Force |

**In General** > s.a. anomalous
acceleration; physical theories;
physics teaching;
symplectic structures.

* __Idea__:
The time rate of change of momentum, or the spatial rate of change of energy.

* __History__: There is a
tradition of trying to avoid the use of the concept (e.g., H Hertz, quantum
mechanics, general relativity below).

* __Measurement__: 2000, The smallest measured forces are
femtonewton forces between DNA molecules [@ news pn(00)may];
2006, Measurement of attonewton forces [@ Arlett et al NanoLett(06)] and practical
piconewton force measurement for cell biology [@ news pn(06)dec]; 2011, Measurements down to the yoctonewton (yN) level [@ news nn(11)apr].

@ __General references__: Jammer 57; Manchak PhSc(09)jul [in Cartesian physics];
Lange AJP(11)apr [historical controversy over the force composition law].

@ __Measurement__: Norte et al PRL(16), Reinhardt et al PRX(16) + Teufel Phy(16) [using trampoline resonators to sense extremely weak forces]; Jain et al PRL(16) + Safavi-Naeini Phy(16) [force sensor affected by momentum kicks of individual photons]; Ivanov a1606 [precision beyond the standard quantum limit].

@ __In special relativity__: Coleman NCB(06)
[and gravity]; Kholmetskii & Yarman EJP(07)
[resolution of apparent paradoxes];
Muñoz Díaz a1206 [in Lagrangian mechanics].

**Types and Examples** > s.a. Centrifugal
Force; friction; newtonian gravity [tidal];
Lorentz Force; self-force;
sound [acoustic force].

* __Pseudoforces__: Fictitious
forces that arise when one uses non-inertial frames of reference; Examples are
the Coriolis force (velocity-dependent), and, in Einstein's view, the force of gravity.

@ __Entropic forces__: Sokolov EJP(10) [nature, statistical mechanics]; > s.a. emergent gravity; formulations
of electromagnetism.

@ __Other special types__: Dolgov PRP(99) [long-range]; Lynden-Bell Obs(00)ap [preserving
the magnitude of a vector]; Vallejo mp/06 [inertial].

**In Curved Spacetime / General Relativity** > s.a. geodesic [deviation];
orbits of test bodies and gravitating
objects.

* __Idea__: Given a world-line with velocity *ξ*^{a} and curvature/acceleration
*ξ*^{m}∇_{m} *ξ*^{a}, the force on it is defined by

*F*^{a} := *m* *ξ*^{m}∇_{m} *ξ*^{a}
= *m*^{–1}* p*^{m} ∇_{m}
*p*^{m} ;

This force cannot be given as a vector field and is strongly observer-dependent,
so one often tries to avoid its use.

* __Maximum force or power__: The
principle according to which no force can exceed the "gravitational vacuum tension" *F*_{max} = \(\tau\) =
*c*^{4}/4*G* = 3.0 × 10^{43} N,
and no power can exceed *P*_{max} =
*c*^{5}/4*G* = 9.1 × 10^{51} W;
The values can be obtained easily from Planck units (up to the factor 4), but
physically force or power corresponds to momentum flow, and the maximum can
be attained only through horizons; Numerically, \(\tau \approx 3.026\times 10^{43}\)N.

@ __General references__: De Felice MNRAS(91);
Qadir & Sharif NCB(92)gq/07;
Qadir & Zafarullah
NCB(96); Arminjon AoM(96)gq/06 [and
Newton's second law]; Ridgely EJP(10) [derivation and examples].

@ __Inertial__: Prasanna gq/01;
Foertsch
et al CQG(03)gq [photon
surfaces]; Jonsson CQG(06)-a0708 [general
formalism, and optical geometry].

@ __Centrifugal__: Prasanna MPLA(02)gq/01
[and Coriolis]; Hasse & Perlick GRG(02)gq/01 [and
lensing].

@ __Maximum force or tension__: Schiller IJTP(05)phy/06 [general
relativity
from maximum power or force]; Gibbons FP(02)ht [maximal
tension = *c*^{4}/4*G*]; Barrow & Gibbons MNRAS(14)-a1408 [maximum force, examples and ramifications]; Dąbrowski & Gohar PLB(15)-a1504 [theories in which it does not hold]; Bolotin & Cherkaskiy a1507 [and the holographic principle]; > s.a. deformed uncertainty principle.

@ __Related topics__: Semerák NCB(95)
[decomposition];
Price GRG(04)gq [normal
to
4-velocity].

**In Quantum Theory** > s.a. interaction.

@ __References__: Mullin & Blaylock AJP(03)dec
[particle statistics and "exchange
force"].

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2016