How does that relate to Newton's model?
General Relativity didn't make Newtonian gravity false. Newton's model is still a good model for most measurements and calculations of the motion of planets and rockets and so forth, and General Relativity must agree with Newton's model in that limit. Here is proof that these two very different models agree on the motion of the Earth around the Sun but disagree in the region where a new relativistic phenomenon - the black hole - is present in Einsetin's model but not in Newton's.
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Newtonian and relativistic potentials are almost the same
at large distance scales like the radius of the Earth's orbit
R = 1.5 1013 cm.
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The Earth and Sun in the Newtonian model
In Newton's model of nature based on the math he invented, the differential calculus, the primary equation by which motion of objects is calculated is
Force = mass x acceleration
This gives a set of second-order differential equations and when we solve that system, we get the motion of the object in question when subjected to the force in question.
If the object in question is the Earth and the force in question is the gravitational force of the Sun on the Earth, then after writing the above equation in spherical coordinates, we get the following equation for the change in time of the radial coordinate r for the position of the Earth:
m (dr/dt)2 = 2 (E - VN(r))
VN(r) = - M m G/r + m (L/m)2/(2 r2)
G is Newton's gravitational constant, M stands for the mass of the Sun and m is the (approximately) the mass of the Earth. Because the Newtonian force law for gravity depends only on radial distance, not on time or angle, there are two constants of motion called angular momentum L and energy E.
The function V(r) is plotted above for the Earth/Sun system. Notice that whenever V(r) = E, the radial coordinate stops changing because dr/dt = 0. These are called turning points. The turning points classify the type of orbit, as shown in the above figure.
Notice that it is the L2/r2 term in the potential that causes the turning points for smaller r. Angular momentum acts almost like a force of repulsion to counter gravitational attraction. That will be important later as an agent of causality violation in the relativistic model.
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Newtonian and relativistic potentials begin to disagree very close
to the center of attraction.
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The Earth and Sun in the relativistic model
In General Relativity, the first ingredient in model-making
is the spacetime metric. This metric must solve the Einstein equation that relate the spacetime curvature to the matter and energy present.
If the matter and energy present is idealized by a
pointlike object of mass M, then the spacetime metric that
solves the Einstein equation is called the Schwarzschild metric.
We won't display that metric here for the sake of brevity. The timelike geodesics are easy to calculate for the Schwarzschild metric. We wind up with an equation that looks a lot like the Newtonian formula above:
m (dr/dt)2 = 2 (E - VR(r))
VR(r) = - M m G/r + m (L/m)2/(2 r2)
- M G m (L/m)2/(2 c2 r3)
At large distances from the Sun, the last term in the potential VR(r) can be neglected,
and the classification of orbits is the same as for the Newtonian case.
But if the Sun really were a pointlike mass, the last term in the potential VR(r) would give us trouble. Notice in the bottom figure the red line, which represents the relativistic potential, veers down to become infinitely negative, instead of infinitely positive as with the Newtonian potential. This behavior is the signal of a black hole. We can't go into that in depth right here, but you can read about it in books specializing in black holes.
The Sun is not pointlike, the mass of the sun is spread within a radius
of about 700,000 kilometers. The black hole behavior would not set in unless the mass of the Sun were confined to a within radius of about three kilometers.
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