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From: Mikko <mikko.levanto@iki.fi>
Newsgroups: sci.physics.relativity
Subject: Re: Vector notation?
Date: Mon, 29 Jul 2024 12:35:09 +0300
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On 2024-07-28 09:27:30 +0000, Stefan Ram said:

>   (The quotation below is given in pure ASCII, but at the end of this
>   post you will also find a rendition with some Unicode being used.)
> 
>   I have read the following derivation in a chapter on SR.
> 
> |(0) We define:
> |X := p_"mu" p^"mu",
> |
> |(1) from this, by Eq. 2.36 we get:
> |= p_"mu" "eta"^"mu""nu" p_"mu",
> |
> |(2) from this, using matrix notation we get:
> |                       (  1  0  0  0 ) ( p_0 )
> |= ( p_0 p_1 p_2 p_3 )  (  0 -1  0  0 ) ( p_1 )
> |                       (  0  0 -1  0 ) ( p_2 )
> |                       (  0  0  0 -1 ) ( p_3 ),
> |
> |(3) from this, we get:
> |= p_0 p_0 - p_1 p_1 - p_2 p_2 - p_3 p_3,
> |
> |(4) using p_1 p_1 - p_2 p_2 - p_3 p_3 =: p^"3-vector" * p^"3-vector":
> |= p_0 p_0 - p^"3-vector" * p^"3-vector".
> 
>   . Now, I used to believe that a vector with an upper index is
>   a contravariant vector written as a column and a vector with
>   a lower index is covariant and written as a row. I'm not sure
>   about this. Maybe I dreamed it or just made it up. But it would
>   be a nice convention, wouldn't it?
> 
>   Anyway, I have a question about the transition from (1) to (2):
> 
>   In (1), the initial and the final "p" both have a /lower/ index "mu".
>   In (2), the initial p is written as a row vector, while the final p
>   now is written as a column vector.
> 
>   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?
> 
>   Here's the same thing with a bit of Unicode mixed in:
> 
> |(0) We define:
> |X ≔ p_μ p^μ
> |
> |(1) from this, by Eq. 2.36 we get:
> |= p_μ η^μν p_ν
> |
> |(2) from this, using matrix notation we get:
> |                   (  1  0  0  0 ) ( p₀ )
> |= ( p₀ p₁ p₂ p₃ )  (  0 -1  0  0 ) ( p₁ )
> |                   (  0  0 -1  0 ) ( p₂ )
> |                   (  0  0  0 -1 ) ( p₃ )
> |
> |(3) from this, we get:
> |= p₀ p₀ - p₁ p₁ - p₂ p₂ - p₃ p₃
> |
> |(4) using p₁ p₁ - p₂ p₂ - p₃ p₃ ≕ p⃗ * p⃗:
> |= p₀ p₀ - p⃗ * p⃗
> 
>   . TIA!

As "eta" or the dot product is an essential element of the structure
of the world as described by SR the distinction between contravariant
and covariant vectors (or vectors and covectors) is not necessary.

Instead of p_μ p^ν or p_μ η^μν p_ν you can simply write p_μ p_ν. If
you want to use a non-orthonormal basis you need to uses a different
matrix for "eta" in p_μ p_ν but the principle is same. Usually there
is no need to use a non-orthonormal basis in SR but in GR using one
may simplify other things.

-- 
Mikko