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Say we have a metric $g_{\mu\nu}$ and we consider a light-like geodesic. If we integrate over the time coordinate $t$ and find that it takes a finite amount of time for light to travel to $r= \infty $ and then when we integrate over an affine parameter $\tau$ it takes inifinite amount of time to travel to $r= \infty $, then what does this suggest about our spacetime ?

My (GR) intuition tells me that there is a "coordinate" singularity at $r=\infty$(for coordinate $\tau$). Is that what is happening here?

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Coordinates don't mean anything. There is not necessarily a coordinate that is "the time coordinate." If you have a coordinate whose gradient happens to be in a timelike direction, you could say it's a timelike coordinate, but the finiteness of a change in such a coordinate doesn't necessarily have any physical significance.

If we integrate over the time coordinate t and find that it takes a finite amount of time for light to travel to r=∞

It doesn't have the interpretation of time to travel.

and then when we integrate over an affine parameter τ it takes inifinite amount of time to travel to r=∞

Nor does this have the interpretation of time to travel.

My (GR) intuition tells me that there is a "coordinate" singularity at r=∞(for coordinate τ).

This affine parameter is not a coordinate.

In the situation you describe, the interpretation is probably that $t$ just doesn't blow up very fast as you approach null infinity. That's not something that has a physical interpretation, it's just a fact about your coordinate system.

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  • $\begingroup$ Oh, okay well in that case thanks! This answer actually solved a few misconceptions I had. $\endgroup$ Jan 7 '20 at 20:50

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