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 $$\frac{d^2 x'^{\lambda }}{d\tau ^2 } = 0$$

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 $$\frac{d^2 x^{\lambda }}{d\tau ^2 } + \Gamma ^{\lambda}_{\mu \nu} \frac{dx^{\mu }}{d\tau }\frac{dx^{\nu }}{d\tau}= 0$$

Where $$\Gamma ^{\lambda}_{\mu \nu}$$ 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 $$f^{\lambda}_{Fict} = -m\Gamma ^{\lambda}_{\mu \nu} \frac{dx^{\mu }}{d\tau }\frac{dx^{\nu }}{d\tau}$$

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 $$f^{\lambda}_{Fict} = m\frac{d^2 x^{\lambda }}{d\tau ^2 }$$

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.