Einstein Field Equation
We obtain the Einstein field equation by equating the acceleration required for circular orbits with the acceleration due to gravity
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This is the relationship between curvature of spacetime and the stress-energy tensor.
The Ricci tensor becomes
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The trace of the Ricci tensor is
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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
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This gives
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The field equation can be written
where
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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.
Read more about this topic: Theoretical Motivation For General Relativity
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