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My idea to attempt to proceed along these lines would be to generalize the idea of angle, but I don't have anything that I feel I can point to.

As a starting point, I'd like to ask - if we have a flat Minkowskii spacetime, and form a triangle form one timelike geodesic and two null geodesics, is there a meaningful concept of the "angles" of this triangle that sum to 180 degrees? Possibly based on using the dot product to determine the angle?

One example of such a triange would be setting the three points of the triangle as (t,x) given by (0,0) (2,0) (1,1)