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Member (Idle past 483 days) Posts: 179 From: Sydney, NSW,Auistralia Joined: |
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Author | Topic: Black Holes, for Eta Carinae | |||||||||||||||||||||||
Eta_Carinae Member (Idle past 4374 days) Posts: 547 From: US Joined: |
As I alluded to earlier I am really busy right now so I'll be brief.
Mike Holland: No, the observer falling through the event horizon does not see the universe age - he sees their clocks run slow just as they see his clock run slow. NosyNed: You can argue that the speed of light changes near a gravitational field (you often hear people say the metric changes the effective refractive index of space since measured with respect to flat space the geodesic is not straight but curved hence an analogy to the refraction of light in a material medium.) But most physicists don't like this analogy as it is coordinate dependent. The velocity of light only changes when you measure it using your time coordinate. If you use the proper time coordinate of the photon (i.e. along the null geodesic itself) then it's velocity is still c. To All: The event horizon has no physical reality. Nothing special happens there if you are passing it. It is a consequence of the coordinate system you are using. Yes you observe an object falling in to slow down (and get redshifted) BUT the object in reality fell in long ago. It is just the photons emitted from the falling object have a hard time climbing out of the potential well. Remember the object will rapidly fade from view as the photons as well as slowing down get redshifted out of existence. Remember that if you transform to certain other coordinate systems (Kruskal, Finkelstein) then you don't see anything special from a physical standpoint at the R=2M position but only at R=0 the singularity itself. Thus as I stated above the event horizon is an artificial effect of the coordinate system you are using (typically spherical polar in a Schwarzchild solution for a point mass in a flat spacetime.)
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Mike Holland Member (Idle past 483 days) Posts: 179 From: Sydney, NSW,Auistralia Joined: |
Yes, Percy, it gets confusing, because we have Special Relativity and General Relativity effects combining here.
If we didn't have General Relativity sticking its oar in, then the falling object would reach the speed of light at the event horizon (assuming it fell from infinity), and we would see normal Special Relativity time dilatation (dilation?) effects for both reference frames before this moment. But the gravitational field adds a non-relative GR effect. Contrary to Eta's post, atomic clocks in orbit HAVE been observed to run FASTER than those on the ground - after the effects of orbital motion have been eliminated. If an observer was suspended just above the event horizon by a very powerful rocket motor, and we were suspended in a similar manner, but a million kilometres away, then there would be no relative motion to bother about, and he would see us (and the rest of the universe) running fast. But if he is falling at high velocity, then SR adds its effects, and I don't know how they add up, but I am certain the GR time dilatation will slow him up just before he reaches the event horizon or speed of light. NB. Imagine an object falling through an event horizon. Just as it reaches there it reaches the velocity of light, acquires infinite mass, and sucks the whole universe in with infinite acceleration. Sorry, I don't buy it. I am working on a reply to Eta to cover this. Some research needed. Mike.
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Mike Holland Member (Idle past 483 days) Posts: 179 From: Sydney, NSW,Auistralia Joined: |
For Eta and Nosyned:
Yes, the speed of light does decrease, but as Eta says, it depends on your reference frame. An observer down there with the slow light would have his clock slowed in the same ratio, so his measurement would still be c. Replying on behalf of ‘All’, There are many entries on the web describing the problems with timing GPS systems due to the GR effect, which causes clocks in orbit to tick over FASTER than those down here - by 442.5 parts in 10**12, or 45 microseconds per day. A couple of references that I looked up arehttp://www.phys.lsu.edu/mog/mog9/node9.html Page Not Found | PhysicsCentral Some other references for the time effects near an event horizon (quotes from 'authorities') - If the observer follows a stone falling into a black hole, he will find that close to the Schwarzschild sphere the stone starts to ‘decelerate’ and gets to the black hole boundary only after an infinitely long timeA distant observer will see a similar picture in the very process of generation of a black hole, when the stellar matter is pulled by gravitation towards the centre. For this observer, the surface of the star takes an infinitely long time to reach the Schwarzschild sphere as if stalling at the gravitational radius. The red shift due to time dilatation caused by a strong gravitational field is aggravated by reddening due to the Doppler effect. Igor Novikov Black Holes and the Universe As Betty hovers closer to the Schwarzschild radius, so the timewarp grows ever larger. She will see Ann’s clock racing faster and faster ahead of her own. ..... But, although in principle Ann from afar would see events in the spaceship running very slowly, in practice she has a hard time seeing anything at all, because of the spiralling red shift....what looks like a black hole is in reality a star frozen in the very late stages of collapse. But all the properties of this collapsing star become very rapidly (typically in milliseconds or less from the onset of collapse) indistinguishable from a genuinely empty, already-formed black hole. This means that, in the few microseconds that Betty takes to whiz across the horizon, all of eternity will have passed by outside. The fast-forwarded images will accelerate to infinite speed. Betty will know, once inside the black hole, that the universe outside is ‘over’, even if it has lasted forever. Paul Davies About Time ... the star’s implosion, as seen by a stationary observer who remains outside, would slow and ultimately freeze as the star’s surface approached the critical Schwarzschild circumference.John Preskill and Kip S. Thorne in Forward to Feynman Lectures on Gravitation. Is that enough to back up my claim that there is a GR time dilatation in addition to Special Relativity effects, and this is a slowing down of clocks near the event horizon, and not simply a red-shift due to the light taking longer to reach us; and also that the observer in the strong gravitational field will see the clock of the remote observer running fast? Mike.
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Sylas Member (Idle past 5259 days) Posts: 766 From: Newcastle, Australia Joined: |
quote: The issue of falling into a black hole is reasonably well discussed by a number of physicists on the web. There is no particular problem with falling past the event horizon. You just can't see things past the event horizon, because light does not get out. I understand you can express this as an object taking infinite time to fall through the event horizon from the perspective of an observer at rest and an infinite distance away. However, this is not the same as saying that it does not fall through the event horizon, and it is not as simple as the SR case of an observer's frame. In GR you don't speak of a frame so much as as a metric. The perception that things can't fall past the horizon is an error about what the metrics mean. A common metric (the Schwartzschild metric) diverges at the Schwartzchild radius. There are other metrics which are better suited for tracking what happens to an infalling object, such as the Kruskal-Szekeres metric, obtained by a suitable co-ordinate transformation, which also removes the divergence. In any case, a falling object itself certainly passes through the event horizon without dramas, and reaches the central singularity within a finite time. At that point, we don't really have a good grasp of what happens. But conventional physics handles falling past the event horizon fine. The singularity at the event horizon is more an artifact of the chosen co-ordinates or metric; not an indication that anything actually fails to get through. Check out
More about the Schwarzschild Geometry
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Eta_Carinae Member (Idle past 4374 days) Posts: 547 From: US Joined: |
Read my earlier post more carefully.
You jumped the gun and assumed something I did not say. Of course a clock runs faster in a plane at higher altitude than lower altitude. I was talking about when the observer 'observes' the other persons clock. OK think about it like this. As you fall through the event horizon you can only observe events in which the light can get to you. i.e. only light rays that end up on your PAST light cone can you observe. Distant future events cannot get in your past light cone THUS you do not see the universe live and die in your eyes as you fall through. Sorry if I wasn't clearer. This is hard to explain without drawing light cones and embedding diagrams. See Wald's book.
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Mike Holland Member (Idle past 483 days) Posts: 179 From: Sydney, NSW,Auistralia Joined: |
Well, you have presented me with a lot of contrary views, but have not provided evidence that my interpretation is wrong.
Eta, in message 16 you said:No, the observer falling through the event horizon does not see the universe age - he sees their clocks run slow just as they see his clock run slow. but when I said in message 17But the gravitational field adds a non-relative GR effect. Contrary to Eta's post, atomic clocks in orbit HAVE been observed to run FASTER than those on the ground - after the effects of orbital motion have been eliminated. you responded in message 20 withYou jumped the gun and assumed something I did not say Of course a clock runs faster in a plane at higher altitude than lower altitude. Do you accept that an observer in Earth’s gravitation field sees the same speeding up (blueshift) in an orbiting clock as an observer hovering above an event horizon observing a remote clock? Is there just a confusion because of the relative velocity of a falling observer? Thank you, cjhs, for referring me to that interesting website Page RedirectionFascinating animated diagrams. Must have taken an age to set it up. I will have to spend some time on it to get a feel for Kruskal-Szekeres coordinates. But it does support my arguments, as per the following extracts from some paragraphs - Gravitational slowing of timeThis time dilation factor tends to zero as r approaches the Schwarzschild radius rs, which means that someone at the Schwarzschild radius will appear to freeze to a stop, as seen by anyone outside the Schwarzschild radius. Gravitational redshiftThat is, an outside observer will observe photons emitted from within a gravitational potential to be redshifted to lower frequencies, or equivalently to longer wavelengths. Conversely, an observer at rest in a gravitational potential will observe photons from outside to be blueshifted to higher frequencies, shorter wavelengths. That the redshift factor is the same as the time dilation factor (well, so one's the reciprocal of the other, but that's just because the redshift factor is, conventionally, a ratio of wavelengths rather than a ratio of frequencies) is no coincidence. Photons are a good clocks. When a photon is redshifted, its frequency, the rate at which it ticks, slows down. So a hovering observer near an event horizon will see remote light blueshifted, which means that remote clocks run fast (just as observed by GPS systems), and the outside universe appears accelerated. You cannot separate these effects - they are all the result of General Relativity time dilatation. Schwarzschild Spacetime DiagramUnfortunately, I could not cut and paste this diagram, but if you have a look at it you will see that all the light ray world lines approaching the event horizon curve upwards and approach it asymptotically - they never reach it. The exact same thing applies to the world lines of any object falling in - it never reaches the event horizon (or does after an infinite time, if you like). So, in our time frame, where time in measured by our clocks and calendars, nothing has ever fallen into a block hole, and nothing ever will. Eta, you say that the event horizon has no physical reality, but it defines a volume from which no light can escape into the outside universe. Surely that is a physical reality! Cjhs, you say that it all depends on the metric, anda falling object itself certainly passes through the event horizon without dramas, and reaches the central singularity within a finite time Of course you can fall into a black hole. There is nothing to stop you. In fact, the fall is almost instantaneous - but by your clock, not ours. But in OUR universe it never happens. You have to remember than we have two observers, with very different space and time coordinates. I don’t care if he meets up with angels and fairies on the way, it doesn’t happen in our universe. No changing of metrics will allow light to pass out through an event horizon. We only have our clocks and metre-rules. Whatever metric we are in, we are stuck with it. I have never seen a clock calibrated for Kruskal-Szekeres coordinates, which look even worse than a logarithmic scale! I am sorry if I seem obtuse to you, but neither of you have produced anything to contradict my views yet. Many thanks for the time you have devoted to this cause. Mike. NB. How does one use italics, boldsface, etc in these posts? I can't find any options! [This message has been edited by Mike Holland, 01-30-2004]
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Eta_Carinae Member (Idle past 4374 days) Posts: 547 From: US Joined: |
Eeeeeeeeeeeeek!
OK I realise something you pointed out and I made a boo boo in my first post answering. I meant to say the person at the event horizon observes the distant observers clock run FAST. (I said slow - sorry). I was tired but heck not that tired. LOL But there is a competing effect here - the fact of how fast you are falling in. I'll be back over the weekend to (hopefully) clear this all up. I'll crank out some calculations - it's been years since I have thought about some of this stuff. But I stand by my later comment about not observing the entire universe age in front of your eyes - though I am wondering as to how much you can see any effect of this - hence I need to crunch some numbers and get out my copy of Misner/Thorne/Wheeler. When I said the event horizon has no physical reality I meant from the perspective of the person passing through it. He/she notices nothing special - only if they try to leave it then they find that their future time coordinate now points to R=0 - meet the singularity - CRUNCH!
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AdminAsgara Administrator (Idle past 2302 days) Posts: 2073 From: The Universe Joined: |
Hi Mike,
If you click on the edit button at the bottom of this message, you will see how I make items bold, or italics, or underlined.
if you want to learn more about the options on this forum, look to the left of a reply window and find the HTML is ON link and also the UBB Code is ON link. AdminAsgara Queen of the Universe
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Mike Holland Member (Idle past 483 days) Posts: 179 From: Sydney, NSW,Auistralia Joined: |
Thanks Asgara. I have edited my last post, and set up all the pasted quotes in italics. Much better. There will be no holding me back now.
Mike.
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Mike Holland Member (Idle past 483 days) Posts: 179 From: Sydney, NSW,Auistralia Joined: |
Thanks for your time and effort, Eta. I am not serious about the falling observer actually seeing the universe fade and die, because it happens in the last billionth of a second (by his clock) before he passes the event horizon.
But please have a look at the diagram on the website referred to by cjhs. It shows the paths of light rays curve up and approach the event horizon asymptotically. This is exactly the point I am making. Light rays never reach the event horizon (in our universe), and neither do falling objects. A falling object would be moving slower than light, so its world-line would be more vertical than those of the light rays. As it approaches the event horizon light rays will keep intersecting its worldline, and it 'sees' light approaching it from the outside universe for the rest of (our) time, but all in an instant in its time. Mike.
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Sylas Member (Idle past 5259 days) Posts: 766 From: Newcastle, Australia Joined: |
quote: If you look at the page I cited, there are several diagrams. Not all of them have the discontinuity of the Schwartzchild metric. When you say "our universe", you are using terms in a rather unusual way... which is a nice way of saying "incorrectly". The schwartzchild radius is not the boundary of a "universe"; it is a horizon of visibility. That's all. Rather than saying "light rays never reach the boundary (in our universe)" you should be saying "the schwartzchild metric cannot be used to show the event of light rays crossing the schwartzchild radius (so use a different metric)". The point you are making is not the same point being made by the diagrams. You are making a point that the point of no return from the black hold marks a boundary of the universe. That's not what the term "universe" means. Cheers -- cjhs
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Mike Holland Member (Idle past 483 days) Posts: 179 From: Sydney, NSW,Auistralia Joined: |
Sure, you could choose a metric with a time scale based on the reciprocal of our time. Then the object passes the event horizon at "time" zero. So what! If you look at those other diagrams you will see that time for us is no longer linear. In fact, as I remarked, one looks worse than logarithmic.
The point is, using these metrics, our clocks still read infinity when the object reaches the event horizon. The metrics do not change this fact. They simply distort the picture. Mike.
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Sylas Member (Idle past 5259 days) Posts: 766 From: Newcastle, Australia Joined: |
quote: What you call "facts" are relative to your chosen co-ordinate system; not absolute statements about spacetime. Your terms are not defined. What do you mean by "when" the object reaches the event horizon? For this statement to make sense, you need to have some kind of relationship between times in different locations in space. But there is no absolute relationship of this kind; and metrics emphatically do not distort the picture. What could that even mean? The distortion is the claim that there is some kind of absolute which the metrics are distorting. That appears to be your error in a nutshull. There is no one correct metric; and you can pick your metric and do appropriate transforms from one set of co-ordinates to another. It is all still the same spacetime. The Schwartzchild metric has a singularity at the event horizon. However, there is no special privileged status for that metric; and the distcontinuity is a property of the metric; not a break down in relativistic physics. For a sufficiently large black hole, a spacecraft flying though the event horizon would not notice any special discontinuity or sudden effect as they fly past the point of no return. For an outside observer, there are various ways (metrics) they might choose to apply co-ordinates to the space. With a suitable transform to more appropriate metrics (more appropriate for discussing what happens as an object falls through the event horizon) there is no singularity or discontinuity. And there is most certainly no absolute basis for matching up events in different locations as being "simultaneous". The singularity which appears at the center of the black hole is another thing entirely. That singularity appears in all the metrics, and it indicates that our current relativistic physics breaks down at the central singularity. We will need a better physical model (a mix of quantum mechanics and relativity) to describe that point in space. But the event horizon presents no such problem. It is described just fine with classical relativistic physics, and there is no reason to think anything special is going on at the event horizon which breaks existing physics. Cheers
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Mike Holland Member (Idle past 483 days) Posts: 179 From: Sydney, NSW,Auistralia Joined: |
Cjhs, you seem to have a conviction that all my concerns can be resolved by considering alternative metrics. You seem to think that event horizons have no real existence, and are theoretical constructs of particular metrics. This is wrong. Matter does not escape from a black hole, in any metric.
Let me try and handle your points one at a time: What you call "facts" are relative to your chosen co-ordinate system; not absolute statements about spacetime. Your terms are not defined. What do you mean by "when" the object reaches the event horizon? For this statement to make sense, you need to have some kind of relationship between times in different locations in space. But there is no absolute relationship of this kind; and metrics emphatically do not distort the picture. What could that even mean? So when we look at a Schwarzschild spacetime diagram, it is a ‘fact’ that light approaches the event horizon asymptotically, but when we look at an Eddington-Finkelstein diagram we see the ‘fact’ that light goes up the diagram at 45 degrees and crosses the event horizon. Fine. I have no problem with this. It is like the old conundrum ’Do parallel lines never meet, or do they meet at infinity?’. I see that Finkelstein time is given byF = t + ln(r-1), so at the event horizon r =1, and F = t + ln(0) = t - infinity! So the point where Finkelstein time shows a falling object crossing the event horizon is the point where our clocks read infinity. So we agree. A change of metric does not change the facts; it only changes the figures you use to describe them. But I have a problem with that metric because my wristwatch it not calibrated for Finkelstein time. The distortion is the claim that there is some kind of absolute which the metrics are distorting. That appears to be your error in a nutshull. There is no one correct metric; and you can pick your metric and do appropriate transforms from one set of co-ordinates to another. It is all still the same spacetime. As I pointed out above, a change of metric does not change the facts. But you are the one proposing to solve the problems by a change of metric. This is your error in a nutshell. Refer to your own first post on this topic:The perception that things can't fall past the horizon is an error about what the metrics mean. A common metric (the Schwartzschild metric) diverges at the Schwartzchild radius. There are other metrics which are better suited for tracking what happens to an infalling object, such as the Kruskal-Szekeres metric, obtained by a suitable co-ordinate transformation, which also removes the divergence. In any case, a falling object itself certainly passes through the event horizon without dramas, and reaches the central singularity within a finite time. So you propose to resolve the problem by choosing a more suitable metric! The Schwartzchild metric has a singularity at the event horizon. However, there is no special privileged status for that metric; and the discontinuity is a property of the metric; not a break down in relativistic physics. For a sufficiently large black hole, a spacecraft flying though the event horizon would not notice any special discontinuity or sudden effect as they fly past the point of no return. Yes and no! I have always accepted that the infalling observer would not see any space or time deformities, but the Schwarzschild diagram agrees with me that he would see a sudden effect if he looks back at the outside universe, because of the relative time dilatation. I have not analysed the other metrics in this regards, but if they disagree then we have a problem. Have another look at the Schwarzschild diagram. All those outside light rays curve upwards, but as the observer would be falling more slowly than light, his worldline would be more vertical, and all approaching light rays would intersect his worldline, for the rest of forever up the diagram. If you disagree with this reading of the diagram, please explain why. And if any of the other metrics give a different result, we have a problem. For an outside observer, there are various ways (metrics) they might choose to apply co-ordinates to the space. With a suitable transform to more appropriate metrics (more appropriate for discussing what happens as an object falls through the event horizon) there is no singularity or discontinuity. And there is most certainly no absolute basis for matching up events in different locations as being "simultaneous". Did I say ‘simultaneous’? Well, I suppose I implied it by talking about the clock of an outside observer when something happens at the event horizon. The singularity which appears at the center of the black hole is another thing entirely. That singularity appears in all the metrics, and it indicates that our current relativistic physics breaks down at the central singularity. We will need a better physical model (a mix of quantum mechanics and relativity) to describe that point in space.But the event horizon presents no such problem. It is described just fine with classical relativistic physics, and there is no reason to think anything special is going on at the event horizon which breaks existing physics. But these are exactly the problems that my view resolves. I do not believe anything special is going on at the event horizon - I do not believe that objects falling from infinity would reach the speed of light there and acquire infinite mass; I do not believe in a total time dilatation where time comes to a complete stop (in any metric or reference frame); I do not believe in a singularity at the centre where the laws of physics break down. It is you guys who believe in all these weird things. Have another look at my original post. My position is that the slowing of time as gravity increases means that these situations are never quite reached, however close conditions may approximate to them. I propose a universe with no such absolutes or singularities or breakdowns of physics. And I hold that this is the universe implied by Einstein’s theories. Mike.
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Sylas Member (Idle past 5259 days) Posts: 766 From: Newcastle, Australia Joined: |
quote: I don't expect all your concerns to be resolved quickly and easily. This is a difficult subject; and I am right at the limits of my ability to explain. No personal criticisms are intended. I do think it would help for you to consider alternative metrics, and why some diverge in some places and others do not. What I think about event horizons is given previously. They are a horizon of visibility. They are a point of no return. Once anything, even light, is inside the event horizon, it will inevitably reach the central singularity within a small finite time. This is even stronger than saying it can't get out. It will proceed to the centre; and at that central point current physics does not have a good description of things. Physics has a good description of things on either side of the event horizon, but not at the central singularity. The major problem of modern physics concerns reconciling quantum mechanics and classical relativity; and the point at which this becomes significant is the centre. I have been quite clear that what is different between the metrics is the divergence of co-ordinates. The event horizon is a point of no return, and metrics make no difference to that. But the horizon is not a singularity in all metrics. That is, comments about "can't get through" the event horizon and being frozen in time are relative to the metric, and not reflecting any absolute reality. Here are some very simple points, which need to be reconciled in any consistent perspective on this matter.
quote: Your facts refer to the diagrams and to co-ordinate systems. The diagrams look different, but they describe the same physical situation. Here is a quote from right next to the diagram.
The Schwarzschild spacetime geometry appears ill-behaved at the horizon, the Schwarzschild radius (vertical red line). However, the pathology is an artefact of the Schwarzschild coordinate system. Spacetime itself is well-behaved at the Schwarzschild radius, as can be ascertained by computing the components of the Riemann curvature tensor, all of whose components remain finite at the Schwarzschild radius. From More about the Schwarzschild Geometry by Prof. Andrew Hamilton Does light go through the event horizon? Yes, it does. You can continue to see the rest of the universe from inside the event horizon. That one of the metrics happens to diverge at the point of crossing the horizon is a statement about the metric; not about whether light gets through or not.
quote: Since you are not at the event horizon, you can't speak sensibly about what your clock reads "when" something else crosses the event horizon without using some co-ordinate system relating times at your location to times somewhere else. That was the major important question in my extract above, which you have quoted, but not answered. Indeed, you continue to make the same mistake by speaking of "the point when our clocks read infinity". Metrics are not ways to calibrate wrist watches. They are ways of giving time and space co-ordinates to events which are not next to you. What your wristwatch reads is time at your location (in any metric). It is a nonsense to speak of calibrating your watch to a spacetime metric.
quote:quote: Sheesh... Yes, I do resolve the "problem" of infinite time at the event horizon by choosing a metric where time is not infinite at the event horizon. So does anyone else who knows physics. If you don't even recognize that, I can't help you. Be careful not to read more in to that more than what I have said explicitly.
quote:quote: To the best of my knowledge, the Schwarzchild metric and time dilation does not imply any sudden effects at all for what is seen of the outside universe by an infalling observer. You are still mixing up divergence of lines in one particular metric with an inability for light to pass the horizon. That’s wrong. An observer who is actually inside the event horizon can continue to see light from the outside universe. Here is another page you may find of interest. Falling to the Singularity of the Black Hole. It includes movies of what can be seen as someone falls into the black hole. The associated text describes what is going on at different stages, in fractions of the Schwarzchild radius. Those fractions are 1.5, 1.0, 0.95, 0.68, 0.35, 0.01, 0.000000001, 0. Four of these views and descriptions refer to observations being made from well inside the event horizon, and the last refers to the singularity where current physics can't help us.
quote: Of course other metrics give different co-ordinates. The Eddington-Finkelstein diagram, for example, has incoming light paths being nice straight lines which don't diverge at all. There's nothing wrong with that. It is just a different allocation of co-ordinates to events. The "problem" is just one of learning the physics of relativity; not a problem with physics itself which is troubling to cosmologists. Humbling as it may be, you badly need to see the problem as one of understanding physics for a student. I'm in the same position, by the way.
quote:quote: Yes, that is all I mean. Using your watch to time events somewhere else requires a notion of simultaneity. I'm not trying to trick you with any of this.
quote:quote: Modern physics is weird, yes. You will never ever understand it if you insist on making it not weird. Note that infalling objects do not acquire infinite mass in any metric, and that divergence of time co-ordinates is an artefact of one allocation of time co-ordinates to events.
quote: If you approach this under the impression that you have a new theory of physics which resolves all the various breakdowns which are spoken of in the pages cited, then you are doomed to be yet another crank. The relativity groups are full of them. If you approach this with the recognition that you are student grappling to understand a difficult subject, then you can make progress. I won’t be able to help very much at all; since I am not an expert in relativity. Reading though the tutorials supplied by Professor Hamilton would be a good start. He is an expert. Best wishes -- Chris [This message has been edited by cjhs, 02-01-2004]
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