# Theory of relativity/Fictitious force

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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

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

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

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

Where ${\displaystyle \Gamma _{\mu \nu }^{\lambda }}$ 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

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

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

${\displaystyle f_{Fict}^{\lambda }=m{\frac {d^{2}x^{\lambda }}{d\tau ^{2}}}}$

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.