## Section 2.5: Understanding Spacetime Diagrams

Animation 1 | Animation 2 | Animation 3

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One of the most useful ways to visualize moving objects in special relativity is with spacetime diagrams. Restart. Consider the animation of a woman walking **(position given in meters and time is given in seconds)**. The motion of the woman in Animation 1 is rather ordinary and is plotted on a position versus time graph as well as in a data table. Notice that the speed of the woman can be determined by the slope of the position versus time graph (in this case 1 m/s).

Now consider what is being represented in Animation 2. In this animation, time is plotted versus position. The graph is the same as Animation 1 with the axes flipped. This way to represent the motion of the woman is *almost *what physicists would call a spacetime diagram. Two things are missing: we want to treat the time on the same footing (as far as units) as the position, and we need to take into account the universal speed limit, *c*.

Animation 3 puts time on equal footing with position by multiplying time by the speed of the woman. Therefore, velocity if the woman times time is plotted versus position. This converts the unit of time into meters.

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Finally, we need to take into account the universal speed limit of the speed of light. Restart. For a true spacetime diagram, we multiply the time by the speed of light. The unit of the *y* axis now becomes the amount of time it takes for light to travel one meter or 3.33 × 10^{−9} seconds. Select **v**/*c*** **to be zero and then press the set value and play button. Notice that the woman does not move in space but moves in time. Now try a **v**/*c*** **of 0.9. What does her *trajectory* or *worldline* on the spacetime diagram look like now? As |**v**/*c*| gets bigger (approaches 1) the trajectory of the woman on the spacetime diagram approaches the line of v = *c*. This is the 45-degree line of slope 1 that appears on the graph. Now try a** v**/*c*** **of −0.9. Since nothing can travel faster than the speed of light, an object that begins at the origin is forced to have a worldline between the two lines on the graph. The only object that can have a worldline on either of those lines is light. If we let the woman move in two dimensions her motion would be constrained to move within a cone which is called the lightcone. The cone's boundaries mark the possible worldlines that light can have if it starts at the origin at *t* = 0 m.

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