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Path: ...!fu-berlin.de!uni-berlin.de!not-for-mail From: ram@zedat.fu-berlin.de (Stefan Ram) Newsgroups: sci.physics.relativity Subject: Vector notation? Date: 28 Jul 2024 09:27:30 GMT Organization: Stefan Ram Lines: 60 Expires: 1 Jul 2025 11:59:58 GMT Message-ID: <vector-20240728102344@ram.dialup.fu-berlin.de> Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit X-Trace: news.uni-berlin.de /TIfnEUintmmkCCPiGs68woGFf85zZyLvILQyaMgMGMmBW Cancel-Lock: sha1:owDrl++oz2KN26bq3ZLB9/vxDfA= sha256:uYeXntzeD0Tn1TRZqHmEYhsWBrMQrZGJvDfFH480Q2Q= X-Copyright: (C) Copyright 2024 Stefan Ram. All rights reserved. Distribution through any means other than regular usenet channels is forbidden. It is forbidden to publish this article in the Web, to change URIs of this article into links, and to transfer the body without this notice, but quotations of parts in other Usenet posts are allowed. X-No-Archive: Yes Archive: no X-No-Archive-Readme: "X-No-Archive" is set, because this prevents some services to mirror the article in the web. But the article may be kept on a Usenet archive server with only NNTP access. X-No-Html: yes Content-Language: en-US Bytes: 3375 (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!