I am new to General Relativity and confused by the geodesic equation and MiSaTaQuWa equation. Most of the book saying that the geodesic equation is the motion of a particle in curved-spacetime. However, I read somewhere about this MiSaTaQuWa equation of motion. What is the difference between geodesic equation and this MiSaTaQuWa equation? If I want to describe the motion of a particle in curved-spacetime, which one should I use? The geodesic equation or the MiSaTaQuWa equation? Thanks for the help.
The geodesic equations describe the motion of a 'test particle' which is assumed to be very like a point. The MiSaTaQuWa equations are for a small extended body which itself has an interaction with the field. From Samuel E Gralla and Robert M Wald (2008) Class. Quantum Grav. 25 205009
http://relativity.livingreviews.org/Articles/lrr-2011-7/index.html "It should be noted that Eq. (19.84) is formally equivalent to the statement that the point particle moves on a geodesic in a spacetime ..." http://arxiv.org/abs/gr-qc/0202086 "While geodesic motion has been demonstrated in the past, only in our case is the reference metric ... a vacuum solution of the Einstein equations through O(μ)."
In that case I would think it better to say that it co-determines the field as opposed to interact with the field.
Does that means that geodesic equation describe the motion of a particle that doesnt interact with the field and the MiSaTaQuWa equation describe the motion of particle that interact with the field? Since any particle with mass will interact with the field, so does that means that geodesic equation only for massless or at least negligible particle?
yep. Think of a real, finite-sized body that is small compared to its surroundings, and "Taylor expand" its motion in [size of body] / [scale of variation of external universe]. At lowest order you get geodesic motion, which is accurate enough for almost everything. The first correction is the gravitational self-force (described in a particular gauge by the MiSaTaQuWa equation).