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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?

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

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