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Tensors and Relativity: Chapter 1

# Spacetime diagrams and the Lorentz transformations

Probably the easiest way to understand what the physical consequences of the above postulates are, is through the use of simple spacetime diagrams . In Figure 1.5 we illustrate some of the basic concepts using a two dimensional slice of spacetime. Figure 1.5: A simple spacetime diagram

A single point, of fixed x and t is called an event  . A particle or observer moving through spacetime maps out a curve x=x(ct), and so represents the position of the particle at different times. This curve is called the particles world- line  . The gradient of the world- line is related to the particle's velocity, so light rays [ v=c ] move on lines on this diagram.

Suppose an observer uses coordinates ct and x as in Figure 1.5, and that another observer, with coordinates and , is moving with velocity v in the x direction relative to . It is clear from the above discussion that the axis corresponds to the world- line of in the spacetime diagram of [ see Figure 1.6 ]. We will now use Einstein's postulates to determine where the axis goes in this diagram. Figure 1.6: The axis of a frame whose velocity is v relative to .

Consider the following three events in the spacetime diagram of shown in Figure 1.7 [ A, B and C ] defined as follows. A light beam is emitted from the point A in  [ event A ]. It is then reflected at [ event B ]. Finally it is received at [ event C ]. How do these three events look in the spacetime diagram of ? Figure 1.7: Light reflection in We already know where the axis lies [ see Figure 1.6 ]. Since this line defines , we can locate events A and C [ at and ]. The second of Einstein's postulate states that light travels with speed c in all frames. We can therefore draw the same light beam as before, emitted from A and traveling on a line in the spacetime diagram of . The reflected beam must arrive at C, so it is a line with negative gradient which passes through C. The intersection of these two lines defines the event of reflection B in . It follows therefore, that the axis is the line which passes through this point and the origin [ see Figure 1.8 ]. Figure 1.8: Light reflection in as measured by .

One of the most startling results which follows from this geometrical construction is that events simultaneous to are not simultaneous to !

Let us now derive the Lorentz transformations  using the geometrical arguments above and the principle of Special Relativity discussed in the last section. Assuming that we orient our axes so that moves with speed v along the positive x axis relative to , the most general linear transformations we can write down are where , , and depend only on the velocity v. Looking at Figure 1.8, we see that the and axes have the following equations: Together with (10), these straight line equations imply which simplify the first two transformation equations giving For the speed of light to be the same in both and we require that Dividing top and bottom by t gives We are therefore left with the following transformation law for and : The principle of Special Relativity implies that if , , and , . This gives the inverse transformations Substituting for and from (16) in (17) gives, after some straightforward algebra, We must choose the positive sign so that when v=0 we get an identity rather than an inversion of the coordinates. The complete Lorentz transformations are therefore, 