Changing the signature of the metric and consecuences in the causal structure of the spacetime

Question

I was reading about the implications in the causal structure due to a change in the signature of the metric.

I know that if you choose a spacetime $(M,g_{ab})$. With $g_{ab}$ a Lorentzian metric, of signature $(-,+,+,+)$, in the Minkowski space time you can construct globally the causal structure of the spacetime, and that structure is given by the light cones.

With causal structure I mean that: If one event $q\in M$ can causal influence another event at the point $p\in M$, then $q$ is inside (or over) the past light cone of $p$.

However, if you choose another type of signature, for example $(-,-,+,+)$, and if you analize this "spacetime" you can't recover the causal structure because "It is not possible to distinguish a past from a future time-like direction and hence order events, even locally". But I can't see why this happens. Can someone help me?

Answer 1

You can now have a closed curve (like a circle) in the timelike directions. Take a light-cone with 2 spatial dimensions, but now switch time and space by turning the cone so that time is now 2D and space 1D.

The timelike axes are now t and y, and the spatial axis is just x.