Trajectories of Particles

Consider a distribution of matter. Isolate a particle small enough that the gravitational field of this particle has negleible effect on the rest of the distribution. Let R(s) be the location of the particle at proper time s. R(s) describes a curve in space-time; at each point of this curve, define the tangent vector T(s)=dR/ds. T(s) is the instaneous four-velocity of the particle. Transorm to a coordinated system that moves along the partcle so that the tangent vector is purely time like.The same rotation quaternion that defines the field also serves as the transformation:

During a proper time interval ds the particle moves along the tangent vecor

Together, these two equations determine the trajectory of the particle.

To see how these equations work, we apply them to the Schwarzchild metric. First, expand the equations:

To keep the example simple, consider only radial motion in the Schwarzchild metric:

Equating components we get four equations for the two non-zero components of the tangent vector and the particle location::

Eliminate both components of T and use the chain rule together with the equation for dt/ds to get

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