Simple Nature - Light and Matter

time into different regions according to certain measurements of relationshipsbetween points (events). Similarly, a circle in Euclideangeometry cuts the plane into two parts, an interior and an exterior,according to the measurement of the distance from the circle’s center.A circle stays the same when we rotate the plane. A light conestays the same when we change frames of reference. Let’s build upthe analogy more explicitly.Measurement in Euclidean geometryWe say that two line segments are congruent, AB ∼ = CD, if thedistance between points A and B is the same as the distancebetween C and D, as measured by a rigid ruler.Measurement in spacetimeWe define AB ∼ = CD if:1. AB and CD are both spacelike, and the two distances are equalas measured by a rigid ruler, in a frame where the two eventstouch the ruler simultaneously.2. AB and CD are both timelike, and the two time intervals areequal as measured by clocks moving inertially.3. AB and CD are both lightlike.The three parts of the relativistic version each require some justification.Case 1 has to be the way it is because space is part of spacetime.In special relativity, this space is Euclidean, so the definitionof congruence has to agree with the Euclidean definition, in the casewhere it is possible to apply the Euclidean definition. The spacelikerelation between the points is both necessary and sufficient to makethis possible. If points A and B are spacelike in relation to oneanother, then a frame of reference exists in which they are simultaneous,so we can use a ruler that is at rest in that frame to measuretheir distance. If they are lightlike or timelike, then no such frameof reference exists. For example, there is no frame of reference inwhich Charles VII’s restoration to the throne is simultaneous withJoan of Arc’s execution, so we can’t arrange for both of these eventsto touch the same ruler at the same time.The definition in case 2 is the only sensible way to proceed ifwe are to respect the symmetric treatment of time and space inrelativity. The timelike relation between the events is necessary andsufficient to make it possible for a clock to move from one to theother. It makes a difference that the clocks move inertially, becausethe twins in example 1 on p. 391 disagree on the clock time betweenthe traveling twin’s departure and return.Case 3 may seem strange, since it says that any two lightlikeintervals are congruent. But this is the only possible definition,402 Chapter 7 Relativity