Measurement of distances in the presence of a black hole

[notknowing]

In SR and GR, length or distances are obtained by what I call a "procedure" using light and clocks. This definition or procedure deviates from the more familiar ruler distance but it is the most practical solution to obtain distances between two objects in relative motion. Consider now an observer outside of the Schwarzschild radius of a black hole and another "observer" inside the Schwarzschild radius. How will the outside observer measure the distance between him and the inside observer (as no light signal can be sent back from the inside observer) ? The procedure to measure distance seems no longer to work. So for the outside observer, even the whole concept of geometry seems to have lost its meaning for the region inside the Schwarzschild radius. Probably, I'm making some mistake here. Can someone help me out ?

 

[NateTG]

So for the outside observer, even the whole concept of geometry seems to have lost its meaning for the region inside the Schwarzschild radius.

Aside from the notion of 'outside observer' making no sense either, I think you're getting it. That's why people talk about 'singularity'.

 

[notknowing]

Aside from the notion of 'outside observer' making no sense either, I think you're getting it. That's why people talk about 'singularity'.

As far as I know, singularities have to do with infinite curvature or forces (leaving aside coordinate singularities), so this would (with the present equations) only occur in the centre of the black hole and not at some arbitrary radius smaller than the Schwarzschild radius.

 

[pervect]

In SR and GR, length or distances are obtained by what I call a "procedure" using light and clocks. This definition or procedure deviates from the more familiar ruler distance but it is the most practical solution to obtain distances between two objects in relative motion. Consider now an observer outside of the Schwarzschild radius of a black hole and another "observer" inside the Schwarzschild radius. How will the outside observer measure the distance between him and the inside observer (as no light signal can be sent back from the inside observer) ? The procedure to measure distance seems no longer to work. So for the outside observer, even the whole concept of geometry seems to have lost its meaning for the region inside the Schwarzschild radius. Probably, I'm making some mistake here. Can someone help me out ?

It's wortwhile to consider this problem in more detail.

case 1). An observer outside the event horizon hovers on his rockets, while a free-falling observer falls through the event horizon. In this case, there is no way to measure the "radar distance".

case 2) An observer outside the event horizon free-falls through the event horizon chasing an observer who has alread passed throuh the horizon. in this case, there IS a way to measure the radar distance.

The two cases strengthen to similarity between the event horizon of a black hole, and the Rindler horizon of an accelerating observer. It's not really the geometry of the BH at the horizon that's confusing - it's understanding the geometry seen by an accelerated observer (the Rindler metric, and the Rindler horizon).

 

[notknowing]

It's wortwhile to consider this problem in more detail.

case 1). An observer outside the event horizon hovers on his rockets, while a free-falling observer falls through the event horizon. In this case, there is no way to measure the "radar distance".

OK, but if we concentrate on case 1 only, you agree there is no way to measure the "radar distance". If there is no other way of measuring distances for this observer (to a point inside the Schwarzschild radius), then, strictly speaking, to the outside observer, space itself actually "ends" at the Schwarzschild radius. It is as if a volume a space has been taken out of the original space (compared before the formation of the black hole) and any thoughts, equations, etc, about the inside are meaningless (to this observer).

 

[robphy]

OK, but if we concentrate on case 1 only, you agree there is no way to measure the "radar distance". If there is no other way of measuring distances for this observer (to a point inside the Schwarzschild radius), then, strictly speaking, to the outside observer, space itself actually "ends" at the Schwarzschild radius. It is as if a volume a space has been taken out of the original space (compared before the formation of the black hole) and any thoughts, equations, etc, about the inside are meaningless (to this observer).

"strictly speaking, to the outside observer, space itself actually "ends" at the Schwarzschild radius" might be a little too strong. Nothing prevents an outside observer from crossing through the horizon. More precisely, there are events within the horizon that are in the causal future of an outside observer.

In accord with your observation about not having a way for an outside observer to measure interior events via radar, the events within the horizon have causal futures that are always within the horizon. So, interior observers also can't use radar to measure events outside the horizon. But it might also be too strong to say that space ends for them at the horizon... since stuff can cross over from the outside.

 

[pervect]

Let's say I take of in a space-ship, and accelrate at 1G. Does the Earth "cease to exist" or become part of a region where "space-time ends" 1 year into my journey, when the Earth falls behind the Rindler horizon of the accelrating rocket ship? This is somewhat of a philosophical question, I suppose, but I would basically say the answer is no.

If you want to address it experimentally, by ceasing his acceleration the Rindler observer can again observe the Earth, though he won't be able to if he continues to accelerate.

As robphy points out, the situation is rather similar for the observer hovering at the event horizon - if he "cuts his jets", he will also see the object on his radar - but only after he crosses the event horizon. The main difference is that we expect that there is a significant downside to crossing the event horizon of a black hole.

 

[notknowing]

"strictly speaking, to the outside observer, space itself actually "ends" at the Schwarzschild radius" might be a little too strong. Nothing prevents an outside observer from crossing through the horizon. More precisely, there are events within the horizon that are in the causal future of an outside observer.

In accord with your observation about not having a way for an outside observer to measure interior events via radar, the events within the horizon have causal futures that are always within the horizon. So, interior observers also can't use radar to measure events outside the horizon. But it might also be too strong to say that space ends for them at the horizon... since stuff can cross over from the outside.

I'm not so sure that my statement is too strong. Physics deals with the observable universe and observations are always made from the viewpoint of a particular observer at a particular time and place and in a particular situation or environment. The laws of physics should make sense to an arbitrary localised observer. It doesn't help to say "yes, but the other observer (which falls into the black hole) sees another thing". If we remain strictly logical and consistent, we should stick to our point of view. The same applies to a lot of other situations, such as for length contraction (one observer in a spaceship can see the other spaceship shorter while the observer in the other spaceship concludes the opposite and nevertheless both are right, from their own perspective) or a lot of situations in quantum physics. Therefore, for the outside observer, space inside the Schwarzschild radius is not defined or measurable and therefore non-existent - TO this local observer. I will even go a step further (and probable shock some people) : Since the space inside the Schwarzschild radius (R) is non-existent (to the outside observer), it can also not contain energy (again to this outside observer). This means that every mass which seems to disappear from him at R is in fact no disappearing from him at all (because conservation of energy remains a valid law to him) and must therefore be converted into gravitational energy - outside of R. So, if one could somehow calculate the integral of the gravitational energy from R to infinity one should find that it equals exactly the mass of the black hole. I do not know how to calculate the energy in the gravitational field, but it should be there. Gravitational energy is a real thing, otherwise it would not be possible to radiate energy away from a rotating star system in the form of gravitational waves.

 

[robphy]

I'm not so sure that my statement is too strong. Physics deals with the observable universe and observations are always made from the viewpoint of a particular observer at a particular time and place and in a particular situation or environment. The laws of physics should make sense to an arbitrary localised observer. It doesn't help to say "yes, but the other observer (which falls into the black hole) sees another thing".
...snip...
Therefore, for the outside observer, space inside the Schwarzschild radius is not defined or measurable and therefore non-existent - TO this local observer.

Let me expand on this idea: "there are events within the horizon that are in the causal future of an outside observer". If you decide (or are somehow forced) to cross the horizon, those events "inside the horizon that were causally disconnected from you in your past" contribute to the determination of the rest of your future. Should those events have been denied existence because they had not been accessible to you for some period? If their existence is denied, I think you'd lose the ability to fully make predictions [and retrodictions] about your possible future.

Certainly, you could CHOOSE to stay outside the horizon...just as you could CHOOSE not to visit, say, a remote island that you never heard about and would have no impact on your life outside of it. But since nothing prevents you from crossing over and visiting, would you deny the existence of those events [of that island]?

If your causal future and past were completely disconnected from those of the events inside the horizon (so that nothing at all can cross over), then I think your statement has a better footing. [However, there are probably subtle issues concerning
observationally indistinguishable spacetimes.]

 

[notknowing]

Let me expand on this idea: "there are events within the horizon that are in the causal future of an outside observer". If you decide (or are somehow forced) to cross the horizon, those events "inside the horizon that were causally disconnected from you in your past" contribute to the determination of the rest of your future. Should those events have been denied existence because they had not been accessible to you for some period? If their existence is denied, I think you'd lose the ability to fully make predictions [and retrodictions] about your possible future.

Certainly, you could CHOOSE to stay outside the horizon...just as you could CHOOSE not to visit, say, a remote island that you never heard about and would have no impact on your life outside of it. But since nothing prevents you from crossing over and visiting, would you deny the existence of those events [of that island]?
If your causal future and past were completely disconnected from those of the events inside the horizon (so that nothing at all can cross over), then I think your statement has a better footing. [However, there are probably subtle issues concerning
observationally indistinguishable spacetimes.]

Thanks for your comments. I really needed some time to think about this.

In fact we really don't know if its possible to cross the horizon at all. Everybody thinks it is possible and one writes a lot of equations about it, but nobody has actually seen something crossing over. One can observe an accration disk but that's all. And if someone sends his spaceship on a course into a black hole, he could not send signals back to tell us how it is like. It is just like a life-after-dead situation. Maybe there is life after death but when we can not proof it or measure it, it belongs to philosophy (or religion) and not to science. I realize that I take a hard (and unusual) standpoint here but I feel it is defendable. So I could well say that all scientists who are working hard on thinking about the region inside the event horizon are actually philosophers and that they should publish in philosophical journals If you say "there are events within the horizon that are in the causal future of an outside observer", then I could ask "How do you know, were you there ?" or "How do you know there is anything at all there ?". And then you can only say " I suppose it because my equations, bla, bla, bla, ..." again - philosophy.

Now, coming to the island situation. There is some difference to the black hole case because signals and information about the island could reach easily the distant traveler (observer). If the observer has never heard about the island, and no light or signal from this island has the possibility to reach him, then for this observer and at that moment, the island really does not exist - it is not part of his observable universe. He can deny its existence (in that situation). Suppose now that he CAN cross some magical gate and finds the island then at that moment and in this situation, it IS part of his reality. If he crosses now back the magical gate into his previous world, and no signal is able to cross the gate, he might think that there is some island behind it, but he could equally have dreamed about it - he becomes a philosopher about what could be really behind the gate

We are however deviating a lot from my original question, namely how does an outside observer (again black hole case) define or measure the distance between him and another observer inside the Schwarzschild radius ? As far as I can tell now, he can not.

 

[robphy]

Thanks for your comments. I really needed some time to think about this.

In fact we really don't know if its possible to cross the horizon at all. Everybody thinks it is possible and one writes a lot of equations about it, but nobody has actually seen something crossing over.

As you are hopefully alluding to, the equations of GR say that nothing particularly special happens locally at the horizon. In other words, the instant an observer crosses the horizon, no detector will go off and say "all your signal are belong to us". Of course, in all of these discussions, we are discussing the properties of timelike curves in a Lorentzian manifold as model of a hypothetical observer.

If you say "there are events within the horizon that are in the causal future of an outside observer", then I could ask "How do you know, were you there ?" or "How do you know there is anything at all there ?". And then you can only say " I suppose it because my equations, bla, bla, bla, ..." again - philosophy.

GR is a theory that let's us model possible universes. Aspects of that theory have been tested and agree with GR's predictions. Many other aspects have not been tested because of current experimental limitations. However, it's more than philosophy. It's taking a mathematical model that makes predictions that have agreed with experiment and seeing what else the model predicts. If you draw a spacetime diagram of the Schwarzschild spacetime, you'll see that there are [idealized] events (points of the diagram) inside the horizon can influence (because a causal curve can meet) a timelike curve inside the horizon.

Yes, there may be some philosophy involved (along the lines of Wigner's "The Unreasonable Effectiveness of Mathematics in the Natural Sciences")... but the mathematical model is laid out for all to see. You are free to disagree with the model and find your own model with alternate assumptions and predictions.

Now, coming to the island situation. There is some difference to the black hole case because signals and information about the island could reach easily the distant traveler (observer). If the observer has never heard about the island, and no light or signal from this island has the possibility to reach him, then for this observer and at that moment, the island really does not exist - it is not part of his observable universe. He can deny its existence (in that situation). Suppose now that he CAN cross some magical gate and finds the island then at that moment and in this situation, it IS part of his reality. If he crosses now back the magical gate into his previous world, and no signal is able to cross the gate, he might think that there is some island behind it, but he could equally have dreamed about it - he becomes a philosopher about what could be really behind the gate

The implication of the distant island is that it does not have technological advances [or political freedom] to communicate to the outside world... and to push the analogy further... once you enter, you lose the ability to leave or communicate. By crossing over into the island, if you find a civilization there, would you conclude that they [the island, its people, and its history] just appeared for your benefit? Or would you logically conclude that they were there all along, but just out of communication with the outside world?

We are however deviating a lot from my original question, namely how does an outside observer (again black hole case) define or measure the distance between him and another observer inside the Schwarzschild radius ? As far as I can tell now, he can not.

By radar, as you have said, the outside observer cannot measure the distance to an event inside the horizon. However, it seems that it should be possible for the outside observer to travel into the horizon and, using his wristwatch and other measuring devices, determine how far he has traveled to get to an observer inside the horizon. Of course, the information is of no use to an outside observer that remains outside. Nevertheless, the information can be determined.

 

[pervect]

Yes, there may be some philosophy involved (along the lines of Wigner's "The Unreasonable Effectiveness of Mathematics in the Natural Sciences")... but the mathematical model is laid out for all to see. You are free to disagree with the model and find your own model with alternate assumptions and predictions.

Note, though, that here at PF such a model would belong in the "independent research forum" of PF (assuming it met the other requirements) and not the GR forum.

Not that PF is necessarily the best place to publish such a model, as I would guess you're probably aware having asked about other venues to publish papers.

 

[robphy]

Note, though, that here at PF such a model would belong in the "independent research forum" of PF (assuming it met the other requirements) and not the GR forum.

Not that PF is necessarily the best place to publish such a model, as I would guess you're probably aware having asked about other venues to publish papers.

Just to clarify...
although you've quoted my comment,
I believe your comment is directed to the OP and not me.

 

[pervect]

Sorry about that - yes, I quoted you, but my remark was directed to the OP. Since event horizons occur even in SR with accelerated observers, I personally think it will be extremely difficult and perhaps even impossible to eliminate them from GR.

Of course the OP is certainly welcome to disagree and come up with an alternate theory - he won't even be the only one attempting to do so. Ed Schaeffer in Wikipedia is apparently trying to do the same thing, for example. (Of course I suppose it's possible that our OP is Ed Schaffer :-)).

 

[Chris Hillman]

Distance between interior and exterior observers?

Hi again, notknowing,

In SR and GR, length or distances are obtained by what I call a "procedure" using light and clocks. This definition or procedure deviates from the more familiar ruler distance but it is the most practical solution to obtain distances between two objects in relative motion. Consider now an observer outside of the Schwarzschild radius of a black hole and another "observer" inside the Schwarzschild radius. How will the outside observer measure the distance between him and the inside observer (as no light signal can be sent back from the inside observer) ? The procedure to measure distance seems no longer to work. So for the outside observer, even the whole concept of geometry seems to have lost its meaning for the region inside the Schwarzschild radius. Probably, I'm making some mistake here. Can someone help me out ?

I assume you have in mind "radar distance": observer A sends out a radar pip which strikes object B and returns to A; A then divides by twice the elapsed time for the round trip as measured by an ideal clock he carries and calls the result the radar distance. (I am using relativistic units, so that the result can be taken as either a time or a distance.) This is indeed one of the simplest notions of "distance in the large" which one can employ in curved spacetimes (see also the discussion in Landau and Lifschitz, Classical Field Theories). When we state the procedure as I just did, I think the problem is obvious: the exterior observer can send a radar pip past the event horizon, but he ain't going to get it back! On the other hand, the interior observer can't send a radar pip to an object outside the horizon.

As pervect said, with sufficient care, you could construct a scenario in which an infalling observer sends a radar pip after an object which has already fallen through through the horizon just BEFORE he himself falls past the horizon, such that he does receive a return after he falls past the horizon. If you sketch the world lines you'll see that this requires some care, however.

In fact we really don't know if its possible to cross the horizon at all. Everybody thinks it is possible and one writes a lot of equations about it, but nobody has actually seen something crossing over.

There is no question whatever that gtr predicts that event horizons exist and that interiors of black hole solutions are perfectly admissable (and accessible!) regions of spacetime models whch feature a black hole. These are theorems.

However, the question of whether or not gtr agrees even qualitatively with Nature in circumstances where we expect event horizons and black hole interiors is a different kind of question. As I have often pointed out (most recently in another PF thread earlier today!), it is interesting that the notion of a black hole poses a rather strange challenge to a standard (and somewhat naive) picture of how theory relates to experiment in physics, because the theory predicts that a physicist can fall into a hole and make some measurements, but it also predicts that he can't report his findings back to his colleagues in the exterior region!

But I disagree with your suggestion that discussions of black hole interiors are "not physics but philosophy". This simply means that physicists who choose to avoid falling into any black holes must rely on more indirect evidence than we might like, but this is really not so very different from ordinary physics in labs all around the world.

(Ask yourself, after Mach, who has ever seen an atom? Yes, physicists can now image individual atoms, or so they say, but think about all the data processing which produces those nifty images from an electronic signal from a rather complicated instrument which has been designed using physical theory which predicts the existence and properties of atoms, and which produces singals which is interpreted using same. Sure, there is a lot of "circular reasoning" under the surface here, but how could it be otherwise? We need to use SOME theory to interpret our data, and to design any nifty technology, such as a powerful scientific instrument, we need to use some theory. Ultimately, one might say that the practice of experimental physics amounts to making convoluted consistency checks which we hope will ultimately give unambiguous warning that something has broken down, should theory be grossly incorrect in some situation. I consider this hope not unreasonable, but to some extent I guess one could argue that my attitude constitutes an act of faith: the universe is subtle, but there is no reason I can see to think it is "designed" just to mislead us!)

I would also draw your attention to something I mentioned in another PF thread: as Chandrasekhar and his students discovered, according to gtr, in principle it is possible to produce a region of spacetime which is locally isometric to the region of interest (think roughly of m < r < 2m) by finding some quiet region of spacetime and arranging the collision of two carefully crafted gravitational plane waves. In this scenario, signals will always propagate from the inside-analogous region to an outside-analogous region. So experimental exploration of the local geometry of the "near interior" is in principle not out of reach to experimenters who wish to publish their results :-/

Needless to day, right now no-one knows how to actually try such an intriguing experiment, but there is no question about what these special cases of "colliding plane wave" (CPW) models predict.

Chris Hillman