Solution

ID: past-exam-of-the-general-relativity-ii-course-of-the-university-of-oxford/2015/2/b/iii/solution

For there are two distinct positive roots:
Both roots are coordinate singularities of the static chart. In the standard extension, is the outer event horizon and is an inner Cauchy horizon. The curvature singularity is . Between the horizons , so the radial direction is timelike; for , again and the central curvature singularity is timelike. These are the three radial regions of the global structure of a charged spherical black hole.[ref]
For completeness, if and , the Schwarzschild spacetime limit has just the positive horizon and a spacelike curvature singularity at ; the formal root of is not a regular inner horizon. If , the metric tensor is flat and the radial origin is regular in Cartesian coordinates.

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