to detect the force of gravity; hence, freefalling reference frames are equivalent to inertial reference frames (the equivalence principle). Developing this insight, Einstein formulated a theory with gravity as a fictitious force, and attributed the apparent acceleration due to gravity to the curvature of spacetime. This idea underlies Einstein's theory of general relativity. See the Eötvös experiment. {|class="toccolours collapsible collapsed" width="60%" style="text-align:left" !Animation: ball that rolls off a cliff |- | Note: The rain frame perspective here, rather than being that of a raindrop, is more like that of a trampoline jumper whose trajectory tops out just as the ball reaches the edge of the cliff. The shell frame perspective may be familiar to planet dwellers who rely minute by minute on upward physical forces from their environment, to protect them from the geometric acceleration due to curved spacetime. |} Mathematical derivation of fictitious forces General derivation Many problems require use of noninertial reference frames, for example, those involving satellites and particle accelerators. Figure 2 shows a particle with mass m and position vector xA(t) in a particular inertial frame A. Consider a non-inertial frame B whose origin relative to the inertial one is given by XAB(t). Let the position of the particle in frame B be xB(t). What is the force on the particle as expressed in the coordinate system of frame B? To answer this question, let the coordinate axis in B be represented by unit vectors uj with j any of { 1, 2, 3 } for the three coordinate axes. Then The interpretation of this equation is that xB is the vector displacement of the particle as expressed in terms of the coordinates in frame B at the time t. From frame A the particle is located at: As an aside, the unit vectors { uj } cannot change magnitude, so derivatives of these vectors express only rotation of the coordinate system B. On the other hand, vector XAB simply locates the origin of frame B relative to frame A, and so cannot include rotation of frame B. Taking a time derivative, the velocity of the particle is: The second term summation is the velocity of the particle, say vB as measured in frame B. That is: The interpretation of this equation is that the velocity of the particle seen by observers in frame A consists of what observers in frame B call the velocity, namely