Quote by robphy
To draw these light cones in the spacetime diagram of a spacetime, curved or otherwise, one must identify all of the lightlike geodesics (a coordinate invariant notion). At a particular event, its light cone is determined by the tangents to these geodesics.

Sure I follow what you say here.
Quote by robphy
Depending on your choice of coordinates, the image of these geodesics in your coordinate chart may trace out all sorts of crazy looking paths (akin to the distortions one gets from various map projections of the earth). In some cases, the image of these light cones may look tipped or distorted relative to the images of other light cones in your coordinate chart.

Sure in certain coordinates.
Quote by robphy
Regardless of appearances in your chart, the physics is determined by the lightlike geodesics, essentially telling you which events are in the causal future [and causal past] of events in spacetime (i.e. the causal connectivity of events). The worldlines of observers are bounded by these lightlike geodesics.

Completely agree!
Quote by robphy
More correctly, these show that closed time loops are mathematically possible, given the constraints imposed in the situation. In other words, saying that one has a 4manifold with a Lorentziansignature metric places some restrictions on what "physics" is possible. However, by themselves, they don't restrict the possibility of closed time loops or other pathologies. Even imposing the field equations might still allow pathologies. That is why one is led to the notion of "causality conditions" and the study of "causal structure", which were developed using "global methods" (i.e. geometric, coordinatefree methods). One may also impose other conditions like "energy conditions", "asymptotic conditions", etc...

Well at one point the time part of the geodesic has to connect to another time part of the same geodesic while the spatial parts are irrelevant. Apart from a closed spacetime or a wormhole I do not see how that can be the case. Can you?
Quote by robphy
When I find the time, I'll try to address your concern about "switching"

Please do, to me it makes absolutely no sense.