Spacetime Diagrams, Light Cones, Metrics, and Invariant Intervals

Come back when you know tensor calculus and I’ll explain to you about n-dimensional forces and the warping of world-lines.” —a science fiction physicist’s reply after being asked how his time machine works85 It is helpful in discussions about the spacetime of special relativity to use what are called Minkowski spacetime diagrams.

These are plots of the spacetime coordinates of a particle; the resulting curve is called the world line of the particle. Such diagrams are four-dimensional—three space axes and one time axis—and hard to visualize, much less draw on a flat sheet of paper! The convention is to make do, whenever possible, with a simplified spacetime that has just one space axis (horizontal) and one time axis (vertical). As you’ll recall from Chap. 1, physicists often call such a simplified diagram a toy spacetime.

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So, for a particle at rest in some observer’s frame of reference, its spacetime diagram for that observer is a vertical world line. If the particle is not at rest then its world line will tilt away from the vertical; the greater the speed the greater the deviation from the vertical. Accelerated particles will have world-lines that curve away from the vertical.

Straight, uncurved world lines represent unaccelerated particles, that is, particles experiencing no forces and so in free fall. Such a world line is called a geodesic. In Fig. 3.7 the world lines for these various cases are shown on the same axes.

Spacetime diagrams were embraced decades ago by philosophers looking for ‘scientific’ ways to support their position on time travel (whatever it might be), as opposed to the mere verbiage of traditional colleagues. A famous example of this is a 1962 paper by the Harvard philosophy professor Hillary Putnam (note 74 in Chap. 1).


There we are asked to imagine the spacetime diagram of one Oscar Smith who, in Fig. 3.8, is at spatial location A next to his time machine. At time t0 Oscar has not yet gotten into his time machine. A little later, at time t1, we suddenly see not only Oscar at A but also two more Oscars who have appeared (apparently out of thin air) moving away from spatial location B! Between t1 and t2 we see the original Oscar at A and the two mysterious Oscars at B (for a total of three Oscars, labeled in the figure as Oscar1, Oscar2, and Oscar3) move forward in time—but one of the new Oscars ( Oscar2) lives a decidedly odd existence in that his life seems to be running in reverse

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