A naive introduction of a dependency of the mass of a black hole on the Schwarzschild time coordinate results in singular behavior of curvature invariants at the horizon, violating expectations from complementarity. If instead a temporal dependence is introduced in terms of a coordinate akin to the river time representation, the Ricci scalar is nowhere singular away from the origin. It is found that for a shrinking mass scale due to evaporation, the null radial geodesics that generate the horizon are slightly displaced from the coordinate singularity. In addition, a changing horizon scale significantly alters the form of the coordinate singularity in diagonal (orthogonal) metric coordinates representing the space-time. A Penrose diagram describing the growth and evaporation of an example black hole is constructed to examine the evolution of the coordinate singularity.
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Lindesay, J. Coordinates with Non-Singular Curvature for a Time Dependent Black Hole Horizon. Found Phys 37, 1181–1196 (2007). https://doi.org/10.1007/s10701-007-9146-4
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DOI: https://doi.org/10.1007/s10701-007-9146-4