Velocity| 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 IJGMP(18)-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];
Emtsova & Toporensky a1901 [velocities of remote objects].
> 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(x−vpt)},
or
vp:= ω / k = Re [1/n(ω)] .
* Signal velocity: The speed
of information vs; It satisfies
vg ≤ vs
≤ vf, 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 [= \(v_e\)];
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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