Force

=== Special theory of relativity ===
{{Main | Relativistic mechanics#Force}}
In the [[special theory of relativity]], mass and [[energy]] are equivalent (as can be seen by calculating the work required to accelerate an object). When an object's velocity increases, so does its energy and hence its mass equivalent (inertia). It thus requires more force to accelerate it the same amount than it did at a lower velocity. Newton's Second Law,
<math display="block">\vec{F} = \frac{\mathrm{d}\vec{p}}{\mathrm{d}t},</math>
remains valid because it is a mathematical definition.<ref name=Cutnell/>{{rp|855–876}} But for momentum to be conserved at  relativistic relative velocity, <math>v</math>, momentum must be redefined as:
<math display="block"> \vec{p} = \frac{m_0\vec{v}}{\sqrt{1 - v^2/c^2}},</math>
where <math>m_0</math> is the [[rest mass]] and <math>c</math> the [[speed of light]].

The expression relating force and acceleration for a particle with constant non-zero [[rest mass]] <math>m</math> moving in the <math>x</math> direction at velocity <math>v</math> is:<ref name=French1972>{{Cite book |last=French |first=A. P. |title=Special Relativity |date=1972 |publisher=Chapman & Hall |isbn=978-0-17-771075-9 |edition=reprint |series=The MIT introductory physics series |location=London}}</ref>{{rp|216}}
<math display="block">\vec{F} = \left(\gamma^3 m a_x, \gamma m a_y, \gamma m a_z\right),</math>
where
<math display="block" qid=Q599404> \gamma = \frac{1}{\sqrt{1 - v^2/c^2}}.</math>
is called the [[Lorentz factor]]. The Lorentz factor increases steeply as the relative velocity approaches the speed of light.  Consequently, the greater and greater force must be applied to produce the same acceleration at extreme velocity. The relative velocity cannot reach <math>c</math>.<ref name=French1972/>{{rp|26}}<ref name=FeynmanVol1/>{{rp|at=§15–8}}
If <math>v</math> is very small compared to <math>c</math>, then <math>\gamma</math> is very close to 1 and
<math display="block">F = m a</math>
is a close approximation. Even for use in relativity, one can restore the form of
<math display="block">F^\mu = mA^\mu </math>
through the use of [[four-vectors]]. This relation is correct in relativity when <math>F^\mu</math> is the [[four-force]], <math>m</math> is the [[invariant mass]], and <math>A^\mu</math> is the [[four-acceleration]].<ref>{{cite web
|first=John B.
|last=Wilson
|title=Four-Vectors (4-Vectors) of Special Relativity: A Study of Elegant Physics
|work=The Science Realm: John's Virtual Sci-Tech Universe
|url=http://SciRealm.com/4Vectors.html
|archive-url=https://web.archive.org/web/20090626152836/http://www.austininc.com/SciRealm/4Vectors.html
|archive-date=26 June 2009
|url-status=dead
|access-date=2008-01-04
}}</ref>

The [[general relativity|''general'' theory of relativity]] incorporates a more radical departure from the Newtonian way of thinking about force, specifically gravitational force. This reimagining of the nature of gravity is described more fully [[#Gravitational|below]].