Falling into a black hole and then shining a light?

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So I asked my teacher what would happen if I fell into a black hole and then shined a light back towards the event horizon. And he said the light would go away from me at c, but also go towards the center of the black hole. OK I understand this would happen if i was in free fall towards the earth, If I threw a ball up while i was in free-fall it would go away from me at a constant speed but also still falling towards earth in a rest frame from earth. But light always travels at c and it seems like it would still travel towards the event horizon and get heavily red-shifted?
 

bcrowell

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One question to ask yourself is which observer observes the red-shift you're imagining. You can't have stationary observers inside the horizon.

When you say that light always travels at c, that isn't the same as saying that light has a coordinate velocity of c when you use the Schwarzschild coordinates.
 
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Thanks for your response . Could I have a stationary observer at the center of the black hole.
 

bcrowell

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Thanks for your response . Could I have a stationary observer at the center of the black hole.
Interesting thought :-) GR doesn't even treat the singularity as a point on the manifold, so GR may not be able to answer that. But I would think that any observer would have to have finite spatial extent, so I would think that such an observer would only be able to exist for some finite time before not being an observer anymore. Also, it seems to me that any two points in an observer (say two neurons in his brain) should be able to have round-trip communication at least once, or else it's dubious that the collection of neurons really constitutes a single observer -- but is this possible for points on opposite sides of the singularity?

I'm actually much more interested by this issue than by the original question. It doesn't really seem to me to have any logical implications for the original question, since there doesn't seem to be any sensible way to find a finite, single-valued result for the Doppler shift as measured by such an observer.
 

Bill_K

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Could I have a stationary observer at the center of the black hole.
It is impossible to have a stationary observer anywhere inside the hole. You can always consider a trajectory that remains at r = constant. However at the surface of the hole r = 2m this trajectory is a null curve; that is, it's the world line of a particle moving at the speed of light. And for r < 2m the trajectory is a spacelike curve, corresponding to a motion faster than light. In other words, inside the hole a stationary observer would have to be a tachyon (and tachyons do not exist.)
 
Hi People
Could a black hole have a region of "normal" space inside it?
 

bcrowell

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It is impossible to have a stationary observer anywhere inside the hole. You can always consider a trajectory that remains at r = constant. However at the surface of the hole r = 2m this trajectory is a null curve; that is, it's the world line of a particle moving at the speed of light. And for r < 2m the trajectory is a spacelike curve, corresponding to a motion faster than light. In other words, inside the hole a stationary observer would have to be a tachyon (and tachyons do not exist.)
I think he was proposing either an observer entirely contained at the singularity or an observer with finite spatial extent centered on the singularity. In either case, your argument doesn't apply.
 

George Jones

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I think he was proposing either an observer entirely contained at the singularity or an observer with finite spatial extent centered on the singularity. In either case, your argument doesn't apply.
The singularity is not part of the spacetime manifold, so it is not possible to be contained at or centred on the singularity. There are no timelike Killing vectors inside the black hole, so no events inside the black hole are in stationary regions. Even if r = 0 were part of the manifold, (loosely) r = 0 is an infinite line line, not a point, so again, I don't see how it is possible to be contained at or centred on r = 0.
 
Remember that the singularity r=0 is spacelike. It's a moment in time, not a point in the middle, and you can't aim anything away from it, any more than you could fire a bullet back in time. Black holes only look like point masses from the outside! Inside, the singularity is spread out over all of space. It's in your future :).

A penrose diagram is usually the easiest way to see what is going on in these types of situations.
 
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OK what if i had someone go into the black hole first then wait a little bit and then send some one else behind him, and then have the first person send a pulse of light back towards him, if this is even possible . Or have the second person send a light pulse towards the center at the first person. And before we go into the black hole each person will know the frequency of our light source and then they will measure the frequency that they receive.
 

pervect

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Well, as MTW points out, [itex]\partial / \partial r[/itex] is a timelike vector in Schwarzschild coordinates inside the event horizon. Another less-precise but more understandable way of saying this is that 'r' is a time coordinate inside the event horizon, not a space coordinate.

So, to borrow a bit from starwars, it is your destiny to fall into the center of the black hole - it's also the destiny of any light beam you may emit. The direction towards r=0 is the direction of your future.

If you look at the space-time plot of r vs t, you won't get very far interpreting it correctly until you realize that r and t have "switched roles" in this manner. Confusing, but that's the way it is. A quote from my textbook, for those who want references (which is good!).

MTW said:
The most obvious pathology at r = 2M is the reversal there of the roles of t and r as timelike and spacelike coordinates.
....
What does it mean for r to change in character from a spacelike coordinate to a timelike one? The explorer in his jet powered spaceship prior to arrival at r=2M always has the option to turn on his jets and change his motion from decreasing r (infall) to increasing r (escape). Quite the contrary is the situation when he has once allowed himself to fall inside r=2M. Then the further decrease of r represents the passage of time. That unseen power of the world which drags everyone forward willy-nilly from age twenty to forty, and from forty to eighty also drags the rocket in from time coordinate r=2M to the later value of the time coordinate r=0. No human act of will, no engine, no rocket, no force (see exercise) can make time stand still. As surely as cells die, as surely as the traveler's watch ticks away "the unforgiving minutes,", with equal certainty, and with never one halt along the way, r drops from 2M to 0.
As far as shining lights out - while they won't reach the event horizon, ever, you would still be able to see the back end of a space-ship falling into a black hole from the front end. You can solve for the paths that light takes - to correctly interpret them, though, you need to realize that inside the event horizon, "the future" is represented by r going to zero, and that you'd measure the 't' coordinate inside the black hole with a ruler, not with a clock.
 
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thanks for your response pervect. your text book is mtw, I should start reading it .
So you measure time with a ruler inside the BH interesting . So I take it my clock would stop inside the black hole . Do you have any recommendations on books that i could read to start to learn about the crazy physics inside a black hole?
 
thanks for your response pervect. your text book is mtw, I should start reading it .
So you measure time with a ruler inside the BH interesting . So I take it my clock would stop inside the black hole . Do you have any recommendations on books that i could read to start to learn about the crazy physics inside a black hole?
No, no, it's the coordinates that change their meaning, not your clocks and rules. Your clock will be increasing with decreasing r coordinate inside the event horizon, and your ruler is capable of probing infinitessimal changes in t at fixed r.
 

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