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

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PeterDonis
#37
Mar6-12, 11:40 PM
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Quote Quote by questionpost View Post
How does the event horizon, which is symmetrical to the singularity, expand before matter has reached the singularity? Wouldn't that imply the object and the singularity are the same object if they have the same gravitational field?
Did you read my post #33? You are assuming that the "mass" of the black hole is somehow "located" at the singularity, and doesn't increase until the infalling object arrives there. That is false.
Nabeshin
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Mar7-12, 12:06 AM
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Quote Quote by PAllen View Post
How about some nice pictures:

http://www.black-holes.org/explore2.html

search, e.g., for merging event horizons.
Yessss, more traffic for our website :P
pervect
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Mar7-12, 05:45 AM
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Quote Quote by PAllen View Post
None of these are contradictory, though Thorne is at least misleading in an attempt a drama.

Let's take them one at a time:

"There is no such thing as a stationary clock at the event horizon." Here, you are rather naturally confused by ambiguity in English language. Pervect is here referring to stationary in the sense of motionless relative to distant observers, not rate of time flow on a clock. The two senses of stationary juxtaposed this way lead to false perception of contradiction. Sorry about that. English is a .... <forum rules> sometimes.
Yes, sorry if this wasn't clear. A stationary observer is basically an observer with constant r, theta, and phi Schwazschild coodinates.

In order to qualify as an observer, his worldline must be timelike. (Which is another technial term from special relativity). An photon isn't an observer, for instance.

Thorne's comments about "a direction you would have thought was spatial" and a "downwards direction" are misguided. The only one expecting this would be someone who interpreted coordinates according the letter used to name them rather than their physical characteristics.
]

I don't see why you say it's misguided. Though I think it may be confusing the OP, because Thorne's approach isn't based on the "clock slowing" paradigm.

My basic impression is that the OP is stuck in a Newtonian view of absolute time, and is also interpreting the whole "clock slowing" down thing as some sort of scalar function that modifies how fast absolute time flows at a given position.

And this is just not compatible with special relativity at all (mostly because of the absolute time idea).

At the risk of possibly causing more confusion, Thorne's view is more like saying that the time doesn't really "stop" (as per the stopped time idea), it's just bent to point in a spatial direction.

In standard Schwarzschild coordinates, the coordinate called 'r' is spacelike outside the horizon and timelike inside the horizon. This means nothing except that 'r' is a bad label for the coordinate inside the horizon. If you instead use the local Fermi-Normal coordinates of a infaller, all of this nonsense disappears.
questionpost
#40
Mar7-12, 07:22 AM
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Quote Quote by PeterDonis View Post
Did you read my post #33? You are assuming that the "mass" of the black hole is somehow "located" at the singularity, and doesn't increase until the infalling object arrives there. That is false.
Ok, when black holes merge, then I see how the event horizon increases, however I don't see how a *not* infinitely dense object does the same thing, so shouldn't the object first have to have an infinite density like the singularity in order to have an event horizon and then merge that event horizon with the black hole its falling into? And since it can only have an infinite density by merging with the singularity, shouldn't the even horizon not increase until then even if the gravitational pull does?

Quote Quote by PAllen View Post
"Time slows to a stop for an infaller" is a statement that should always be joined to: "from the point of view of a static observer further away; not from the point of view (for example) an infaller just ahead of a given infaller".
search, e.g., for merging event horizons.
Ok, then "why" does it stop just because it's at a boundary where the escape velocity happens to be light? Also, you said before that other escape velocities don't matter, so does that mean once inside the event horizon, even if I traveled 99% the speed of light away from the singularity, it wouldn't slow down my in-fall? It seems related to hypothetically traveling at the speed of light.
Why is that too? We can calculate what happens when you travel at the speed of light with an equation yet right next to it have another equation that says you can never travel at or faster than the speed of light, within the same theory known as GR.
PAllen
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Mar7-12, 07:31 AM
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Quote Quote by pervect View Post

I don't see why you say it's misguided. Though I think it may be confusing the OP, because Thorne's approach isn't based on the "clock slowing" paradigm.



At the risk of possibly causing more confusion, Thorne's view is more like saying that the time doesn't really "stop" (as per the stopped time idea), it's just bent to point in a spatial direction.
Well, time pointing in a spatial direction is a non-sequitur. At any point in a manifold there is a light cone defining time like directions, light like directions, and space like directions. Any small region looks just like Minkowski space, including a region where the horizon is passing by at c. There is nothing spatial about a timelike direction inside an event horizon except that it is labeled r in some coordinate schemes. It's labeled U or V in Kruskal (depending on your convention). It's labeled t in local Fermi-Normal coordinates. I think it is genuinely misleading to attach significance to a letter used in interior Schwarzschild coordinates for what essentially are historic reasons.
PAllen
#42
Mar7-12, 12:19 PM
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Quote Quote by questionpost View Post


Ok, then "why" does it stop just because it's at a boundary where the escape velocity happens to be light? Also, you said before that other escape velocities don't matter, so does that mean once inside the event horizon, even if I traveled 99% the speed of light away from the singularity, it wouldn't slow down my in-fall? It seems related to hypothetically traveling at the speed of light.
Why is that too? We can calculate what happens when you travel at the speed of light with an equation yet right next to it have another equation that says you can never travel at or faster than the speed of light, within the same theory known as GR.
Why? That's not really a question of physics. A distant observer would see a a clock slow and light red shift on approach to a neutron star. Approach to a horizon is the same thing only more extreme. Asymptotically stopping just reflects that force needed to escape becomes infinite on approach to horizon. This stoppage is only observed by someone remaining further away from the horizon. There is no stoppage for the infaller.

I don't know what you are referring to in claiming I said escape velocities don't matter. I don't know what you are going on about traveling at or faster than the speed of light. I keep repeating this is all nonsense.

The singularity is a point in time not in space. Once inside the horizon, you can shine a flashlight any direction, and fire bullets in any direction, but all light and any projectiles you fire, in any direction, move forward in time toward the singularity. Poetically, you can say the singularity is a point in time where space ceases to exist for you. (In fact, you will be subject to enormous (ultimately infinite) compression and stretching, but you can always define a tiny enough region where everything is momentarily normal - until the moment of reaching the singularity).
pawprint
#43
Mar7-12, 04:46 PM
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The OP thanks you all. This has been a most interesting thread and I have achieved the desired 'intuitive' breakthrough that I was seeking. Just a few notes in special appreciation starting with post #16-

Quote Quote by pervect View Post
A stationary clock near a singularity would run slow when compared with another stationary clock that's far away from any singularity, and it'd approach stopping as the stationary clock got closer and closer to the event horizon.

There's no such thing as a stationary clock at the event horizon, however. In fact, any clock crossing the event horizon must be moving at the speed of light - or rather, since the event horizon can be thought of as trapped light, any physical infalling clock, which is stationary in its own frame, will see the event horizon approaching it at the speed of light.

This motion causes signficant SR effects. If you neglect the velocity effects, it would be correct to say that from the point of view of an infalling observer, a clock at infinity would run faster and faster, without bound, as one approached the event horizon,.

When you include the velocity effects, though, the clock at infinity doesn't run infinitely fast.

You will run into the usual special relativity (SR) issues associated with the twin paradox when you include the velocity effects - I'm not sure what yoru background is in SR.
I've been quite comfortable with the implications of SR for about 37 years now pervect. I like your analysis but you made an error in the first paragraph.

I won't quote quotes but I found the two in post #19 by Naty1 very helpful. In Post #20-

Quote Quote by questionpost View Post
...It doesn't even make sense that time would stop, because then how would anything ever reach the singularity to add to its mass?
questionpost has expressed the same thought that has been concerning me for a long time, and which prompted this thread. But I'm no longer concerned. The answer I have found from the many responses is that although observers see objects slow effectively to a stop before passing the event horizon, the observation is dependant not on a single effect (time dilation) as I had previously thought but also on gravitationally caused 'slowness of light' from the object back to the observer.

The same statement rephrased: Were an 'instantaneous' link available from the falling clock to a slave clock held by an observer, it would indeed show the falling clock to be slowing at a rate depending on the gravitational gradient of the particular black hole. BUT IT WOULD NOT SLOW TO ZERO at the event horizon. The appearance of this effect to an observer without a simultaneous link, while real enough, is caused by the slowness of light (or other EM signal) returning to the observer from the intense gravity field. The 'simultaneously' linked clock would only approach zero rate of change as it approached the singularity.

The clock itself behaves exactly as it would aboard a vessel approaching light speed, with all the same implications for local and distant observers. After all, although nothing can be seen to 'break' the speed of light this doesn't change the fact that an intrepid traveller accelerating at 1 g will subjectively do so after about three years.

As I have said before, I seek intuitive understanding without math. Of course I know that simultaneous links are thought to be impossible, and that infinite anythings are rare. Indeed the only infinite 'physical' thing I can think of is the depth of a gravity well in which sits a singularity.

Once again thanks to everybody who participated in this thread. You have settled demons which have been of growing concern to my intuition for some time.
pawprint
#44
Mar7-12, 05:10 PM
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OP here again. Not having refreshed my browser I had missed this quoted post. I now must take issue with paragraph 1-
Quote Quote by PAllen View Post
...That's not really a question of physics. A distant observer would see a a clock slow and light red shift on approach to a neutron star. Approach to a horizon is the same thing only more extreme. Asymptotically stopping just reflects that force needed to escape becomes infinite on approach to horizon. This stoppage is only observed by someone remaining further away from the horizon. There is no stoppage for the infaller...
I now hold the view expressed in the previous post that the clock does not asymptotically stop at the event horizon, it only LOOKS as though it does...
PeterDonis
#45
Mar7-12, 05:55 PM
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Quote Quote by questionpost View Post
Ok, when black holes merge, then I see how the event horizon increases, however I don't see how a *not* infinitely dense object does the same thing, so shouldn't the object first have to have an infinite density like the singularity in order to have an event horizon and then merge that event horizon with the black hole its falling into? And since it can only have an infinite density by merging with the singularity, shouldn't the even horizon not increase until then even if the gravitational pull does?
Your picture of a black hole is not an accurate one. Several issues:

(1) The "black hole" is not just the singularity. The term is used to refer to the entire region of spacetime inside the event horizon. When people talk about two black holes merging, they are talking about two regions inside event horizons merging into one region inside an event horizon.

(Strictly speaking, there is only a single event horizon, and only a single region of spacetime inside it; that region just happens to be shaped like a pair of trousers instead of a tube, so to speak.)

(2) A black hole is not "infinitely dense". The singularity itself can be thought of as "infinitely dense", but the singularity has no causal effect on anything else in the spacetime, so its characteristics are irrelevant for understanding what happens elsewhere.

(Strictly speaking, the singularity is not even "in" the spacetime--the spacetime itself "ends" at the singularity, meaning there are events arbitrarily close to the singularity but none actually "at" it.)

(3) The event horizon is defined "teleologically"--it is the boundary of the region of the spacetime (as above, there is only *one* such region, but it may be shaped like a pair of trousers instead of a tube) that cannot send light signals to "infinity" (strictly speaking, to "future null infinity"). That definition requires you to know the entire history of the spacetime to pin down exactly where the horizon is. So when an object of non-negligible mass falls into a black hole, the horizon starts to move outward from its old radius to its new radius even *before* the infalling object reaches it, because the horizon is defined in terms of where light signals go all the way into the infinite future. A light signal sent from outside the "old" horizon radius may still be trapped behind the new horizon even if it is sent *before* the infalling object reaches the "new" horizon radius--if it is sent a short enough time before, so that it doesn't have time to make it past the new horizon radius before the infalling object arrives.

Take a look at the diagrams on this page:

http://casa.colorado.edu/~ajsh/collapse.html

Particularly the Kruskal and Penrose diagrams of the star collapsing to a black hole. It may help to visualize what I'm saying above.
pawprint
#46
Mar7-12, 06:15 PM
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Quote Quote by PeterDonis View Post
Take a look at the diagrams on this page:

http://casa.colorado.edu/~ajsh/collapse.html

Particularly the Kruskal and Penrose diagrams of the star collapsing to a black hole. It may help to visualize what I'm saying above.
That's a great page PeterDonis, and it entirely confirms my new paradigm.
PAllen
#47
Mar7-12, 06:26 PM
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Quote Quote by pawprint View Post
OP here again. Not having refreshed my browser I had missed this quoted post. I now must take issue with paragraph 1-


I now hold the view expressed in the previous post that the clock does not asymptotically stop at the event horizon, it only LOOKS as though it does...
I don't understand what you disagree with, but a fact is that the whatever you say about a clock sitting in a dense planet or neutron star (gravitational time dilation and red shift) you must say the same thing about a clock hovering near the event horizon, because they are exactly, in every way, the same phenomenon in GR. Note that a clock hovering near the event horizon sees distant clocks going extremely fast. An infaller is different because (see Pervect's post a little earlier) because you have SR speed effects as well as gravitational time dilation.
pawprint
#48
Mar7-12, 06:36 PM
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Quote Quote by PAllen View Post
I don't understand what you disagree with, but a fact is that the whatever you say about a clock sitting in a dense planet or neutron star (gravitational time dilation and red shift) you must say the same thing about a clock hovering near the event horizon, because they are exactly, in every way, the same phenomenon in GR. Note that a clock hovering near the event horizon sees distant clocks going extremely fast. An infaller is different because (see Pervect's post a little earlier) because you have SR speed effects as well as gravitational time dilation.
There is one significant difference PAllen. The VIEW of the clock sitting on a neutron star is not subjected to the near 100% redshift that the clock near the event horizon is.
PAllen
#49
Mar7-12, 06:43 PM
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Quote Quote by pawprint View Post
There is one significant difference PAllen. The VIEW of the clock sitting on a nuetron star is not subjected to the near 100% redshift that the clock near the event horizon is.
It is just a matter of degree. The clock on the neutron star is extremely redshifted. If more matter fell into the neutron star until it collapsed into a black hole, the redshift of the clock would smoothly grow arbitrarily large (assuming it maintained position on the collapsing surface, then hovers just outside the freshly formed event horizon). The phenomena are absolutely identical in GR. You cannot claim they are different (except for degree) unless you reject GR - in which case you should say so.
pawprint
#50
Mar7-12, 06:58 PM
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Clarification:

A clock near an event horizon would appear to have slowed to almost nothing, considering red-shift alone and excluding gravitational effects. In 'reality', as far as it can be applied in these circumstances, the gravitation slows the clock to near zero at the singularity. The redshift, by different means, makes the clock appear to have stopped at the event horizon.
PeterDonis
#51
Mar7-12, 07:03 PM
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Quote Quote by pawprint View Post
In 'reality', as far as it can be applied in these circumstances, the gravitation slows the clock to near zero at the singularity.
This statement doesn't even have a well-defined meaning. There are no "static" observers inside the horizon; that is, no observers who "hover" at a constant radius. So the interpretation of "rate of time flow" that works outside the horizon, and according to which a clock "hovering" near the horizon "runs very slow" compared to a clock far away, does not even work inside the horizon. Unless you can come up with some alternate way of comparing the "rate of time flow" near the singularity with that far away from the hole, you can't say anything at all about how gravitation "slows clocks" near the singularity.
pawprint
#52
Mar7-12, 07:05 PM
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Quote Quote by PeterDonis View Post
...Unless you can come up with some alternate way of comparing the "rate of time flow" near the singularity with that far away from the hole, you can't say anything at all about how gravitation "slows clocks" near the singularity.
I have (see post #43). I'm sorry you don't agree. And I don't dispute GR.
PeterDonis
#53
Mar7-12, 07:24 PM
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Quote Quote by pawprint View Post
I have (see post #43). I'm sorry you don't agree. And I don't dispute GR.
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:

Quote Quote by pawprint View Post
questionpost has expressed the same thought that has been concerning me for a long time, and which prompted this thread. But I'm no longer concerned. The answer I have found from the many responses is that although observers see objects slow effectively to a stop before passing the event horizon, the observation is dependant not on a single effect (time dilation) as I had previously thought but also on gravitationally caused 'slowness of light' from the object back to the observer.
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.

Quote Quote by pawprint View Post
The same statement rephrased: Were an 'instantaneous' link available from the falling clock to a slave clock held by an observer, it would indeed show the falling clock to be slowing at a rate depending on the gravitational gradient of the particular black hole. BUT IT WOULD NOT SLOW TO ZERO at the event horizon.
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.

Quote Quote by pawprint View Post
The appearance of this effect to an observer without a simultaneous link, while real enough, is caused by the slowness of light (or other EM signal) returning to the observer from the intense gravity field. The 'simultaneously' linked clock would only approach zero rate of change as it approached the singularity.
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".

Quote Quote by pawprint View Post
The clock itself behaves exactly as it would aboard a vessel approaching light speed, with all the same implications for local and distant observers. After all, although nothing can be seen to 'break' the speed of light this doesn't change the fact that an intrepid traveller accelerating at 1 g will subjectively do so after about three years.
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
#54
Mar7-12, 07:35 PM
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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.


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