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Tensors and Relativity: Chapter 1

Spacetime diagrams and the Lorentz transformations

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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.

  figure290
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,

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so light rays [ v=c ] move on tex2html_wrap_inline1210 lines on this diagram.

Suppose an observer tex2html_wrap_inline1118 uses coordinates ct and x as in Figure 1.5, and that another observer, with coordinates tex2html_wrap_inline1218 and tex2html_wrap_inline1220 , is moving with velocity v in the x direction relative to tex2html_wrap_inline1118 . It is clear from the above discussion that the tex2html_wrap_inline1218 axis corresponds to the world- line of tex2html_wrap_inline1124 in the spacetime diagram of tex2html_wrap_inline1118 [ see Figure 1.6 ]. We will now use Einstein's postulates to determine where the tex2html_wrap_inline1220 axis goes in this diagram.

  figure307
Figure 1.6: The tex2html_wrap_inline1218 axis of a frame whose velocity is v relative to tex2html_wrap_inline1118 .

Consider the following three events in the spacetime diagram of tex2html_wrap_inline1124 shown in Figure 1.7 [ A, B and C ] defined as follows. A light beam is emitted from the point A in tex2html_wrap_inline1124 tex2html_wrap_inline1254 [ event A ]. It is then reflected at tex2html_wrap_inline1258 [ event B ]. Finally it is received at tex2html_wrap_inline1262 [ event C ]. How do these three events look in the spacetime diagram of tex2html_wrap_inline1118 ?

  figure319
Figure 1.7: Light reflection in tex2html_wrap_inline1124

We already know where the tex2html_wrap_inline1218 axis lies [ see Figure 1.6 ]. Since this line defines tex2html_wrap_inline1272 , we can locate events A and C [ at tex2html_wrap_inline1278 and tex2html_wrap_inline1280 ]. 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 tex2html_wrap_inline1210 line in the spacetime diagram of tex2html_wrap_inline1118 . The reflected beam must arrive at C, so it is a tex2html_wrap_inline1210 line with negative gradient which passes through C. The intersection of these two lines defines the event of reflection B in tex2html_wrap_inline1118 . It follows therefore, that the tex2html_wrap_inline1220 axis is the line which passes through this point and the origin [ see Figure 1.8 ].

  figure328
Figure 1.8: Light reflection in tex2html_wrap_inline1124 as measured by tex2html_wrap_inline1118 .

One of the most startling results which follows from this geometrical construction is that events simultaneous to tex2html_wrap_inline1124 are not simultaneous to tex2html_wrap_inline1118 !

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 tex2html_wrap_inline1124 moves with speed v along the positive x axis relative to tex2html_wrap_inline1118 , the most general linear transformations we can write down are

  eqnarray787

where tex2html_wrap_inline1318 , tex2html_wrap_inline1320 , tex2html_wrap_inline1322 and tex2html_wrap_inline1324 depend only on the velocity v. Looking at Figure 1.8, we see that the tex2html_wrap_inline1218 and tex2html_wrap_inline1220 axes have the following equations:

eqnarray789

Together with (10), these straight line equations imply

equation791

which simplify the first two transformation equations giving

eqnarray793

For the speed of light to be the same in both tex2html_wrap_inline1118 and tex2html_wrap_inline1124 we require that

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Dividing top and bottom by t gives

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We are therefore left with the following transformation law for tex2html_wrap_inline1220 and tex2html_wrap_inline1340 :

  eqnarray801

The principle of Special Relativity implies that if tex2html_wrap_inline1342 , tex2html_wrap_inline1344 , tex2html_wrap_inline1346 and tex2html_wrap_inline1348 , tex2html_wrap_inline1350 . This gives the inverse transformations

  eqnarray803

Substituting for tex2html_wrap_inline1220 and tex2html_wrap_inline1340 from (16) in (17) gives, after some straightforward algebra,

equation805

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,

eqnarray807


next up previous index
Next: The spacetime interval Up: Title page Previous: The principle of Special