacetime diagrams, time is measured in light-distance units ($ct$), allowing both space and time to be represented with the same units.
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 $x'$ 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 $ct'$ represents the time axis for a moving reference frame, suggesting an observer moving with velocity $v$ relative to the original reference frame.
- Many students assume all objects should have vertical worldlines.
- However, moving objects have slanted worldlines, with steeper slopes corresponding to slower speeds and shallower slopes corresponding to faster speeds.
