## Special theory of relativity::Force

### ::concepts

Force::forces    Object::first    Title::physics    Which::newton's    Motion::center    Velocity::constant

Special theory of relativity 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

$\vec{F} = \mathrm{d}\vec{p}/\mathrm{d}t$

remains valid because it is a mathematical definition.<ref name=Cutnell>{{#invoke:Footnotes|harvard_citation_no_bracket}}</ref>:855–876 But in order to be conserved, relativistic momentum must be redefined as:

$\vec{p} = \frac{m_0\vec{v}}{\sqrt{1 - v^2/c^2}}$

where

$v$ is the velocity and
$c$ is the speed of light
$m_0$ is the rest mass.

The relativistic expression relating force and acceleration for a particle with constant non-zero rest mass $m$ moving in the $x$ direction is:

$F_x = \gamma^3 m a_x \,$
$F_y = \gamma m a_y \,$
$F_z = \gamma m a_z \,$

where the Lorentz factor

$\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}.$<ref>{{#invoke:citation/CS1|citation

|CitationClass=web }}</ref>

In the early history of relativity, the expressions $\gamma^3 m$ and $\gamma m$ were called longitudinal and transverse mass. Relativistic force does not produce a constant acceleration, but an ever decreasing acceleration as the object approaches the speed of light. Note that $\gamma$ is undefined for an object with a non-zero rest mass at the speed of light, and the theory yields no prediction at that speed.

If $v$ is very small compared to $c$, then $\gamma$ is very close to 1 and

$F = m a$

is a close approximation. Even for use in relativity, however, one can restore the form of

$F^\mu = mA^\mu \,$

through the use of four-vectors. This relation is correct in relativity when $F^\mu$ is the four-force, $m$ is the invariant mass, and $A^\mu$ is the four-acceleration.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Force sections
Intro  Development of the concept  Pre-Newtonian concepts  Newtonian mechanics  Special theory of relativity  Descriptions  Fundamental forces  Non-fundamental forces  Rotations and torque  Kinematic integrals  Potential energy  Units of measurement  Force measurement  See also  Notes  References  Further reading  External links

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