A Maximizing survival time when falling into a black hole

[Ibix]

Thanks @PeterDonis.

I think @PAllen makes good points. I would only make one additional point (with a degree of sophistry, it must be said) which is that the crossing point of the time-like curves doesn't exist in a realistic black hole.

I think I see the point about there needing to be spacelike geodesics in my spacelike volume to be able to call it "space" rather than a space-like foliation. If the natural concept of "going straight ahead not advancing in time" takes you out of the volume then it's a bit difficult to argue that the volume represents a "slice of spacetime at a single time".

Would you argue that the volumes in Minkowski spacetime defined by equal Rindler time aren't "space" on this basis?

 

[PAllen]

In cosmology, pick any comoving timelike geodesic. Then pick any space like geodesic 4 orthogonal to it. This geodesic will not remain in a slice of constant cosmological time. No one pretends this makes the standard cosmological foliation useless or uninformative.

 

[PeterDonis]

I still think the killing vector based foliation of the interior is the least arbitrary and most informative

Ultimately this is a matter of preference, since as you note no physics is affected by a coordinate choice. I agree that the points you bring up are valid ones to consider when making such a choice.

 

[PeterDonis]

crossing point of the time-like curves doesn't exist in a realistic black hole.

Yes, that's correct; it only exists in the maximally extended manifold, which is not physically realistic.

 

[PeterDonis]

Would you argue that the volumes in Minkowski spacetime defined by equal Rindler time aren't "space" on this basis?

These correspond to "space" outside the horizon, not inside it. And this particular foliation is a geodesic one (in flat spacetime, not the curved spacetime outside a black hole).

 

[PAllen]

Just another thought on the importance of spatial 3-cylinders in the SC BH interior. In flat spacetime, and in the asymptotically flat BH exterior it is impossible construct/embed a spacelike 3-cylinder at all. That this is possible in the interior is telling us something significant about the difference in geometry between the interior and exterior.

 

[PeterDonis]

in the asymptotically flat BH exterior it is impossible construct/embed a spacelike 3-cylinder at all

If we don't include the origin of the Kruskal diagram in the "exterior", this is true. But spacelike surfaces passing through the origin of the Kruskal diagram, and extending to infinity in both directions (the right-hand wedge and the left-hand wedge of the Kruskal diagram) are spacelike 3-cylinders. The origin is on the horizon, so it's not strictly speaking "exterior", but it's not "interior" either.

In the more realistic geometry of an actual black hole formed by gravitational collapse, the point corresponding to the origin of the Kruskal diagram is not there, nor is any of the left-hand wedge (the region occupied by the collapsing matter is there instead), so in that geometry, yes, the only spacelike 3-cylinders are in the BH interior. (These 3-cylinders "end" in one direction inside the collapsing matter, but they extend infinitely in the other direction, at least for the classical case where Hawking radiation is excluded.)

 

[PAllen]

If we don't include the origin of the Kruskal diagram in the "exterior", this is true. But spacelike surfaces passing through the origin of the Kruskal diagram, and extending to infinity in both directions (the right-hand wedge and the left-hand wedge of the Kruskal diagram) are spacelike 3-cylinders. The origin is on the horizon, so it's not strictly speaking "exterior", but it's not "interior" either.

By geometric 3-cylinder, I am including the fact that the area of 2-sheres is constant. So, no, the best you can do including the whole horizon is to have a 3-cylinder whose axis is lightlike. In fact, what you are describing, as you get further and further from the origin, becomes geometrically more and more just concentric spheres about a common origin (as the spacetime becomes asymptotically flat).

 

[PeterDonis]

By geometric 3-cylinder, I am including the fact that the area of 2-sheres is constant.

Ah, ok.