The amount of radiation would depend on the amount of gas present, but in any case the Earth would become inhabitable. 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. There would be dramatic changes in the emission spectrum. In any case, it is likely to be negligibly small with no practical gravitational consequences for the Earth. I will defer to the experts to clarify if this effect would be any different from the existing frame dragging produced by the Sun. This rotation would produce a frame dragging effect. When the star gets close enough, the black hole's gravitational grasp violently rips it apart and sloppily devours its gasses while belching out intense radiation. They lie in wait until a hapless star wanders by. The typical speed of rotation of the event horizon is from a half the speed of light to near the speed of light. Hubble Finds Hungry Black Hole Twisting Captured Star Into Donut Shape Black holes are gatherers, not hunters. However, more realistic is the rotating Kerr solution. Specifically, the Schwarzschild (non rotating) solution does not change the gravitational field outside of the current Sun's size (as already mentioned). Other than that, gravitational changes would be minimal, if any. 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. The theoretical lower limit for a star to become a black hole via a stellar collapse is about 2.7 solar masses. 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.
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