A spacetime diagram showing the location of three events.

Understanding Spacetime Diagrams

One of the most useful ways to analyze objects moving in one dimension in special relativity is with spacetime diagrams.  In a spacetime diagram, time is plotted on the vertical axis and position is plotted on the horizontal axis.  A stationary particle produces a vertical trajectory in this diagram as time advances.  An object moving with constant velocity produces a straight line trajectory with slope equal to 1/velocity.  The unit of time on a spacetime diagram is chosen to be the time it takes for light to travel one unit of length.  This choice of units puts time and space are on an equal footing.  For example, if the unit of time is one year, then the unit of distance is one light year 9,460,730,472,580.8 km. If the unit of length is one foot, then the unit of time is approximately one nanosecond.

 

A spacetime diagram is an important visualization tool in special relativity because the speed of light is a universal constant.  Consequently, light is always represented by a line with unit slope. The two red lines in the simulation represent light traveling to the right and left along the stick after the explosion at t=0.  In special relativity, the slope of these lines is always plus or minus one because light travels one unit of distance in one unit of time in all reference frames.  The yellow line in the spacetime diagram shown should not be confused with a spatial view of the stick.  The vertical axis is time (not space) and the thickness of the line was chosen for visibility.  The length of the line does, however, represent the length of the stick in the chosen reference frame.  Points on a spacetime diagram represent events and are drawn as small circles for visibility.

 

In order to compare the predictions of special relative and Galilean relativity, the Simultaneity Spacetime Diagram model also shows a pseudo-spacetime diagram in which speed of light obeys the principles of Galilean invarience.  The home reference frame is assumed to be a special reference frame such that the speed of light is the same to the right and left.  According to Galileo and Newton, an observer traveling with a right-moving meter stick will measure a smaller speed for light traveling to the right and a larger speed for light traveling to the left.  Countless experiments have shown that this classical physics concept of an absolute space and an absolute time is incorrect.

Exercise

When we are dealing with moving reference frames, we must modify our idea of simultaneity to include the idea that events that are simultaneous in one reference frame are not simultaneous in another reference frame.  This is perhaps one of the most important things to keep in mind when considering the apparent paradoxes that arise in special relativity.  Almost all of these apparent paradoxes can be understood by remembering that events simultaneous in one reference frame are not simultaneous in another reference frame.

 

Set the stick velocity to +0.2 and drag the light source in the home frame to a location such that the right- and left-traveling light reach the detector at the same time.  Use geometric (spacetime) arguments to predict if the right or left detector will be the first to detect the explosion.  Does your argument depend on the length of the stick?  Check your prediction in the other reference frame.

Questions

Run the model with different stick velocities and compare the Home and Other spacetime diagrams.  What changes and what remains constant in each of these spacetimes when the velocity is changed?  You can right-click within a diagram to make a snapshot if you wish to compare spacetime diagrams with different initial conditions.

 

Repeat the spacetime comparison if Galilean invariance is assumed.  What changes and what remains the same?