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?