A four-vector has certain transformation properties. Those terms involving Christoffel symbols do not. Even in a flat space using spherical coordinates, you would get apparent acceleration for a free particle.
If you simply interpret the RHS of the geodesic equation as a four-acceleration, can't we just treat gravity as a force?
Murphrid is correct. You can certainly move the term involving Christoffel symbols to the other side from the term involving the coordinate acceleration, but the resulting equation is no longer a tensor equation and does not transform as a tensor does.
However, the non-tensor equation that you have can certainly then be interpreted as an acceleration due to a gravitational force. The gravitational force, however, is clearly a fictitious force and even in this situation cannot be locally distinguished from other fictitious forces such as those due to acceleration or rotation of the coordinate system.
Additionally, how would you split the Einstein field equations into a source providing force equation? The Geodesic equation has enough problems as is, interpreting one side as a force, the Einstein field equations would have just infinitely more problems on that front...
Reading this thread triggered an interest in fictitious forces and I googled to get more information.Some promising stuff seemed to come up but there was something which confused me.Writing about fictitious forces for "Scientific American" David Pollitzer referred to General relativity and amongst other things stated
"The cornerstone of Einsteins theory,however,is the proposition that gravity is itself a fictitious force(or, rather,that it is indistinguishable from a fictitious force)"
So what is it? Is it considered to be a fictitious force or just indistinguishable from a fictitious force or something else? Thanks if anyone answers.
Try googling What is a "fictitious force"
Caltech 2004 Nobel laureate David politzer writing for "Scientific American"
