Can light near a black hole travel in -t in external coords?

[TGlad]

Many diagrams show light cones tipping over when closer to a black hole singularity, such that emitted light can have a downwards (negative time) component in the distant observer coordinate frame. e.g this diagram:

or this one:

or this one:

However, other diagrams show that the light cone gets narrower towards the singularity, such that it looks like it emissions never have a downwards component:

So my question is, which version is correct? (for a Schwarzschild black hole, using coordinates of an observer at infinity). Can the light cone ever have a -t component in the distant observer's t coordinate?

 

[PeterDonis]

Many diagrams

Please give a source for these diagrams.

 

[TGlad]

In order:
https://plato.stanford.edu/entries/spacetime-singularities/lightcone.html
http://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/black_holes/index.html
http://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/black_holes_picture/index.html

http://asymptotia.com/2008/03/10/tipping-the-light-cone-black-holes/
http://visualrelativity.com/LIGHTCONE/schwarzschild.html
http://www.scienceforums.net/topic/77581-is-a-black-hole-interior-similar-to-the-universe/

 

[Orodruin]

All of them are correct, just using different coordinates.

 

[PeterDonis]

All of them are correct, just using different coordinates.

I'm not sure the first two diagrams correspond to any coordinates that I'm aware of for Schwarzschild spacetime. The third one seems almost correct for Schwarzschild coordinates, but the light cone placed on the horizon is wrong: it should be squashed to a single line.

The rest of the diagrams look like either Eddington-Finkelstein or Painleve coordinates.

 

[Orodruin]

I'm not sure the first two diagrams correspond to any coordinates that I'm aware of for Schwarzschild spacetime. The third one seems almost correct for Schwarzschild coordinates, but the light cone placed on the horizon is wrong: it should be squashed to a single line.

I think that even if we do not know of any such coordinates, they could in principle be defined. I may be wrong of course. My main point was that how the diagrams look will depend on the choice of coordinates.

What I find misleading in all cases are the general time and space arrows that seem to indicate those directions are always time/space.

 

[PeterDonis]

I think that even if we do not know of any such coordinates, they could in principle be defined.

That may be, but it would be really nice if the articles that showed these diagrams would say what coordinates they are actually using. I strongly suspect that at least the first three diagrams were not constructed using actual coordinate charts, but just by handwaving.

 

[PAllen]

That may be, but it would be really nice if the articles that showed these diagrams would say what coordinates they are actually using. I strongly suspect that at least the first three diagrams were not constructed using actual coordinate charts, but just by handwaving.

I can explain the first three as drawing Schwarzschild coordinates as if they Cartesian, with 't' coordinate (whatever its meaning in different parts of the chart) vertical, putting interior and exterior on the same chart, and ignoring the misbehavior of the metric on the horizon. Whether you approve of such a practices is another question ...

[oops: I didn't see Peter's earlier post - yes, in SC coordinates the cones would narrow towards being lines near either side of the horzion. However, close to the singularity, which is what I was looking at, they are fine.]

 

[TGlad]

I'm not sure the first two diagrams correspond to any coordinates that I'm aware of...
The rest of the diagrams look like either Eddington-Finkelstein or Painleve coordinates

That makes sense. Maybe the first three are somehow Kruskal–Szekeres coordinates.
Anyway, thanks for the information, I think that is useful enough for my level of understanding.

 

[PeterDonis]

Maybe the first three are somehow Kruskal–Szekeres coordinates.

They're not; in Kruskal coordinates the horizon would be a 45-degree line, not vertical, and the singularity would be a hyperbola at the top of the diagram.

 

[Orodruin]

They're not; in Kruskal coordinates the horizon would be a 45-degree line, not vertical, and the singularity would be a hyperbola at the top of the diagram.

Like this: