section_path: Revisions of the force concept/Special theory of relativity
section_name: Special theory of relativity
paragraph_text: 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, F = d p d t, {\displaystyle \mathbf {F} ={\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} t}},} remains valid because it is a mathematical definition. But for momentum to be conserved at relativistic relative velocity, v {\displaystyle v}, momentum must be redefined as: p = m 0 v 1 − v 2 / c 2, {\displaystyle \mathbf {p} ={\frac {m_{0}\mathbf {v} }{\sqrt {1-v^{2}/c^{2}}}},} where m 0 {\displaystyle m_{0}} is the rest mass and c {\displaystyle c} the speed of light.
  
  The expression relating force and acceleration for a particle with constant non-zero rest mass m {\displaystyle m} moving in the x {\displaystyle x} direction at velocity v {\displaystyle v} is: F = (γ 3 m a x, γ m a y, γ m a z), {\displaystyle \mathbf {F} =\left(\gamma ^{3}ma_{x},\gamma ma_{y},\gamma ma_{z}\right),} where γ = 1 1 − v 2 / c 2. {\displaystyle \gamma ={\frac {1}{\sqrt {1-v^{2}/c^{2}}}}.} 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 c {\displaystyle c}. If v {\displaystyle v} is very small compared to c {\displaystyle c}, then γ {\displaystyle \gamma } is very close to 1 and F = m a {\displaystyle \mathbf {F} =m\mathbf {a} } is a close approximation. Even for use in relativity, one can restore the form of F μ = m A μ {\displaystyle F^{\mu }=mA^{\mu }} through the use of four-vectors. This relation is correct in relativity when F μ {\displaystyle F^{\mu }} is the four-force, m {\displaystyle m} is the invariant mass, and A μ {\displaystyle A^{\mu }} is the four-acceleration.
  
  The 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 below.