Lorentz transformation

Proposition

edit

Given that the interval   is invariant under a Lorentz transformation, prove that the Lorentz transformation is orthogonal.

1.  . Given.
2.   metric tensor
3.   Lorentz transformation
4.   Lorentz transformation
5.   Substitute 3 and 4 into 2.
6.   metric tensor
7.   From 5, 1, and 6.
8.   Rearrange 7.
9.   From 8, since   and   may be arbitrary.
10.   Kronecker delta
11.   Multiply both sides of 9 by  , then apply 10.
12.   Contracting the indices   in 11.
13.   Contracting the indices   in 12.
14.   Swap the order of indices in order to transpose the first   of 13.
15.     14 may be paraphrased as  .

Reference: http://www.physics.gla.ac.uk/~dmiller/lectures/RQM_2008.pdf, page 9.