# Can force be defined as the rate of change of four-momentum in General Relativity?

In Newtonian physics, the force acting on a particle is defined as the rate of change of momentum $$F=\frac{dp}{dt}.$$

Also, the force can be defined as the derivative of the potential $$F=-\frac{dV}{dr}.$$

However, I am curious whether in General Relativity, we can define the four-force as the rate of change of the four-momentum as follows: $$F^r=\frac{dp^r}{d\tau},$$ where $$p^r$$ is the radial component of the four-momentum and $$\tau$$ is the proper time. If this definition is correct, it would be helpful if someone provide links to relevant references.

The equations of Special and General Relativity are often not linear like Newton's Second Law is. To fit Special Relativity, you have to modify it to the form: $$$$\mathbf{F} =\gamma(\mathbf{v})^3m\mathbf{a}_\parallel +\gamma(\mathbf{v})m\mathbf{a}_\perp,$$$$ where $$\mathbf{a}_\parallel$$ and $$\mathbf{a}_\perp$$ is a decomposition of the acceleration $$\mathbf{a}$$ into a parallel and perpendicular part to the velocity $$\mathbf{v}$$. You can find this derivation here.
In General Relativity, the geodesic equation is also not linear, so you can't define such a four-force vector for gravity, but you can do so for all the other forces (for example electromagnetic ones) using said geodesic equation: $$$$F^\sigma =m\left(\ddot{x}^\sigma+\Gamma_{\mu\nu}^\sigma\dot{x}^\mu\dot{x}^\nu\right).$$$$