What is the sum of forces equation in curved spacetime?

Question

I've been thinking about the Sum of Forces equation and the fact that gravity isn't really a force. So what is the proper way to think about this equation when dealing with objects in a gravitational field? My thinking goes like this:

The sum of forces equation for flat (Lorentzian) spacetime is: $$\sum_i F=ma$$ Where $\sum_i F$ is the sum of forces and $a$ is the acceleration due to the sum (imbalance) of forces. So what is the correct form of this for positively curved spacetime? Is it proper to think of it like this? $$\sum_i F=m(a-A)$$ Let's say the imbalance of forces was due to a proton's attraction to an electron at some distance in a gravitational field. $\sum_i F$ would represent the actual electromagnetic force between the particles and $A$ would be the acceleration of the curvature (assuming the curvature was constant). The same would go for negatively curved spacetime.

So is it proper to think of this: $$\sum_i F=m(a+A)$$ As the proper form of the Sum of Forces equation in negatively curved spacetime?

Answer 1

The manifestly covariant equation is $\Sigma F^{\mu}=D_{\tau}P^{\mu}$ where $F^{\mu}$ is the four force, $D_{\tau}$ is the covariant derivative, and $P^{\mu}$ is the four momentum.

Curvature is a rank four tensor, so it cannot be described as simply positive or negative. However, if you wish to include gravity or fictitious forces, that can be done by writing the above expression in terms of the Christoffel symbols: $\Sigma F^{\mu}= \frac{d}{d\tau}P^{\mu} + \Gamma^{\mu}{}_{\nu\eta}U^{\nu}P^{\eta}$