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As for the Sun becoming a Black Hole and the earth's orbit should go unchanged. I don't think our sun has sufficient mass in the first place, and if it did, I believe their would certainly be a change in the gravitational field as a Black Hole's field strength is far greater, even if their was no increase in the amount of stellar material between the sun and the Black Hole. Can anyone confirm this, and if so, is the Scalar Curvature responsible for the increase expression of the field?
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The strength of the field is a function of density rather than mass. A stellar black hole will always have a smaller mass than the progenitor star (assuming of course there is no excretion matter to increase black hole mass). Yet the field strength of the black hole is far greater than the progenitor star despite having a smaller mass.
It's the metric rather than the scalar curvature that one looks at that is an indicator of field strength.
The two defining metrics for black holes are Schwarzschild metric for non rotating black holes and the Kerr metric for rotating black holes.
The Schwarzchild metric when used for low gravity field objects such as the Earth breaks down into a simple spherical metric.
Regards
Steven