# What is an equation which tells you if two points are time-like separated in curved spacetime?

I was thinking about how to tell if two points in a general spacetime are time-like separated. I think the only way of doing this is checking every path from $$x$$ to $$y$$ and seeing if one of the paths remains time-like on the entire path.

Checking every path means using a path integral. So I want to find a functional $$F[g,y,z]$$ which takes a metric $$g_{\mu\nu}(x)$$ and two space-time points $$y$$ and $$z$$ and is equal to say $$1$$ if the points are time-like separated, $$-1$$ if they are space-like and $$0$$ if they are on a light cone.

So I'm thinking it would look something like this (Just a wild guess as an example of the kind of thing I'm looking for):

$$F[g,y,z] = \theta\left(\int\limits_{x(0)=y}^{x(1)=z} \exp\left( \int\limits_{0}^{1}\sqrt{g}g_{\mu\nu}(x(\tau))\dot{x}^{\mu}(\tau)\dot{x}^{\nu}(\tau)d\tau\right) Dx\right)$$

Where $$\theta$$ is some as yet unspecified function.

Can you help me find such an equation?

• That's a problem in global analysis. – Qmechanic Mar 19 at 16:51
• @Qmechanic Interesting. So is there a well known solution? – zooby Mar 19 at 17:33
• You asked essentially the same question with a different approach: physics.stackexchange.com/q/536962 – user195162 Mar 19 at 18:29