Theoretical motivation for general relativity

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. 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: 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 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 π {displaystyle pi } 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, v {displaystyle mathbf {v} } is the 3-velocity, and γ {displaystyle gamma } is