# Theory of relativity/Fictitious force

For a rectilinear inertial frame a test particle in the absence of a force will remain in a constant velocity state in accordance with Newton's first law of motion

However in transforming to a noninertial or more generally curvalinear coordinate system the equation of motion for the test particle in the absence of a real force transforms into

Where are the Christoffel symbols. An observer using such coordinates may interpret the test particle's motion with respect to these coordinates as described by it experiencing a force of

So that he may interpret the equation of motion as being the result of a force due to Newton's second law of motion as

This "force" is fictitious in that it can be transformed away by going to a rectilinear inertial frame, so it is called a fictitious force. It is also commonly referred to as an inertial force or force of affine connection. Commonly experienced examples of a fictitious force would be a centrifugal force or the Coriolis force that a spinning observer will observe as acting on free falling test particles.