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The spacetime interval

  Consider the effect of the Lorentz transformations on the spacetime interval 

equation809

Substituting for x and t from the above Lorentz transformations one obtains

eqnarray811

Generalizing to four dimensions we see that the spacetime interval

equation813

is invariant  under the Lorentz transformations.

The most general transformation between tex2html_wrap_inline1118 and tex2html_wrap_inline1124 will be more complicated but it must be linear. We can write it as:

equation817

where tex2html_wrap_inline1370 . This linear transformation is called the generalized Lorentz transformations. It contains ten parameters: four correspond to an origin shift tex2html_wrap_inline1372 , three correspond to a Lorentz boost [ which depends on tex2html_wrap_inline1156 ] and three to the rotation which aligns the axes of tex2html_wrap_inline1118 and tex2html_wrap_inline1124 . The last six are contained in the tex2html_wrap_inline1380 matrix tex2html_wrap_inline1382 [ six because tex2html_wrap_inline1382 is symmetric i.e. tex2html_wrap_inline1386 .

Later we will show that the Poincaré transformations preserve Maxwells equations as well as light paths.


Peter Dunsby
Mon Jul 22 05:46:38 ADT 1996