A theoretical motivation for general relativity, including the motivation for the geodesic equation and the Einstein field equation, can be obtained from special relativity by examining the dynamics of particles in circular orbits about the earth. A key advantage in examining circular orbits is that it is possible to know the solution of the Einstein Field Equation a priori. This provides a means to inform and verify the formalism. General relativity addresses two questions: How does the curvature of spacetime affect the motion of matter? How does the presence of matter affect the curvature of spacetime? The former question is answered with the geodesic equation. The second question is answered with the Einstein field equation. The geodesic equation and the field equation are related through a principle of least action. The motivation for the geodesic equation is provided in the section Geodesic equation for circular orbits. The motivation for the Einstein field equation is provided in the section Stress–energy tensor. Geodesic equation for circular orbits Kinetics of circular orbits For definiteness consider a circular earth orbit (helical world line) of a particle. The particle travels with speed v. An observer on earth sees that length is contracted in the frame of the particle. A measuring stick traveling with the particle appears shorter to the earth observer. Therefore, the circumference of the orbit, which is in the direction of motion appears longer than times the diameter of the orbit. In special relativity the 4-proper-velocity of the particle in the inertial (non-accelerating) frame of the earth is where c is the speed of light, is the 3-velocity, and is . The magnitude of the 4-velocity vector is always constant where we are using a Minkowski metric . The magnitude of the 4-velocity is therefore a Lorentz scalar. The 4-acceleration in the earth (non-accelerating) frame is where is c times the proper time interval measured in the frame of the particle. This is related to the time interval in the Earth's frame by . Here, the 3-acceleration for a circular orbit is where is the angular velocity of the rotating particle and is the 3-position of the particle. The magnitude of the 4-velocity is constant. This implies that the 4-acceleration must be perpendicular to the 4-velocity. The inner product of the 4-acceleration and the 4-velocity is therefore always zero. The inner product is a Lorentz scalar. Curvature of spacetime: Geodesic equation The equation