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From: ram@zedat.fu-berlin.de (Stefan Ram)
Newsgroups: sci.physics.research
Subject: Re: Matrix Multiplication in SR
Date: 1 Aug 2024 07:01:38 GMT
Organization: Stefan Ram
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ram@zedat.fu-berlin.de (Stefan Ram) wrote or quoted:
>[[Mod. note -- I think that last subscript "mu" should be a "nu".
>That is, equations (0) and (1) should read (switching to LaTeX notation)
>$X := p_\mu p^\mu
>   =  p_\mu \eta^{\mu\nu} p_\nu$
>-- jt]]

  Thanks for that observation! 

  In the meantime, I found the answer to my question reading a text
  by Viktor T. Toth.

  Many Textbooks say,

              ( -1  0  0  0 )
eta_{mu nu} = (  0  1  0  0 )
              (  0  0  1  0 )
              (  0  0  0  1 ),
              
  but when you multiply this by a column (contravariant) vector,
  you get another column (contravariant) vector instead of a row,
  while the "v_mu" in
  
eta_{mu nu} v^nu = v_mu

  seems to indicate that you will get a row (covariant) vector!

  As Viktor T. Toth observed in 2005, a square matrix (i.e., 
  a row of columns) only really makes sense for eta^mu_nu (which is
  just the identity matrix). He then clear-sightedly explains that
  a matrix with /two/ covariant indices needs to be written not
  as a /row of columns/ but as a /row of rows/:
  
eta_{mu nu} = [( -1 0 0 0 )( 0 1 0 0 )( 0 0 1 0 )( 0 0 0 1 )]
 
  . Now, if one multiplies /this/ with a column (contravariant)
  vector, one gets a row (covariant) vector (tweaking the rules for
  matrix multiplication a bit by using scalar multiplication for
  the product of the row ( -1 0 0 0 ) with the first row of the
  column vector [which first row is a single value] and so on)!

  Exercise  Work out the representation of eta^{mu nu} in the same
  spirit.