Value of g near a black hole (re-visited)

[PAllen]

Let's be clear about one thing. The statement that no one outside the horizon sees, or is causally influenced, by anything inside, does not imply that nothing exists inside. Let me describe Rindler horizon scenario. Two ships are accelerating together at 1 g for a long time. They have reached extremely near c. One of them runs out of fuel and stops accelerating. The other ship will see the out of fuel ship fall a little behind, but then become red shifted, clocks on it slow down and asymptotically stop; red shift grows to infinity.The empty ship becomes invisible. The empty ship is never seen to be farther than a short distance away from accelerating ship, as long as it can be seen at all. It is 'trapped' on the Rindler horizon. Of course, for the empty ship, nothing strange has happened. The other ship accelerates away from it, getting ever further away. The empty ship can receive signals from the accelerating one, but any signals it sends can never reach the accelerating ship (because the accelerating ship stays ahead of the light; no contradiction because it had a head start and keeps accelerating - no superluminal implication).

This scenario has much similarity to the black hole event horizon.

 

[PeterDonis]

The generally agreed position is that we, in the 'external' universe, cannot observe a clock, or anything else except perhaps another black hole, enter an event horizon in finite time

There's another distinction here that you are not making, which I think I have mentioned, but which it's worth making explicit. The model that says that light from an object falling into a black hole will take longer and longer to get out to a distant observer (until ultimately light from the object just as it crosses the horizon takes an "infinite time" to get out to a distant observer) only applies, strictly speaking, to a "test object" falling into the hole. That is, it only applies if the mass of the object falling in is so small that any effect its mass has on the curvature of the spacetime as a whole is negligible. The scenarios we have been discussing here, where two black holes merge, or where an object of non-negligible mass (such as a star) falls into a black hole, do not meet this requirement; so strictly speaking, the argument that it will seem to take an "infinite time" for an object to reach the horizon, according to a distant observer, does not apply to these scenarios.

It is still true that light from events near the horizon takes longer to get out, even in the scenarios we have been discussing. But the fact that the infalling objects are of non-negligible mass means that the horizon's radius changes during the process, which changes the rules, so to speak, that determine what information can escape.

 

[pawprint]

I submit. I set a trap for the Seniors and it has been turned upon me.

In my last post I brought real black holes (as opposed to ideal ones which are likely to be rare) back into play by resorting to cosmological arguments. I can find no fault in PAllen's response. Once again I have to agree with everything said in reply, or at the least admit I cannot combat the arguments. I cannot feel comfortable with any description of gravity except 'that which locally distorts spacetime'. The only way I'll win this argument is if my view is supported by gravitational observations, which so far are non-existent.

Indeed it is the sublime silence of LIGO which has brought me into the gravitational fray over the last couple of years. But that's another thread. I have been interested in gravitation as distortion of spacetime for a long time, and it has become important to me to better understand as many of its implications as I can manage.

Overnight (for me) I see PeterDonis has also posted. Thank you PD. I specifically had that point in mind when I included 'or anything else except perhaps another black hole' in my last post. I'm glad you pointed out that the exception applies to all non-negligible masses.

There is only one (non-mathematical) point from the entire thread that I have not truly grasped. I accept the concept that, as almost infinite energy is required to hover near the event horizon, gravity can be inferred to become infinite at the EH. My difficulty is that the singularity must be at a more distorted (i.e. deeper, in the image below) position in spacetime than the event horizon, and if this is so then spacetime at the EH cannot truly be said to be infinitely warped. I alluded to this when I spoke of the intrepid explorer subjectively exceeding the speed of light.

My problem may simply be due to the impossibility of representing spacetime in three dimensions.

(This image is believed to be unencumbered by copyright)​

My memory is that not many years ago such diagrams of black holes showed an infinitely deep gravity well at the singularity. However now a search reveals the vast majority of such images to resemble that shown here, with almost flat bottoms. Does this represent a paradigm shift or just lazy artists?

 

[pawprint]

A quick addition to my last post. I don't dispute GR but neither do I believe it to be a complete theory yet. I expect a new class of experiments will shake a few ideas within the next decade. In particular I expect the constancy of G to be seriously questioned. If G is not constant either in space or time then all inferences from astronomical observations will need to be reconsidered.

 

[pervect]

The curvature of space-time is non-singular at the event horizon. Being a bit more specific, we can say that the tidal forces on a body freely falling through the event horizon are finite. They're calculated in most textbooks, for instance you'll find the detailed calculations in MTW if you look. The tidal forces on a free-falling body are equal to and given by the appropriate components of the curvature tensor.

The tidal forces at the central singularity are not finite, nor are the components of the curvature tensor. Both go to infinity there.

The tidal forces on an accelerating body are, confusingly, not quite the same as the curvature tensor. So the equivalence between the components of the curvature tensor and the measured tidal forces is strictly true only for a non-acclerating body.

The details get technical, but are a result of Bell's spaceship paradox, where the front and tail of a rigid spaceship have to accelerate at different rates if the spaceship is to remain rigid. If the front and tail accelerated at exactly the same rate, the spaceship would have to stretch (this is like the string breaking between two space-ships that accelrate at the same proper accelration).

The difference between the accelerometer readings at the head and tail of the space-ship could reasonably be interpreted as definiing a sort of tidal force. But there is no actual curvature of space-time in this scenario, the spaceships are in perfectly flat space-time.

 

[pawprint]

Thank you pervect. I like Bell, and Wheeler too. But I can only read them in English translation (from the math).

 

[PeterDonis]

There is only one (non-mathematical) point from the entire thread that I have not truly grasped. I accept the concept that, as almost infinite energy is required to hover near the event horizon, gravity can be inferred to become infinite at the EH.

"Gravity" in the sense of "the proper acceleration required to hover at a constant radius". But there are other senses of "gravity" that are *not* infinite at the horizon, as several posters have pointed out. Curvature in the sense of the Riemann curvature tensor, for example, or various scalars derived from it, is perfectly finite at the horizon, but becomes infinite at the singularity.

The key thing you appear to be struggling with is that you are trying to find one single "thing" that can be thought of as "gravity". There isn't. "Gravity" encompasses multiple phenomena, and they don't all "go together" the way one's intuition thinks they ought to. But our intuition is based on a very narrow set of conditions where speeds are small and all aspects of "gravity" are very weak, so they all kind of "look the same". GR has to handle a much wider range of cases, where "gravity" gets a lot stronger and the various phenomena associated with it start acting differently (like proper acceleration vs. curvature at the black hole's horizon).

My problem may simply be due to the impossibility of representing spacetime in three dimensions.

This is a problem, yes, but I would put it slightly differently. I think you are having problems because you are trying to deduce *everything* about gravity from a single diagram. To really get a complete picture, you have to look at multiple representations of the spacetime, each of which picks out a different aspect of it. Then you have to put all the different viewpoints together and understand how they interact. The page I linked to earlier, showing diagrams in Finkelstein, Kruskal, and Penrose coordinates in addition to Schwarzschild coordinates, is an excellent resource for doing that.

My memory is that not many years ago such diagrams of black holes showed an infinitely deep gravity well at the singularity. However now a search reveals the vast majority of such images to resemble that shown here, with almost flat bottoms. Does this represent a paradigm shift or just lazy artists?

Probably lazy artists if they are really intending to show "flat bottoms". But I suspect that what look to you like "flat bottoms" are really infinitely deep wells that just get cut off by the edge of the drawing.

It's worth noting, however, that the "infinitely deep well" idea has problems too. The underlying issue is the temptation to think of the singularity as a "place"--a location "in space". In reality, the singularity is a *spacelike surface*--which means that the closest thing to it in our intuitions is an *instant of time*--a "slice" of the universe (more precisely, of the portion of the universe that's behind the horizon) at a particular time. You can't represent "the universe at an instant of time", or "a portion of the universe at an instant of time" as a spatial point on a spatial diagram. It should really be a *separate* "spatial diagram" all its own.

If you look at the Kruskal or Penrose diagrams on the page I linked to earlier, you will see that they make this obvious: the singularity is a hyperbola that goes from left to right in the Kruskal diagram, and it is a horizontal line in the Penrose diagram. (This is also why we say that the singularity is "in the future", and why it's impossible to avoid the singularity once you're inside the horizon--because you can't avoid moving into the future.)

 

[pawprint]

The key thing you appear to be struggling with is that you are trying to find one single "thing" that can be thought of as "gravity". There isn't. "Gravity" encompasses multiple phenomena, and they don't all "go together" the way one's intuition thinks they ought to. But our intuition is based on a very narrow set of conditions where speeds are small and all aspects of "gravity" are very weak, so they all kind of "look the same". GR has to handle a much wider range of cases, where "gravity" gets a lot stronger and the various phenomena associated with it start acting differently (like proper acceleration vs. curvature at the black hole's horizon)...

...This is a problem, yes, but I would put it slightly differently. I think you are having problems because you are trying to deduce *everything* about gravity from a single diagram.

I first read of time dilation (in science fiction) at the age of 8 or 9 and it took another 12 years before I had intuitively grasped the combined SR implications of mass, time, length and redshift associated with it. I will certainly revisit the Penrose page referred to and spend more time there. GR is stretching my mind, but once it is 'well mixed in' I hope for intuitive understanding.

As an aside I attended a spontaneous lecture by Roger 23 years ago. Everybody was expecting him to talk of matters cosmological but he spoke for 100 minutes on quantum effects in the brain's microtubules! I remember it all the better because it was the first important public event I attended wearing paws rather than shoes.

I missed the multiquote button re pervect's post above, but I smiled when I read of Bell's spaceship paradox.

No, it does not imply that... Further, the enlarged event horizon 'rings' for a while, emitting more GW. These are all oscillations of the metric (or geometry) outside the event horizon.

Can PAllen or somebody else who knows tell me the sort of frequency ranges expected of these 'ringing' waves once they reach flat spacetime? LIGO is said to be unresponsive to frequencies below 200 Hz, but newer instruments are hoped to have much wider bandwidths.

 

[Spin-Analyser]

Pervect's statement about the horizon moving at c past any infaller is simply true. To the infaller it simply appears as the light of prior infallers reaching them. Thus the moment they cross the horizon is the moment they can see all prior infallers. I don't see the tension with any other statements.

Could you clarify this please? Are you saying that an in-faller would see the in-fallers in front of them ( who were of course outside the horizon from a distance ) cross the event horizon as they approach, at which point they disappear from this this observers perspective only to reemerge once the event horizon is reached? This doesn't seem right?

 

[PAllen]

Could you clarify this please? Are you saying that an in-faller would see the in-fallers in front of them ( who were of course outside the horizon from a distance ) cross the event horizon as they approach, at which point they disappear from this this observers perspective only to reemerge once the event horizon is reached? This doesn't seem right?

As long as an infaller is outside the horizon, they see prior infallers as they were closer to the horizon than they are. Note that distances perceived by this infaller are very different from the r coordinate value - there is "lot's of room". Passing the horizon by our chosen infaller is experienced as seeing prior infallers pass the horizon.

There is no disappearance or reappearance.

I came up with an analogy on another thread. Imagine a chain of infalling observers. Imagine a pink flashbulb goes off beyond one end of this chain. Prior to the pink light reaching (say) the last observer in the chain, all prior infallers are seen as before the flash reached them. The moment the flash reaches the last observer is exactly the moment when this observer sees all prior observers flash pink. Thus, the moment the this observer crosses the horizon is the moment they see earlier infallers as of when they crossed the horizon. Factoring light delay, you deduce they all got hit with the flash before you, but you only see them flash pink the same time you do. Similarly, factoring in light delay, this trailing infaller deduces the earlier infallers crossed the horizon before they did.

 

[Spin-Analyser]

Okay I kind of see, but if the in-faller were to get very close to the horizon (say one plank length away) and then move away, would that mean that they observed the previous in-fallers crossing the event horizon twice, once on the way towards it and them reemerging on the return journey?

 

[PAllen]

Okay I kind of see, but if the in-faller were to get very close to the horizon (say one plank length away) and then move away, would that mean that they observed the previous in-fallers crossing the event horizon twice, once on the way towards it and them reemerging on the return journey?

No. Until the moment the event horizon 'passes them' they sell all prior infallers as of before they passed the horizon. In my analogy, if the pink flash is 1 Planck length from the trailing observer:

- no earlier observer is seen to have flashed pink
- the trailing observer can still, in principle, accelerate away from the light, and without ever quite exceeding c, stay ahead of it: see Rindler Horizon.

At precisely the moment the flash reaches the trailing observer, all prior observer's flash pink, and no acceleration at all will catch light that has already passed.

Sufficiently locally, all 'near horizon' phenomena are accurately described by a passing flash of light - because the event horizon is a light like surface.

 

[Passionflower]

As long as an infaller is outside the horizon, they see prior infallers as they were closer to the horizon than they are. Note that distances perceived by this infaller are very different from the r coordinate value - there is "lot's of room".

You are wrong.

A free falling observer (free falling from infinity) will measure his distance to the EH to be exactly equal to r - rs (where rs is the event horizon).

Contrast this with a stationary observer close to EH he will measure his distance from the EH to be more than r - rs

 

[Spin-Analyser]

What if the in-faller stops one plank length away? They will see the event horizon as just in front of them, but also just in front of every other object that hasn't crossed the horizon yet but who are closer to the singularity than they are? This seems very paradoxical!

 

[PAllen]

You are wrong.

A free falling observer (free falling from infinity) will measure his distance to the EH to be exactly equal to r - rs (where rs is the event horizon).

Contrast this with a stationary observer close to EH he will measure his distance from the EH to be more than r - rs

I wasn't actually thinking of infalling from infinity. In these scenarios of changing mind at the last second, I think in terms of observers hovering near the horizon, then shutting off fuel. However, I failed to specify this, and certainly you are right about an infaller from infinity.

[Edit: this observation does clarify that I needn't have said anything about comparative distances, as it is not relevant to the main issues - see next post.]

 

[PAllen]

What if the in-faller stops one plank length away? They will see the event horizon as just in front of them, but also just in front of every other object that hasn't crossed the horizon yet but who are closer to the singularity than they are? This seems very paradoxical!

No, they will see earlier infallers a 'normal' distance away, as of before they passed the horizon. Think of it this way: if they could know where the horizon was just before it hit them, they would see earlier infallers at a distance such that they could deduce they must be already inside; however, if each earlier infaller had a watch, the image they would see on the watch would be of a moment just before each earlier infaller crossed the horizon. Think more about the passing flash of light example. Further, if at this last moment, they accelerated away frantically, they would never see the earlier infaller's watches reach their infall moment.

 

[Spin-Analyser]

So let me clarify.

1). It is perfectly possible to observe objects crossing the event horizon of a black hole.
2). It is not possible to observe them at a time after they reached the event horizon.

Think very carefully about your next answer.

 

[PAllen]

So let me clarify.

1). It is perfectly possible to observe objects crossing the event horizon of a black hole.
2). It is not possible to observe them at a time after they reached the event horizon.

Think very carefully about your next answer.

Both your statements are not quite right.

It is perfectly possible to see objects cross the horizon when you cross the horizon. It will be obvious (at that moment) that they crossed before you. Further, you can deduce for possible infallers ahead of you, that 'if they are still where they appear to be', and you know where the horizon is, they are inside the horizon. However, since you are seeing an 'old' image of them, you cannot tell if they made a last moment decision to escape (and thus are closer to you than they appear) unless you also make such a decision, and later see that they did. Further, as long as you remain outside the horizon, you cannot tell for sure whether they crossed or not.

 

[Spin-Analyser]

It is perfectly possible to see objects cross the horizon when you cross the horizon. It will be obvious (at that moment) that they crossed before you.

I'm talking about an observer who never reaches the horizon themselves. As they approach they will see objects crossing the event horizon, but they will be seeing them as they were before they reached the horizon?

 

[PAllen]

I'm talking about an observer who never reaches the horizon themselves. As they approach they will see objects crossing the event horizon, but they will be seeing them as they were before they reached the horizon?

Let's talk about a supermassive BH, and let's say you are following 10 feet behind your partner, approaching the horizon. Let's say you know exactly where it is all times (by computation and knowledge of the region). Let's say you fall to 3 feet from the horizon and stop. At this moment, it is possible to still see an image of your partner that looks 10 feet from you. Now consider two cases:

1) Your partner crossed the horizon. You will see their image fade to black, and their wristwatch will never quite reach the time they crossed the horizon.

2) Your partner stopped 1 foot from the horizon. Some time after you stop, you will see that your partner started accelerating to hover before you did, getting closer to you in the process, and are now stopped 2 feet away.


The closer to the horizon your partner makes decision (2), the longer before you can distinguish it from (1).

 

[Spin-Analyser]

1) Your partner crossed the horizon. You will see their image fade to black, and their wristwatch will never quite reach the time they crossed the horizon.

So if you are close enough to the horizon you can observe objects crossing it?

2) Your partner stopped 1 foot from the horizon. Some time after you stop, you will see that your partner started accelerating to hover before you did, getting closer to you in the process, and are now stopped 2 feet away.

They both hover just above the horizon. One of them turns off their engines. Does the other one see them disappear passed the horizon?

 

[PAllen]

So if you are close enough to the horizon you can observe objects crossing it?

This is getting repetitive. You are seeing an old image from before they crossed. It is from when they were still (in my prior example) 10 feet from you but right near the horizon. This light takes a while (up to forever) to reach you.

They both hover just above the horizon. One of them turns off their engines. Does the other one see them disappear passed the horizon?

The one that remains hovering sees the other one approach the horizon, effectively going black before reaching it. The image history here is quite different from tandem infallers. The image history (past world lines + light null paths) makes for the difference these cases.

 

[Passionflower]

Basically an observer can detect a signal from another observer who passed the event horizon if he passes the event horizon as well in time. He will observe the signal only after he passed the horizon himself.

An observer who never passes the event horizon cannot receive a signal from an observer passed the event horizon.

 

[Spin-Analyser]

This is getting repetitive. You are seeing an old image from before they crossed. It is from when they were still (in my prior example) 10 feet from you but right near the horizon. This light takes a while (up to forever) to reach you.

I'm just trying to be absolutely clear that you're saying you can see an object turn black, effectively crossing the horizon. But at this point you're still not sure if they reached the horizon or not? If they accelerated at the last moment they would presumably reappear from the others perspective after a time. And if they cross the horizon they would suddenly see all the other objects that had fell in earlier?

The one that remains hovering sees the other one approach the horizon, effectively going black before reaching it. The image history here is quite different from tandem infallers. The image history (past world lines + light null paths) makes for the difference these cases.

The one that remains hovering will see them approaching the horizon as they are falling towards the singularity, meaning the event horizon will always be falling in ahead of them?

 

[PAllen]

I'm just trying to be absolutely clear that you're saying you can see an object turn black, effectively crossing the horizon. But at this point you're still not sure if they reached the horizon or not? If they accelerated at the last moment they would presumably reappear from the others perspective after a time. And if they cross the horizon they would suddenly see all the other objects that had fell in earlier?

I think I will have to let someone else answer your questions after this. Somehow, I think I'm being clear and you get something quite different from what I said out of it. Someone else may express it in a way you get it.

Repeating yet again: You don't see them actually cross the horizon if you remain outside. No exception. The turning black is just a matter of infinite red shift and time dilation relative to you if you are hovering further away.

If they divert from crossing at the last minute, sometime before infinite redshift, you see them turn on their thrusters and (as in my tandem example) get closer to you (you having already hovered). All this is due to light delay. You never see turning fully black and reappearing[edit: you can see someone have arbitrarily close to infinite redshift, then approach you becoming less redshifted, even pass you]. Ultimately, after infinite time, you can infer they crossed if you never detect that they stopped and hovered.

Finally, yes, the moment you cross you see prior infallers as of the moment they crossed.

The one that remains hovering will see them approaching the horizon as they are falling towards the singularity, meaning the event horizon will always be falling in ahead of them?

I don't understand this at all. The one that is hovering simply sees the one that turns off thrusters fall towards the horizon, get redder, ultimately black, just outside the horizon. Nothing about the history from horizon to singularity can be seen by the one remaining outside.

 

[Spin-Analyser]

I think I will have to let someone else answer your questions after this. Somehow, I think I'm being clear and you get something quite different from what I said out of it. Someone else may express it in a way you get it.

Repeating yet again: You don't see them actually cross the horizon if you remain outside. No exception. The turning black is just a matter of infinite red shift and time dilation relative to you if you are hovering further away.

If they divert from crossing at the last minute, sometime before infinite redshift, you see them turn on their thrusters and (as in my tandem example) get closer to you (you having already hovered). All this is due to light delay. You never see turning fully black and reappearing[edit: you can see someone have arbitrarily close to infinite redshift, then approach you becoming less redshifted, even pass you]. Ultimately, after infinite time, you can infer they crossed if you never detect that they stopped and hovered.

Finally, yes, the moment you cross you see prior infallers as of the moment they crossed.

Then you can see objects crossing the horizon (because you're one plank length away and they're in front of you), so light is escaping from inside the horizon?

I don't understand this at all. The one that is hovering simply sees the one that turns off thrusters fall towards the horizon, get redder, ultimately black, just outside the horizon. Nothing about the history from horizon to singularity can be seen by the one remaining outside.

The bit I'm having trouble with is seeing distance between you (hovering a plank length above the horizon) and another object but niether of you have crossed the horizon. You could therefore move alongside the other observer and niether of you would have crossed the event horizon, so you can't have been next to the horizon in the first place?

 

[PAllen]

Then you can see objects crossing the horizon (because you're one plank length away and they're in front of you), so light is escaping from inside the horizon?

The bit I'm having trouble with is seeing distance between you (hovering a plank length above the horizon) and another object but niether of you have crossed the horizon. You could therefore move alongside the other observer and niether of you would have crossed the event horizon, so you can't have been next to the horizon in the first place?

I say "you never see x" . You respond: "Then you can see x". We will never get anywhere this way.

Classically, Planck length is irrelevant. Quantum mechanically, nobody knows. Take your pick depending on approach to a partial theory of quantum gravity: (a) there is nothing resembling a horizon (and surface of smallest visibility is smaller than EH as predicted by GR); (b) there is something that is not a horizon microscopically, but it looks a lot like it macroscopially; (c) there is a horizon, but with some difference in properties from the classical picture; (d) a horizon never forms and matter is always outside what would be the horizon radius.

I don't understand your second paragraph at all.

 

[Spin-Analyser]

Let me be clearer. You're saying you can observe light coming out to your eye from inside the horizon as you hover just above it (because you see distance between you and objects ahead of you), but you're seeing light that hasn't reached the horizon yet?

 

[PAllen]

Let me be clearer. You're saying you can observe light coming out to your eye from inside the horizon as you hover just above it (because you see distance between you and objects ahead of you), but you're seeing light that hasn't reached the horizon yet?

Nope, never said this, said the opposite several times. I said if you are outside, you only see light that was emitted outside as well. It may look like it comes from a distance such that if the object is still that distance from you it would be inside. But the light is old, from outside the horizon.

 

[Spin-Analyser]

It's old light from outside the horizon coming at you from inside the horizon of light that hasn't reached the horizon yet?