Vector notation?
Sujet : Vector notation?
De : ram (at) *nospam* zedat.fu-berlin.de (Stefan Ram)
Groupes : sci.physics.relativityDate : 28. Jul 2024, 10:27:30
Autres entêtes
Organisation : Stefan Ram
Message-ID : <vector-20240728102344@ram.dialup.fu-berlin.de>
(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!
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