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

pawprint

I have (see post #43). I'm sorry you don't agree. And I don't dispute GR.

PeterDonis

That's good (meaning not disputing GR). But your post #43 does not propose a valid way of defining "rate of time flow". Here is what I take to be the relevant part of your post #43, with comments interspersed:

For the "rate of time flow" of a *static* observer hovering close to the horizon, this viewpoint works OK. It does *not* work (at least, not as stated) for the "rate of time flow" of an observer falling *into* the hole.

Here you are trying to reason about the "rate of time flow" of an infalling clock, but you are depending on this idea of an "instantaneous link" between the infalling clock and a "slave clock" hovering far away. But you haven't defined *how* this "instantaneous link" is specified--in other words, you haven't told me, if I'm looking at a spacetime diagram of the hole, showing the worldlines of the infalling object and the "slave" clock, how to draw "lines of simultaneity" between them to define which pairs of events are "linked" by the instantaneous link. Once you do that, then you can try to define a "relative clock rate" that way; it still won't work, but at least you could try, and perhaps trying it will help you to see the problems.

No, it wouldn't. See comments above; this is one of the things that might become more evident to you if you actually tried to explicitly define a "simultaneous link".

You have this backwards. From the standpoint of GR, the *hovering* clock--the clock that is static at a constant radius r, above the horizon--is the one that is "accelerating". The observer that is freely falling into the hole is not "accelerating" at all; he's in free fall. So if you are trying to make an analogy with an observer accelerating in a rocket, that observer is analogous to the *hovering* clock, *not* the infalling clock.

pawprint

Others have spoken of infalling observers and I agree with them unconditionally. I also agree with eveything you said about them in your last post. But I have not mentioned them in this thread.

As for the "instantaneous link" it will be a sad day for physics when thought experiments are disallowed.

PAllen

This is factually wrong. Gravitational time dilation and gravitational redshift are the same phenomenon. There is no redshift for a clock hovering near the the horizon that can be separated or distinguished in any way from the redshift of a clock sitting on a neutron star (except for degree). These are mathematical facts, not subject differences of opinion.

pawprint

I know it does not fit the current paradigm. Let me try it another way: Only an infinitely deep gravitational well slows clocks infinitely. The well at the event horizon is not infinitely deep. It is certainly less deep than at the singularity. You are effectively asserting that clocks are slowed infinitely by gravity equivalent to the escape speed of light.

PAllen

It is precisely infinitely deep in the sense that the thrust required to escape from near the horizon goes to infinity as you approach the horizon. This 'escape thrust' requirement is the exact equivalent of g force on a neutron star, extrapolated to the limit. Inside the horizon, there is no escape at all, and no ability to define a reasonable notion of g force. Note an earlier post, where I described that progress toward the singularity inside the horizon is progress in time, not toward a spatial point. Shoot a bunch of bullets away in all directions, and they will move away from you in all directions (spatially), while all move forward in time towards the singularity.

PeterDonis

Yes, you have, though you may not have realized it. You talked about observers approaching the singularity. Such observers *have* to be infalling; there are no timelike (or null) worldlines inside the horizon that do not move continuously inwards towards the singularity.

I wasn't disallowing your thought experiment; I was pointing out that it was incompletely specified, and saying what a proper specification would have to look like.

pawprint

OK. The difference between the Senior members and myself is spacetime related, and furthermore is now well defined. It need not be restated here. I think we can all agree on at least that much :{)

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, let alone 'rapidly'. I think this position has been locked in pretty firmly by PAllen, PeterDonis and others. This position necessarily implies that gravitational wave detectors can never detect such events.

Given this I think it reasonable to consider two cosmological implications of the agreed view.

1) Several billions of dollars have been spent on gravitational wave antennas. At least some cosmologists (a large majority, perhaps) expect the antennas to detect black hole events within their limit of sensitivity. If the Senior forum members are right then those cosmologists are mistaken. The events will occur so slowly (in relation to the detectors) that the detectors will at best see them, in electomagnetic terminology, as smoothed DC.

2) The current cosmological paradigm assumes that black holes have grown in the past and continue to grow today. What alternatives exist? Did they all spring out of the primal event full blown?

A deep dichotomy is felt. The insight (or madness perhaps) I am defending agrees with the cosmologists' expectations. I cannot see how the opposing view can accommodate them.

My position has not changed since my opening post, but it has certainly become better defined from a cosmological perspective. Perhaps it would be more appropriate for a new thread to be started, possibly in a different forum, if members would like to continue in this new cosmological vein. Would a Senior member like to adjudicate (or arbitrate) on the proposition? In the meantime we can perhaps agree on a clearly shared opening position here.

Thank you all.

"Disability access is a Dalek Plot"

PAllen

No, it does not imply that. Whether a star is torn apart and mostly absorbed by a black hole, or two black holes merge, enormous gravitational radiation (GW) will be emitted. It's (the GW) energy content will often carry away over 5% of the total mass of the star or black hole. There is no contradiction because the GW is generated by activity outside the initial event horizon. 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.
There is no disagreement. See above. These are difficult concepts. The only 'issue' is your level of understanding, which you are trying to improve - great!.
Models for the formation of stellar black holes are pretty well defined. At present, there are more unknowns, than knowns, about how supermassive black holes came to be. This is an active research area. However, no one believes they are primordial; they grew somehow; it is just that models so far don't show a likely way for the big ones to form.
No need for a new thread. The discussion here is going fine, and I hope is helpful.

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

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.


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

"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).

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.

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

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.

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.