I will give my interpretation of the Einstein Field Equations and if anyone would like to correct any mistakes that I make, please do. I, and others, have much to learn
If my interpretation is correct, the fall of a solitary object to the
ground to one location on the Earth can be locally transformed away by
a correct choosing of a system of accelerating coordinates, but the
fall of objects all over the Earth cannot. Effectively, these objects
can be visualized as a spherical cloud of tiny particles, each
following its own geodesic path, and the multiplicity of paths are all
converging, such, that the cloud's volume is shrinking at an
accelerating rate, and, as the shell/cloud collapses toward the
Earth, the rate of acceleration, which is the second time derivative,
is proportional to the mass of the Earth, in accordance with the
Einstein field equations.
I apologize for the lack of tensor skills below ;)
Einstein says that for empty space, far from any gravitating object,
the spacetime will be Minkowskian [flat], requiring that the Riemann
curvature tensor R_abcd should vanish?
But in a spacetime near a gravitating object, there will be a
non-zero intrinsic curvature, because the total gravitational field of
an object, cannot be simply "transformed away" to the second order.
This necessary condition is given by the non-vanishing Riemann
curvature tensor. Although, at points where the full curvature tensor
R_abcd is non-zero, the contracted curvature tensor of the second rank
tensor, R_ab = g^ab R_abcd = R^c_acb also known as the "Ricci
tensor" , may ...vanish?
Therefore the equations for the gravitational field in the vacuum of
empty space are: Ricci tensor, R_ab = 0
The R_ab = 0 represents ten equations in the ten components of g_ab
at each point in empty spacetime, devoid of matter or electomagnetic
energy, while not eliminating the gravitational metric-field itself.
The equations are generally covariant, thus, given any single
solution, infinitely many others may also be constructed via applying
arbitrary, albeit continuous coordinate transformations. The
conditions to be satisfied by the field equations must be the
vanishing of the covariant derivatives - guaranteeing the conservation
of any energy-momentum source term, that may be placed on the right
side of the equation. Although it seems to make one think about the
energy of the energy of the...of the ... of the ...of the energy of
a gravitational field!
The divergence generalizes to the covariant derivative in tensor
calculus, such, that the covariant derivatives of the metrical field
equations must identically vanish. The Ricci tensor R_ab in and of
itself does not satisfy this necessary requirement, but a new tensor
can be created which does satisfy the requirement via a slight
modification of the Ricci tensor, and, without disturbing the
relation R_ab = 0 for the vacuum of empty space.
Subtracting half the metric tensor times the invariant Ricci scalar R
= g^ab R_ab gives the Einstein Tensor:
G_ab = R_ab - (1/2) g_ab R
Since R_ab = 0
G_ab = 0
The covariant energy-momentum tensor is T_ab , regarded as the cause,
or the "source" of the metric curvature. "Mass tells space how to
curve and space tells mass how to move." It gives the conservation of
energy-momentum, and it implies gravitational energy "gravitates"
just as all other forms of energy do.
G_ab = k T_ab
k = -8pi G where G is Newton's universal gravitational constant.
