What if the Sun became a black hole? 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.... 
 A: 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).
A: First of all, to get this out of the way, the Sun is too small to ever become a black hole or even a neutron star. The theoretical lower limit for a star to become a black hole via a stellar collapse is about 2.7 solar masses.
However, hypothetically, if the Sun suddenly turns a black hole of the same mass, there may be a potential initial burst of gravitational waves depending on the mechanism of how you intend to compress the Sun down. Other than that, gravitational changes would be minimal, if any. Specifically, the Schwarzschild (non rotating) solution does not change the gravitational field outside of the current Sun's size (as already mentioned). However, more realistic is the rotating Kerr solution. The typical speed of rotation of the event horizon is from a half the speed of light to near the speed of light. This rotation would produce a frame dragging effect. I will defer to the experts to clarify if this effect would be any different from the existing frame dragging produced by the Sun. In any case, it is likely to be negligibly small with no practical gravitational consequences for the Earth.
There would be dramatic changes in the emission spectrum. The gas around the black hole would be dragged and puled in producing a black body radiation with the soft x-ray peak plus a hard x-ray radiation from the corona. The amount of radiation would depend on the amount of gas present, but in any case the Earth would become inhabitable.
