Worldsheet metric & event horizon

Question

Given a certain metric $g_{alpha beta}$ (not necessarily diagonal) in which $g_{tau tau}=0$ for a certain function, is there any way of determining if there is a singularity in that point, or if it is just a pathology of my choice of coordinates? Does it always mean that if have $g_{tau tau}=0$ I have an event horizon?

Edit
As i read the answer given by Lubos i came to realize that my question was maybe too generic. So I'm going to expand further on:

Consider in the context of AdS/CFT a given timelike trajectory of a heavy quark in SYM $x^{mu}(tau)$, one knows that the quark is dual to a string/brane ending on that trajectory. The surface described in the AdS side is given in the large N limit by the minimal surface wich extremizes the Nambu-Goto Action. Finding that surface for an arbitrary trajectory is a non trivial problem.

For the case of timelike trajectories the embedding of the string $X^{mu}(tau)$ is known due to Mikhailov, with that embedding one can obtain the worldsheet metric $g_{alpha beta}$ wich is known to have some event horizons at $g_{tau tau}=0$. I would like to know how to characterize the structure (maybe topological?) of that event horizon, for example the Kretschmann scalar would be a good starting point?

Luboš Motl · Answer

If one allows an arbitrary metric $g_{rhosigma}$ and just uses the symbol $tau$ for one of the coordinates, i.e. $sigma^0$, then $g_{tautau}=0$ doesn't imply anything special about the point. It just says that the vectors purely in the $tau$ direction are null.

But in the Minkowski signature, there are many null vectors at each point, anyway. So the metric may still be
$$ g_{tausigma}neq 0, quad g_{tautau}=g_{sigmasigma}=0 $$
which is a normal Minkowski metric because $(a+b)(a-b)=a^2-b^2$. This purely off-diagonal way to write the metric of a 2D Minkowski space is known as the light-cone coordinates.

More generally, you asked whether there is unavoidably an "event horizon". The presence of an event horizon cannot be determined locally at all! The event horizon looks just like any other regular smooth region of the spacetime geometry. Whether some point is at an event horizon may only be decided by investigating the whole causal future of this point i.e. a big chunk of the whole spacetime.

Worldsheet metric & event horizon

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