Does an event horizon ever exist?

[PAllen]

Right. I wrote


If the metric components with respect to a particular coordinate system contain non-zero cross-terms, then that particular coordinate system is not orthogonal. It doesn't necessarily mean that an orthogonal coordinate system doesn't exist, but it might be case that an orthogonal coordinate doesn't exist. I think that general results are a bit tricky to obtain.

To sum up this side discussion:

- for general coordinates, you can have any combination light like, space like, and time like coordinates; further their character can change from place to place.

- For orthogonal coordinates, the only possibility is 3 spacelike, 1 timelike.

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My earlier claims about orthogonal coordinates involving null coordinates were incorrect. This follows from two facts:

- two linearly independent null vectors cannot be orthogonal (funny, I helped someone prove this for homework exercise some time ago, but forgot about it for this discussion).

- you can choose 4 vectors at an event such that 3 are orthonormal spacelike, one is null and orthogonal to two of the spacelike vectors; but then it won't be orthogonal to the last one.

These facts rule out any orthogonal coordinates involving any light like coordinate.