Velocity |

**In Classical Mechanics** > s.a. classical mechanics.

* __Idea__: The time derivative
of position, *v*^{a}:=
d*q*^{a}/d*t*.

@ __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].

> __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 [*v*_{p} < 0].

* __Energy transfer velocity__:
@ in Panofsky & Phillips 62.

* __Front velocity__: For
waves that have a well-defined front, such as shock waves; *v*_{f}
:= lim_{k → ∞} *ω*(*k*)/*k*;
This is the concept that is relevant for causality considerations.

* __Group velocity__:
For a wave packet centered at *k* = *k*_{0},

*v*_{g}:= d*ω*/d*k* |
_{k0} ;

First mentioned by Hamilton (1839), reintroduced by Stokes (1876) and Rayleigh (1877).

* __Group velocity, generalized__: *v*_{φ}:=
*l*/(d*φ*/d*ω*), 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*–*v*_{p}*t*)},
or

*v*_{p}:= *ω* /
*k* = Re [1/*n*(*ω*)] .

* __Signal velocity__: The
speed of information *v*_{s};
Satisfies *v*_{g} ≤ *v*_{s} ≤ *v*_{f},
but no general definition is known.

* __Remark__: Some claim that it is *v*_{s} that should be less than *c*,
others that it is *v*_{f}.

* __Relationships__: For linear waves in a homogeneous medium, *v*_{g} = *v*_{e}.

@ __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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