Editing Four-force (section)

== In general relativity ==
In [[general relativity]] the relation between four-force, and [[four-acceleration]] remains the same, but the elements of the four-force are related to the elements of the [[four-momentum]] through a [[covariant derivative]] with respect to proper time.

<math display="block">F^\lambda := \frac{DP^\lambda }{d\tau} = \frac{dP^\lambda }{d\tau } + \Gamma^\lambda {}_{\mu \nu}U^\mu P^\nu </math>

In addition, we can formulate force using the concept of [[coordinate transformation]]s between different coordinate systems. Assume that we know the correct expression for force in a coordinate system at which the particle is momentarily at rest. Then we can perform a transformation to another system to get the corresponding expression of force.<ref>{{cite book|last1=Steven|first1=Weinberg|title=Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity|date=1972|publisher=John Wiley & Sons, Inc.|isbn=0-471-92567-5|url-access=registration|url=https://archive.org/details/gravitationcosmo00stev_0}}</ref> In [[special relativity]] the transformation will be a Lorentz transformation between coordinate systems moving with a relative constant velocity whereas in [[general relativity]] it will be a general coordinate transformation.

Consider the four-force <math>F^\mu=(F^0, \mathbf{F})</math> acting on a particle of mass <math>m</math> which is momentarily at rest in a coordinate system. The relativistic force <math>f^\mu </math> in another coordinate system moving with constant velocity <math>v</math>, relative to the other one, is obtained using a Lorentz transformation:

<math display="block">\begin{align}
  \mathbf{f} &= \mathbf{F} + (\gamma - 1) \mathbf{v} {\mathbf{v}\cdot\mathbf{F} \over v^2}, \\
         f^0 &= \gamma \boldsymbol{\beta}\cdot\mathbf{F} = \boldsymbol{\beta}\cdot\mathbf{f}.
\end{align}</math>

where <math>\boldsymbol{\beta} = \mathbf{v}/c</math>.

In [[general relativity]], the expression for force becomes

<math display="block">f^\mu = m {DU^\mu\over d\tau}</math>

with [[covariant derivative]] <math>D/d\tau</math>. The equation of motion becomes

<math display="block">m {d^2 x^\mu\over d\tau^2} = f^\mu - m \Gamma^\mu_{\nu\lambda} {dx^\nu \over d\tau} {dx^\lambda \over d\tau},</math>

where <math> \Gamma^\mu_{\nu\lambda} </math> is the [[Christoffel symbol]]. If there is no external force, this becomes the equation for [[geodesic]]s in the [[curved space-time]]. The second term in the above equation, plays the role of a gravitational force. If <math> f^\alpha_f </math> is the correct expression for force in a freely falling frame <math> \xi^\alpha </math>, we can use then the [[equivalence principle]] to write the four-force in an arbitrary coordinate <math> x^\mu </math>:

<math display="block">f^\mu = {\partial x^\mu \over \partial\xi^\alpha} f^\alpha_f.</math>