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 Incoming links: Reissner-Nordstrom Kretschmann scalar

Past exam of the general relativity II course of the University of Oxford / 2015 / 2 / b 2026-10-25

For the three cases, distinguish a curvature singularity from a coordinate singularity. The Reissner-Nordstrom Kretschmann scalar is

K = R_{α β γ δ} R^{α β γ δ} = \frac{48 M ^{2}}{r ^{6}} - \frac{96 M Q ^{2}}{r ^{7}} + \frac{56 Q ^{4}}{r ^{8}} .

It diverges at

r = 0

when either

M

or

Q

is nonzero. Any positive root of

F

instead admits Eddington-Finkelstein coordinates: set

v = t + \int d r / F

to obtain

d s^{2} = - F d v^{2} + 2 d v d r + r^{2} (d θ^{2} + sin^{2} θ d ϕ^{2}) .

The determinant of its

(v, r)

block is

- 1

, including at

F = 0

. Hence positive roots of

F

are removable coordinate singularities of the static chart. The apparent degeneracy at the poles of the angular chart is also a coordinate singularity.

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