By Thomas A. Moore
A common Relativity Workbook is a textbook meant to help a one-semester undergraduate direction on normal relativity. via its special workbook-based layout, it allows scholars to advance an effective mastery of either the physics and the assisting tensor calculus through guiding them to paintings throughout the implications. the math is brought progressively and in a totally actual context. every one bankruptcy, that's designed to correspond to at least one classification consultation, comprises a brief evaluate of the options with no obscuring derivations or info, by means of a sequence of containers that consultant scholars in the course of the technique of operating issues out. This active-learning strategy allows scholars to enhance a safer mastery of the cloth than extra conventional methods. greater than 350 homework difficulties help extra studying.
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5 Frame-independence of the Spacetime Interval Consider two events whose coordinate separations in an IRF S are given by ∆t, ∆x, ∆y, ∆z. 23) independent of the values of the coordinate differences or the value of b. 1 . Show this in the space below. 6 Frame-Dependence of the Time Order of Events If event A causes event B, then it follows that event A must occur before event B in all IRFs; otherwise, in some IRFs, observers would see an event preceding its cause, which is absurd. For the sake of simplicity in our argument, let us define coordinates so that both events occur on the +x axis in a certain frame S.
But this is absurd. 4b) the rails! Our original assumption must therefore be false: observers in two IRFs must in fact agree about the values of distances measured perpendicular to the line of relative motion. ) For two frames in standard orientation, then we must have y´ = y and z´ = z if the principle of relativity holds. Comparing coordinate times . Now we will show that a clock at rest in S´ that is present at two events registers their time coordinate difference to be Tt l = Tt 1 - b2 , where ∆t is the coordinate time difference between those same events as registered by a pair of synchronized clocks in frame S.
We can calculate the proper time Tx measured between two events A and B by a clock traveling along a given worldline connecting those events using a method analogous to the one we would use to calculate the path length along a given curve in ordinary space. (1) We first imagine dividing the worldline into infinitesimally small segments. (2) We then use the metric equation to calculate what an inertial clock would measure while traversing each segment. If the segment is small enough to be reasonably straight, the time dx registered by the actual clock should be essentially the same.