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The Metric Tensor Field Is the Metric That Defines the Geometry of Spacetime
Spacetime has a particular field, the metric tensor field, from which we can calculate its geometry. The thing that has angular momentum is really the value of the metric Tensor field. If you start with a spinning star and it collapses to a black hole, the angular momentum will be the same as the angular momentum that we calculate in the metric Tensor Field. A non-spinning black hole is perfectly spherical; not true for a spinning one.