Velocity  

In Classical Mechanics > s.a. classical mechanics.
* Idea: The time derivative of position, va:= dqa/dt.
@ General references: Lévy-Leblond AJP(80)may [notions, in special relativity, and rapidity]; Braginsky et al PRD(00)gq/99 [speed meter]; Smith SHPMP(03) [v can be considered instantaneous]; Kiang CAA(04)ap/03 [notions, in classical mechanics, special relativity, and general relativity]; Casey AJP(07)aug [areal velocity, for non-planar problems].
@ In curved spacetime: Bolós CMP(07)gq/05 [relative velocity]; Bolós JGP(13) [with respect to stationary observers in Schwarzschild spacetime]; Gharechahi et al a1510 [3-velocities of a test particle in the 1+3 (threading) and 3+1 (foliation) spacetime decomposition formalisms]; Abramowicz a1608 [covariant definitions and confusions].
> Related topics: see dark matter [general relativistic velocity, as alternative]; tachyons.

In Quantum Mechanics > s.a. measurements in quantum theory.
* Weak velocity: The expression \(\langle\)a|v|b\(\rangle\) / \(\langle\)a|b\(\rangle\), where v is the velocity operator and |b\(\rangle\) and |a\(\rangle\) the states of a particle before and after a velocity measurement.
@ References: Rohrlich & Aharonov PRA(02)qp/01 [superluminal weak v].

For Waves > s.a. dispersion; wave phenomena; constants [speed of light]; schwarzschild-de sitter spacetime [vp < 0].
* Energy transfer velocity: @ in Panofsky & Phillips 62.
* Front velocity: For waves that have a well-defined front, such as shock waves; vf := limk → ∞ ω(k)/k; This is the concept that is relevant for causality considerations.
* Group velocity: For a wave packet centered at k = k0,

vg:= dω/dk | k0 ;

First mentioned by Hamilton (1839), reintroduced by Stokes (1876) and Rayleigh (1877).
* Group velocity, generalized: vφ:= l/(dφ/dω), with l a length, and φ the phase change over l.
* Phase velocity: For a wave of wave number k and frequency ω, ψ = A exp{i k(xvpt)}, or

vp:= ω / k = Re [1/n(ω)] .

* Signal velocity: The speed of information vs; Satisfies vgvsvf, but no general definition is known.
* Remark: Some claim that it is vs that should be less than c, others that it is vf.
* Relationships: For linear waves in a homogeneous medium, vg = ve.
@ General references: in Shore CP(03)gq.
@ Phase velocity: Lakhtakia et al PLA(05) [negative, for electromagnetic waves in curved spacetime]; Rousseaux et al NJP(08)-a0711 [in a water tank].
@ Group velocity: in Stratton 41; McDonald AJP(98)aug [and energy]; Dolgov & Khriplovich PLA(98)ht/97 [and front velocity]; Bers AJP(00)may [= ve]; McDonald AJP(01)may [negative]; Amelino-Camelia et al JCAP(03)ht/02 [in non-commutative spacetime].
@ Related topics: Drozdov & Stahlhofen a0704 [local concept]; Mayo & Kerstein PLA(07) [front speed speedup in random media]; Budko PRL(09) [locally negative velocity for electromagnetic waves in vacuum].


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