The equation of motion in General theory of Relativity

In theory of Gravity: the equation of motion is determined by the gravitational field, like in our Newtonian theory a field/force when solved in terms of space and time coordinates gives a differential equation called a equation of motion whose solution gives you the trajectory of motion. In Gravity theory the equation of motion in an ordinary (locally inertial) coordinate system there are two tensors, 1. teh affine connection tensor and 2. the metric tensor, (like in Newtonian mechanics you had vectors these are generalied to tensors). The metric tensor gives you the proper-time and is actually the gravitational potential energy, as usual the derivative of a potential is a field hence the affine connection tensor is the derivative of the metric tensor = gravitational potential energy. SO we call the affine connection tensor which is the derivative of the potential energy, the gravitational field. Then we go on and impose symmetry conditions, such as conservation of momentum and energy which hlps us to solve the tensor equations of motion.

If our objects are mass-haves we do not have a problem with the equations of motion in terms of the propertime. But propertime is zero for zero-mass particles such as photons **and neutrinos, which were thought to be massless for a long time, but assigning them a very small mass as is consistent as per claims of neutrino oscillations from recent experiments one need to see super-luminal neutrinos in a very different set-up, their superluminal attributes can come either from inconsistency or from the detailed nature of space-time metric and their derivative, that is, the gravitational fields which are a result of the inherent structure and geometry of Universe itself**, for zero-propertime objects therefore one can consider a freely falling coordinate system instead of the locally inertial system of our everyday concern. A freely falling coordinate system is one which is free from other influences except purely gravitational forces …