# Worldsheet metric & event horizon

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?

If one allows an arbitrary metric $g_{\rho\sigma}$ and just uses the symbol $\tau$ for one of the coordinates, i.e. $\sigma^0$, then $g_{\tau\tau}=0$ 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 $$g_{\tau\sigma}\neq 0, \quad g_{\tau\tau}=g_{\sigma\sigma}=0$$ 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.