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From: ram@zedat.fu-berlin.de (Stefan Ram)
Newsgroups: sci.physics.relativity
Subject: Re: Vector notation?
Date: 1 Aug 2024 11:13:59 GMT
Organization: Stefan Ram
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ram@zedat.fu-berlin.de (Stefan Ram) wrote or quoted:
>When, in (1), both "p" are written exactly the same way, by what
>reason then is the first "p" in (2) written as a /row/ vector and
>the second "p" a /column/ vector?

  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)!