page_title: Theoretical motivation for general relativity
section_title: Einstein field equation
hierarchical_section_title: Theoretical motivation for general relativity / Einstein field equation
context_page_description: 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
context_section_description: We obtain the Einstein field equation by equating the acceleration required for circular orbits with the acceleration due to gravity
  .
  This is the relationship between curvature of spacetime and the stress–energy tensor.
  The Ricci tensor becomes
  .
  The trace of the Ricci tensor is
  .
  Comparison of the Ricci tensor with the Ricci tensor calculated from the principle of least action, Theoretical motivation for general relativity#Principle of least action in general relativity identifying the stress–energy tensor with the Hilbert stress-energy, and remembering that A+B=1 removes the ambiguity in A, B, and C.
  and
  .
  This gives
  .
  The field equation can be written
  where
  .
  This is the Einstein field equation that describes curvature of spacetime that results from stress-energy density. This equation, along with the geodesic equation have motivated by the kinetics and dynamics of a particle orbiting the earth in a circular orbit. They are true in general.
caption: Two-dimensional visualization of space-time distortion. The presence of matter changes the geometry of spacetime, this (curved) geometry being interpreted as gravity.
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