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This artist’s concept depicts a supermassive black hole at the center of a galaxy. Illustration: NASA/JPL-Caltech
A spinning black hole has a different gravitational field than a non-spinning one. Spin is an intrinsic aspect of the black hole. A spinning black hole will radiate more energy than a non-spinning one. The accretion disk’s temperature also increases the faster the black hole spins. We can observe these differences in the X-ray spectra of accreting black holes.
Another method involves measuring the speed that matter rotates around the black hole.