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For any three points in Minkowski spacetime plausibly regarded as vertices of a triangle, is it possible that the interior angles of the resulting triangle sum to anything other than 180 degrees? I thought the answer to this question was straightforwardly 'no', because Minkowski spacetime is flat (i.e., it has zero curvature). That is, only in spacetimes of positive curvature can the interior angles of a triangle sum to more than 180 degrees, and only in spacetimes of negative curvature can the interior angles of a triangle sum to less than 180 degrees. But I'm confused because I've heard (perhaps mistakenly) that, with respect to Minkowski spacetime, only triangles whose vertices are spacelike separated have interior angles summing to 180 degrees.

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What you heard is right in the following sense:

Define a triangle (for which the notion of angles between sides is meaningful) as any three points which can by the appropriate Lorentz transformation be brought into three points having identical time coordinates. This is possible if and only if the three points which are pairwise spacelike separated. If this triangle can be defined for the given three points, then the same-time triangle incorporating those three spacetime points is unique, and the sum of its angles will be $180^\circ$.

In contrast, consider a triangle having two points with spacelike separation and the third separated timelike separated from each of the other two. The best you can do is go to a reference frame in which points A and B are at the same time, and point C is somewhere along the line from A to B and at a later time. In that case, what would you mean by the "angles" ABC or ACB? The other several combinations of spacelike and timelike separation have similar issue.

So the people who say the angles add up to $180^\circ$ only when the points are spacelike separated, really should add "and when they aren't spacelike separated the angles are meaningless."

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  • $\begingroup$ That's helpful. Does the same situation arise in spacetimes of constant negative (or constant positive) curvature? That is, in such spacetimes is it also the case that the interior angles of triangles add up to less (or more) than 180 degrees only when the points of the vertices are spacelike separated, and when they aren't so separated, the angles are meaningless? $\endgroup$ – bmcdanie Dec 21 '16 at 17:57

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