| dbp:proof
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- Newtonian gravitation can be written as the theory of a scalar field, , which is the gravitational potential in joules per kilogram of the gravitational field , see Gauss's law for gravity
where is the mass density. The orbit of a free-falling particle satisfies
In tensor notation, these become
In general relativity, these equations are replaced by the Einstein field equations in the trace-reversed form
for some constant, , and the geodesic equation
To see how the latter reduces to the former, we assume that the test particle's velocity is approximately zero
and thus
and that the metric and its derivatives are approximately static and that the squares of deviations from the Minkowski metric are negligible. Applying these simplifying assumptions to the spatial components of the geodesic equation gives
where two factors of have been divided out. This will reduce to its Newtonian counterpart, provided
Our assumptions force and the time derivatives to be zero. So this simplifies to
which is satisfied by letting
Turning to the Einstein equations, we only need the time-time component
the low speed and static field assumptions imply that
So
and thus
From the definition of the Ricci tensor
Our simplifying assumptions make the squares of disappear together with the time derivatives
Combining the above equations together
which reduces to the Newtonian field equation provided
which will occur if (en)
- Contracting the differential Bianchi identity
with gives, using the fact that the metric tensor is covariantly constant, i.e. ,
The antisymmetry of the Riemann tensor allows the second term in the above expression to be rewritten:
which is equivalent to
using the definition of the Ricci tensor.
Next, contract again with the metric
to get
The definitions of the Ricci curvature tensor and the scalar curvature then show that
which can be rewritten as
A final contraction with gives
which by the symmetry of the bracketed term and the definition of the Einstein tensor, gives, after relabelling the indices,
Using the EFE, this immediately gives, (en)
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