Consider the following metric
ds2=V(dx+4m(1−cosθ)dϕ)2+1V(dr+r2dθ2+r2sin2θdϕ2),
where V=1+4mr.
That is the Taub-NUT instanton. I have been told that it is singular at θ=π but I don't really see anything wrong with it. So, why is it singular at θ=π?
EDIT:: I have just found in this paper that the metric is singular "since the (1−cosθ) term in the metric means that a small loop about this axis does not shrink to zero at lenght at θ=π" but this is still too obscure for me, any clarification would be much appreciated.
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