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

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?

• upload.wikimedia.org/wikipedia/commons/5/56/… Commented May 6, 2020 at 23:08
• Really good Image, I would like to have more information. Commented May 7, 2020 at 3:08
• Well I do not have more information ...I took from the wiki page about anthropic principle.. Commented May 7, 2020 at 14:59

## 1 Answer

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.

• I don't understand how to turn the cone. Commented May 6, 2020 at 6:55
• I added a diagram. Commented May 6, 2020 at 7:15