error in mathematics behind black holes

See http://arxiv.org/abs/gr-qc/9707012, lecture notes on black holes
written by PK Townsend of Cambridge.

The analytic continuation procedure that he gives on page 17 ignores
the fact that the value of v would be equal to negative infinity when
r=2M for any value of t, so even though it looks like there are no
apparent singularities in the metric in the formula that he created
from the change of coordinates, the fact that v would be negative
infinity makes the analytic continuation formula invalid. (Furthermore,
r and v are not independent variables, as v is directly dependent on r
whenever r=2M.)

Therefore, the assertion that GR predicts black holes is quesitonable.
Anyone out there disagree?

Craig

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