Tensors and Relativity: Chapter 1

The spacetime interval

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Consider the effect of the Lorentz transformations on the spacetime interval 

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

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Generalizing to four dimensions we see that the spacetime interval

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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 .

  • Note that these parameters for a group galled the Poincaré group. 

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