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tech / sci.physics.relativity / Re: Addition of velocity

Re: Addition of velocity

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https://www.novabbs.com/tech/article-flat.php?id=117521&group=sci.physics.relativity#117521

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From: PointedE...@web.de (Thomas 'PointedEars' Lahn)
Newsgroups: sci.physics.relativity
Subject: Re: Addition of velocity
Date: Sat, 10 Jun 2023 14:25:50 +0200
Organization: PointedEars Software (PES)
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by: Thomas 'Pointed - Sat, 10 Jun 2023 12:25 UTC

Thomas 'PointedEars' Lahn wrote:
> Or using the Minkowski metric, γ and "time dilation" jump out more naturally:
>
> −c² dτ² = η_ab dx^a dx^b; [η]_ab = diag(−1, 1, 1, 1), x⁰ := ct, (x^i) = x⃗.
>
> ⇔ dτ² = −1/c² η_ab dx^a dx^b
>
> ⇔ dτ = √(−1/c² η_ab dx^a dx^b) = 1/c √(−η_ab dx^a dx^b)
>
> ⇔ dτ/dt = 1/c √(η_aa dx^a/dt dx^a/dt) = 1/c √(c² − v⃗²) = √(1 − v⃗²/c²)
^
> = 1/γ

I forgot a minus sign there, but the rest is correct.

> ⇒ Δτ = ∫_W dt/γ = Δt/γ.

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Subject	Author
Re: Addition of velocity	Thomas 'PointedEars' Lahn