Upon talking to someone about the concepts of black holes, a question arised that I did not know the answer to.
If the Sun became a black hole, but the mass remained the same as it is now, the Earth would orbit in the same manner that it currently does because of the reason that the mass does not change (gravitational field stays constant).
However, does the curvature of spacetime change in the region where the Sun used to be located? And how would the curvature change in the region of the Earth's orbit? It must have an effect on spacetime, but I cant seem to form a reasonable enough argument for why….
Best Answer
Assuming nothing else changed about the body then the change will have no effect on the curvature of spacetime outside the radius that the sun had had previously.
Inside the sun’s radius, however, is a different story. As it currently stands, the curvature of spacetime increases as one approaches the sun’s radius and then, after passing the boundary, decreases until levelling out in the center of mass of the sun.
Note that the above is only true for a sun with uniformly distributed mass, but it’s enough to get a good picture.
In the case of a black hole, rather than starting to decrease, the curvature of spacetime would continue to increase—all the way down to the singularity (because there would be no counteracting mass once one passed through the sun’s radius).