General Relativity Question

[dimachka]

Im trying to figure out about what happens as an observer falls into a black hole using the swarzschild geometry. Assuming the person is somehow able to not fall apart, I'm sure that they can see their feet before the event horizon, after the event horizon and even during the event horizon. (Drew an eddington-finkelstein diagram). What I am unsure about is whether this observer will ever see her feet hit the singularity assuming she doesn't fall apart.

My thoughts on this are that any two points on the worldline of her feet and head not at r=0 are connected by a null geodesic and thus except at the singularity the observer can always see her feet (of course at a different time than where her head currently is) However, i am not sure how this works as she gets to the singularity. Since any non-zero points are connected, it seems she can get arbitrarily close and still see her feet, but I don't know whether that means she ever actually sees her feet hit the singularity. Thanks for the input.

BTW, this is question 12.13 in Hartle's book.

 

[George Jones]

My thoughts on this are that any two points on the worldline of her feet and head not at r=0 are connected by a null geodesic

Is this what you really meant to say?

There are two different worldlines, one for her feet and one her head.

From any event on the worldline of her feet, is it always possible to draw a null geodesic that intersects the worldline of her head.

Remember, the singularity is spacelike.

 

[dimachka]

the singularity is at r=0, that is why i said that is true for all point r=!0. It seems like although when her head is approaching the singularity, the observer will be able to see her feet, she will see them still away from the singularity. (past the horizon the null geodesics curve into the singularity) Am i thinking about this correctly?

EDIT: i see what you were saying, no i did not mean to say any two points are connected. But any point on the head's worldline, is connected to some point on the foot's worldline by a null geodesic outside of the singularity.

EDIT2: I suppose my difficulty is that general relativity doesn't say anything about what is happening right at the singularity and the question asks whether the observer will see her feet AT the singularity, am i missing what G.R. can say about this question?

 

[George Jones]

Add another timelike worldline to either Figure 12.6 or 12.7. Can, even near the singularity, a 45 degree line, be drawn from the bottom worldline to the top worldline?

Remember, the singularity (the hyperbola r = 0) is spacelike.

 

[George Jones]

Here's another way to look at it.

On a Kruskal-Szekeres diagram, draw a 45 degree line from the "event" of the head crashing into the singularity back to the worldline of the feet.

What does this tell you?