Final answer:
The Kerr metric will appear similar to the Schwarzschild metric at distances far from the black hole's event horizon, where the effects of rotation are negligible. The rotation speed of a black hole can be measured by examining its gravitational influence on surrounding matter, such as through accretion disk behavior and the dynamics of nearby stellar objects.
Step-by-step explanation:
The Kerr metric, representing a rotating black hole, will resemble the Schwarzschild metric, which represents a non-rotating black hole, at distances large compared to the black hole's event horizon.
The Schwarzschild metric is a solution to the equations of general relativity for a spherically symmetric, non-rotating mass, and at a sufficient distance from a rotating black hole where the effects of rotation become negligible, the Kerr metric asymptotically approaches the Schwarzschild metric. However, the exact point at which this occurs depends on the mass and angular momentum of the black hole as well as the observer's angle relative to the rotation axis.
To measure the rotation speed of a black hole from the outside, astronomers look for the effects of the black hole's gravitational influence, such as accretion disk behaviors, relativistic jet formations, or the orbital dynamics of nearby stars or matter. The interaction between the black hole and its surrounding environment can give clues to its rotation rate. For example, relativistic frame-dragging effects in the vicinity of a rotating black hole can be studied to estimate its spin. These observational methods allow scientists to infer the angular momentum and how fast the black hole rotates.