What is the metric of Vaidya black-hole horizon?
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The metric of a Vaidya black hole in outgoing/retarted null coordinates are
ds2=−(1−2m(u)r2)du2−2dudr+r2(dθ2+sin2θdϕ2)ds2=−(1−2m(u)r2)du2−2dudr+r2(dθ2+sin2θdϕ2)
The eveolving horizon locates at r=2m(u)r=2m(u) and thus dr=2m˙dudr=2m˙du with dot denoting dduddu. Hence, I conclude that the degenerate metric of the Vaidya black-hole horizon should be
ds2=−(1−2m(u)r2)du2−2dudr+r2(dθ2+sin2θdϕ2)ds2=−(1−2m(u)r2)du2−2dudr+r2(dθ2+sin2θdϕ2)
The eveolving horizon locates at r=2m(u)r=2m(u) and thus dr=2m˙dudr=2m˙du with dot denoting dduddu. Hence, I conclude that the degenerate metric of the Vaidya black-hole horizon should be
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3 Killing Vectors & Killing Horizons. 30 ... K.S. Thorne, Black Holes and Time Warps, Picador, 1994. • W. Israel, Dark ... The Schwarzschild metric (1916) is a solution to the vacuum Einstein equations
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