5.3 Constructing and interpreting spacetime diagrams

Spacetime diagrams are visual tools that combine space and time, helping us understand how events and objects move through the universe. They show events as points, worldlines as paths, and use light cones to illustrate causality and the speed limit of light.

These diagrams are crucial for grasping Lorentz transformations, which describe how different observers perceive space and time. They visually represent key concepts like simultaneity, proper time, and time dilation, making abstract ideas more concrete and relatable.

Spacetime Fundamentals

Representing Events and Worldlines in Spacetime

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• Spacetime diagrams represent events in a combined space and time coordinate system
• Events are points in spacetime represented by a specific location in space and a specific moment in time
• Worldlines are paths through spacetime that trace the history of an object's motion
• Stationary objects have vertical worldlines while moving objects have slanted worldlines (light has a 45° worldline)

Axes and Units in Spacetime Diagrams

• The time axis is typically the vertical axis in a spacetime diagram and represents the passage of time
  • Time is often measured in units of distance (light-seconds, light-years) to maintain the same scale as the space axis
• The space axis is the horizontal axis and represents the spatial dimension (often just one dimension for simplicity)
  • Space is measured in standard units of distance (meters, kilometers)

Light and Causality

Light Cones and the Speed of Light

• Light cones are regions of spacetime that represent the possible paths of light emanating from or converging to a single event
  • The future light cone contains all events that can be reached by light signals from the central event
  • The past light cone contains all events that could have sent light signals to reach the central event
• The speed of light is the ultimate speed limit in the universe and defines the boundaries of the light cones (45° lines in spacetime diagrams)

Causality and the Ordering of Events

• Causality is the principle that an effect cannot occur before its cause
• Events within an object's future light cone can be affected by the central event (causal connection)
• Events within an object's past light cone can affect the central event (causal connection)
• Events outside the light cones (spacelike separated) cannot have a causal connection with the central event
  • The order of spacelike separated events depends on the reference frame and can appear simultaneous, or one can occur before the other

Simultaneity and Proper Time

Simultaneity and Its Relativity

• Simultaneity refers to events that occur at the same time according to a particular reference frame
• In a spacetime diagram, a line of simultaneity connects events that are simultaneous in a given reference frame
• The relativity of simultaneity means that events simultaneous in one reference frame may not be simultaneous in another frame moving relative to the first
  • Different observers in relative motion have different lines of simultaneity in their spacetime diagrams

Proper Time and Time Dilation

• Proper time is the time measured by a clock that moves along a worldline between two events
  • It is the shortest possible time between the events and is always less than or equal to the time measured in any other reference frame
• Time dilation is the phenomenon where a moving clock appears to tick more slowly than a stationary clock
  • In a spacetime diagram, the proper time (wristwatch time) between two events on a worldline is always less than the coordinate time (measured by a stationary observer)
  • Gamma factor $(\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}})$ represents the amount of time dilation experienced by a moving object