Black Holes - the two points of view.

[PAllen]

There are two different redshift phenomena taking place. If a clock is hovering near the Event Horizon, we will see it ticking slowly, at a rate dependent on the size of the BH and its distance from it. Any photons leaving it will be delayed as they escape the gravity, but ALL the photons will be delayed the same amount as the clock is hovering at a fixed point in the gravitational field. So there is no redshift due to the photons taking a long time to get to us. They all take the same long time to reach us. The only redshift in this case is due to the clock slowing, ie. gravitational time dilation. I agree that in this case the gravitatiional time dilation and the slowing of escaping photons are all part of the metric.

If the clock falls into the gravity field, then successive photons take longer and longer to escape the increasing gravity, and this creates a redshift in addition to that of a hovering clock.

Do you believe that there would be a photon redshift when the clock is not moving? Or do you not accept that as the clock descends into the gravity field, emitted light gets delayed more and more?

OK, I agree that a complete mathematical analysis of the descent should cover both redshifts, but there are still two processes involved. For a falling body, successive photons take longer to be emitted because of the time slowing, and then, in addition, take longer to reach us when they have been emitted. Two steps which both cause redshift.

Mike

For anyone hovering or falling clock emitting light, there is one factor for both redshift and time dilation (rate of the image of the clock) as seen by a distant observer. You can choose to factor this into a gravitational component and a speed component, but this is only possible in ideal geometries that don't exist in the real universe. In fact, the time dilation between two distant locations is a non-physical abstraction. The only thing that is physical is the observation of the rate of clocks of known intrinsic rate - and this will always agree with the redshift factor.

 

[Dale]

Redshift is always calculated the same way: take two timelike worldlines, on one pick two nearby events, calculate the proper time between them, find a null geodesic from each event to an event on the other worldline, calculate the proper time between those, the redshift is the ratio.

This procedure works in flat spacetime, curved spacetime, with or without motion.

 

[Mike Holland]

The time a photon takes to reach us when the clock is at a particular distance from the EH is affected by the delay of the photon in escaping from the gravitational field. If the clock is falling, then, in addition to a SR effect from its speed relative to us, the photons will take longer and longer to escape from the gravity, until at last they cannot escape at all.

There is an ever-increasing delay in the time photons take to get to us, and while this is not time dilation, it is a redshift that looks just the same to us.

Mike

 

[Dale]

I'm just saying that there is only one process for calculating redshift. You don't have to do two calculations and add or multiply them together.

 

[PAllen]

I think this point was made earlier in this thread, but I would like to pose it in a graphic form. This is the point that an eternal black hole as described by SC geometry almost certainly does not exist in our universe. Let's instead look at formation of black hole.

To be able to see the formation better from afar, let's have the far fetched scenario of a trillion stars of some super cluster collapsing with no net angular momentum, no accretion disk forming. I pick the far fetched number of a trillion stars because that allows the black hole to form while the stars are still well separated from each other, and individually resolvable (in principle) up until the last moments. Let's further assume there is a background of galaxies behind this collapsing cluster, but nothing in your line of sight in front of it.

What would you see? As the collapse occurred, you would see the cluster, as a whole, reddening, and more and more extreme Einstein rings from galaxies behind the cluster. Up until the last moments, you would see highly red shifted light from stars throughout the cluster - especially, you could see stars in the center of the cluster. Then, in a relatively brief period of time, the cluster would further redden/darken until it was blacker than even CMB radiation. Against the background galaxies, it would look, quite literally, like a black hole in the sky surrounded by an Einstein ring of light from galaxies behind it.

How would you want to interpret this? It is a mathematical fact that this is what you would see. Would you say that a trillion stars have actually vanished? Would you say that the stars in center magically are compressed invisibly against the not quite yet formed horizon (having jumped billions of miles from the center of the cluster to the edge of this black ball)? You could say there is an invisible ball of a trillion frozen stars, a millimeter larger than the theoretical event horizon. Then, if matter falls in, it soon vanishes and the black region grows slightly (after all settles down). Again, you could say the black ball is still just larger than the theoretical event horizon, with frozen stars throughout, and new matter somewhere at the outer edge.

If you prefer this interpretation, it is, indeed, not determinable from outside observations that further collapse has occurred inside the black region. However, I would than ask:

If look like a duck, ... . Isn't black hole a good description of the this scenario? Then if you ask, what would happen according to a ship orbiting one of those interior stars? GR has only one answer - further collapse (to a singularity), in very finite time for the ship.

 

[Mike Holland]

OK, PAllen, that's a lovely scenario for me to go and think about. I am inclined towards the invisible ball of almost-frozen stars a millimeter larger than the EH, but need to get my mind around it. Maybe add some numbers to the scenario.
Mike

 

[Mike Holland]

Right, we have a very large, clumpy mass of gas collapsing. Most of the calculations using the Schwarzschild metric have been done for a gas of uniforn density, but I doubt that these trilllion tiny clumps in this massive cloud will affect the results. In what way do you think the clumps (stars) will make a difference to the calculations/simulations that have been done for collapsing clouds of gas?

"Up until the last moments, you would see highly red shifted light from stars throughout the cluster - especially, you could see stars in the center of the cluster. Then, in a relatively brief period of time, the cluster would further redden/darken until it was blacker than even CMB radiation."

I rather doubt that. We are talking about a Black Hole about 3x10**12 kilometers in radius (assuming sun-like stars) and weighing in at 2x10**45 grams. One kilometer from the horizon the time dilation would be 1,000,000:1, and a millimeter away it would be 1,000,000,000:1. The extreme reddening and weakening of the light would make it invisible long before it reached this stage.

In Newtonian terms, the volume of the BH would be 10**30 cubic kilometers, where the sun is 10**18 cubic kilometers. So the "clumps" could maintain their integrity as they pass through the EH, except that tidal forces might rip them apart, but all that is in their local timeframe.

I'm sticking with the calculations based on the Schwarzschild metric, a frozen ball of clumpy gas, but the clumps may stick out more than a millimeter!

Mike

 

[PAllen]

Right, we have a very large, clumpy mass of gas collapsing. Most of the calculations using the Schwarzschild metric have been done for a gas of uniforn density, but I doubt that these trilllion tiny clumps in this massive cloud will affect the results. In what way do you think the clumps (stars) will make a difference to the calculations/simulations that have been done for collapsing clouds of gas?

They don't. The purpose for the scenario is the idea of (in principle) seeing inside the collapsing cluster.

"Up until the last moments, you would see highly red shifted light from stars throughout the cluster - especially, you could see stars in the center of the cluster. Then, in a relatively brief period of time, the cluster would further redden/darken until it was blacker than even CMB radiation."

I rather doubt that. We are talking about a Black Hole about 3x10**12 kilometers in radius (assuming sun-like stars) and weighing in at 2x10**45 grams. One kilometer from the horizon the time dilation would be 1,000,000:1, and a millimeter away it would be 1,000,000,000:1. The extreme reddening and weakening of the light would make it invisible long before it reached this stage.

No disagreement with what I said. It is all a matter of what is meant by 'the last moments'. Millimeter was a figure of speech, not a calculation. Also, note that the 1 million factor of redshift would still be quite visible - you would be imaging gamma rays (of which there would be plenty) as light or radio waves. Only when the the highest energy gamma rays are shifted below the lowest detectable radio wave frequency would true invisibility occur. That would happen quite fast on astronomical time scale (but I have not calculate a specific number).

In Newtonian terms, the volume of the BH would be 10**30 cubic kilometers, where the sun is 10**18 cubic kilometers. So the "clumps" could maintain their integrity as they pass through the EH, except that tidal forces might rip them apart, but all that is in their local timeframe.

It is actually well known that for collapsing cluster this size (actually, a good bit smaller) tidal forces in the outer regions (but still well within the event horizon) are less than 1 g.

I'm sticking with the calculations based on the Schwarzschild metric, a frozen ball of clumpy gas, but the clumps may stick out more than a millimeter!

Mike

Agreed - not meant to be literal.

 

[PAllen]

Note, you have ignored the question of what GR predicts for a rocket orbiting a star in this cluster. It is one thing to say that you think this situation is one where GR breaks down. It is quite another to dispute that GR has a completely unambiguous prediction for the rocket inhabitants experience.

Note also, that the weight of evidence (IMO) is that the cosmic censorship hypothesis is false in GR; that is that naked singularities are a prediction of GR. First, sufficiently rapidly rotating collapse is shown to produce them in exact solutions (extreme parameters of the Kerr metric). Second, most numerical simulations of appropriate conditions indicate the formation (for a period of time) of naked singularities. See:

http://prd.aps.org/abstract/PRD/v19/i8/p2239_1

and a number of later studies reaching the same conclusion.

If naked singularities are predicted by GR it is completely untenable to claim that internal region of a collapsing cluster or cloud is not a real prediction of GR. You are in the absurd situation of claiming that if GR says an event horizon would form, it doesn't and the collapse doesn't proceed (just because one class of observer won't visually see it); while if an event horizon is not predicted, the collapse does proceed (because it can be seen), leading to the most extreme form of black hole - the naked singularity.

I suggest you adopt the consistent position (shared by a number of prominent physicists) that GR breaks down before formation of event horizons (let alone naked singularities) rather trying to dispute what GR predicts.

 

[Mike Holland]

Note, you have ignored the question of what GR predicts for a rocket orbiting a star in this cluster. It is one thing to say that you think this situation is one where GR breaks down. It is quite another to dispute that GR has a completely unambiguous prediction for the rocket inhabitants experience.

Good Lord! Have I expressed my views that badly? I don't believe GR breaks down anywhere. And as for the rocket inhabitants experience, I believe that GR has at least two equally valid and compatible predictions - from his point of view and from our point of view. In fact, I believe it covers all points of view. I have not gone into what happens inside an Event Horizon, because the maths is too complicated for me, and because I don't believe it is relevant to my argument which is all about what we see and infer about gravitational collapse, where Schwarzschild coords hold sway.

Note also, that the weight of evidence (IMO) is that the cosmic censorship hypothesis is false in GR; that is that naked singularities are a prediction of GR. First, sufficiently rapidly rotating collapse is shown to produce them in exact solutions (extreme parameters of the Kerr metric). Second, most numerical simulations of appropriate conditions indicate the formation (for a period of time) of naked singularities. See:

http://prd.aps.org/abstract/PRD/v19/i8/p2239_1

and a number of later studies reaching the same conclusion.

OK, I was not aware of these calculations, and need to do some studying. But the main issue for my topic will be - How long does it take for such a singularity to form, from our external point of view. If they form in a finite time, then I guess my whole argument collapses.

If naked singularities are predicted by GR it is completely untenable to claim that internal region of a collapsing cluster or cloud is not a real prediction of GR. You are in the absurd situation of claiming that if GR says an event horizon would form, it doesn't and the collapse doesn't proceed (just because one class of observer won't visually see it); while if an event horizon is not predicted, the collapse does proceed (because it can be seen), leading to the most extreme form of black hole - the naked singularity.

Now you are mis-representing me. All along I have followed the predictions of GR, that a collapsing supermassive star (or whatever) would, in its own timeframe, form an Event Horizon and then collapse to a singularity. I have never denied this. But I have also quoted and agreed with a number of experts who calculate that this collapse to a Black Hole will take an infinite time in our remote observer timeframe.

You are taking the untenable position of claiming that only one timeframe is valid for describing these events - the local one. We don't visually see it because it hasn't happened in our time frame. The collapse proceeds in both timeframes - at different rates. Slower and slower in ours, and very swiftly in the local one. If you can't get your head around time flowing differently from different points of view (and time frames) and both points of view being equally valid, then you will never grasp this.

I can understand that some cosmologists would think GR breakls down at the EH, because the maths disagrees with their belief that Black Holes exist. I am convinced that GR is valid all the way.

Thanks for the info about naked singlarities.

Mike

 

[PAllen]

Good Lord! Have I expressed my views that badly? I don't believe GR breaks down anywhere. And as for the rocket inhabitants experience, I believe that GR has at least two equally valid and compatible predictions - from his point of view and from our point of view. In fact, I believe it covers all points of view. I have not gone into what happens inside an Event Horizon, because the maths is too complicated for me, and because I don't believe it is relevant to my argument which is all about what we see and infer about gravitational collapse, where Schwarzschild coords hold sway.

SC coordinates are just one choice for external observers. Claiming they are the only valid choice for external observers violates the essence of GR - that coordinates are purely a matter of convention, for all observers. It is equally valid for external observers to use any other coordinates.

Consider this phrase: "what we see and infer about gravitational collapse". You insist, actually, that we are not allowed to infer anything we don't see. That is an absurd restriction, throughout physics.

I haven't seen your answer to the difference between 'see' and infer for the following simple SR prediction, that I and several others have raised:

- two rockets accelerate away from each other at 1g. Quickly, each can not see or send signals to each other or to much of the universe. Each may continue to infer about the other rocket or the invisible universe. They are not required to consider that what they don't see doesn't exist for them.

OK, I was not aware of these calculations, and need to do some studying. But the main issue for my topic will be - How long does it take for such a singularity to form, from our external point of view. If they form in a finite time, then I guess my whole argument collapses.

Definitely finite time as seen by external observer. Also, note my wording versus yours:

As seen by some observer = as happens according to some observer

is a misinterpretation of GR. Consider yet again the acceleration rocket examples.

Now you are mis-representing me. All along I have followed the predictions of GR, that a collapsing supermassive star (or whatever) would, in its own timeframe, form an Event Horizon and then collapse to a singularity. I have never denied this. But I have also quoted and agreed with a number of experts who calculate that this collapse to a Black Hole will take an infinite time in our remote observer timeframe.

You have misinterpreted them as shown by examining their quotes in context (in most cases), and especially by looking at scientific writings versus popular writings. There are no global frames in GR. There are local frames and global coordinates. The former represent physics, the latter are matters of convention. The predicitions of GR about a 'universe' include all the coordinate charts (each arbitrary) needed to cover the universe.

You are taking the untenable position of claiming that only one timeframe is valid for describing these events - the local one. We don't visually see it because it hasn't happened in our time frame. The collapse proceeds in both timeframes - at different rates. Slower and slower in ours, and very swiftly in the local one. If you can't get your head around time flowing differently from different points of view (and time frames) and both points of view being equally valid, then you will never grasp this.

Obviously, I think it is you who is taking the untenable position. What we see and what we may conclude happens are different, throughout physics. Note that in addition to saying a distant observer sees clocks approaching an event horizon slow down and stop relative to theirs, it also says the infalling observer sees our clocks proceeding normally (not at the same rate, but differing only by a finite factor) compared to theirs, through the event horizon and up to the singularity. So which is the real time rate comparison between external in infalling observers? GR really says there is no unique answer to to this question, not that an external observer must interpret the comparison using SC coordinates.

I can understand that some cosmologists would think GR breakls down at the EH, because the maths disagrees with their belief that Black Holes exist. I am convinced that GR is valid all the way.

Thanks for the info about naked singlarities.

Mike

Just to be clear, if you think GR is valid all the way, then doesn't that include the validity of what the rocket in my collapsing cluster experiences?

What almost all physicists think is that when quantum effects are taken into account, singularities do not actually form, in contradiction to GR predictions [Even you agree GR predicts a singularity for the rocket observer in my collapse scenario]. As for event horizons, there are a wider range of views about what a quantum corrections to GR would imply, and the issues flow from ideas about Hawking radiation and quantum information. Some think you get something that looks macroscopically like a horizon, but microscopically it is not; some think you get nothing resembling a horizon; and many other variations as well.

 

[TrickyDicky]

What almost all physicists think is that when quantum effects are taken into account, singularities do not actually form, in contradiction to GR .

Are you sure? I don't have any reliable way to confirm that about physicists in general, but I don't think physicists here at PF think that.
In any case if what you say is true, that would mean BH's are no longer mainstream objects in physics.

 

[PAllen]

Are you sure? I don't have any reliable way to confirm that about physicists in general, but I don't think physicists here at PF think that.
In any case if what you say is true, that would mean BH's are no longer mainstream objects in physics.

I've seen almost all experts here at PF who have expressed an opinion state:

- singularities are predicted by GR
- singularities almost certainly don't actually exist, and this prediction shows GR breaks down in these extreme regions.

I have seen the same thing in my reading of professional papers.

However, your conclusion doesn't follow. Most professional physicists I know of, definitely believe BH's are part of our universe and are mainstream physics. They would have many of the macroscopic properties predicted by GR, but differ in details. Among those differences is that there is no actual singularity at the 'center'. As I mentioned, there is a wider range of views about the likely 'real nature' of the horizon. I think (this is more of guess compared to singularity viewpoint) is that a majority think the efforts to visualize a horizon in galactic center BH's will come out positive. This simply means there is something macroscopically behaves a lot like a horizon. There are several speculative theories of BSM physics that have this feature macroscopic horizon-like behavior, while micrcoscopically, there is no horizon.

[Edit: Let me reverse the challenge: I can think of no physicist in history who has stated they believe the GR prediction of actual singularities is true of reality, rather than an indication of breakdown of GR.]

 

[TrickyDicky]

I've seen almost all experts here at PF who have expressed an opinion state:

- singularities are predicted by GR
- singularities almost certainly don't actually exist, and this prediction shows GR breaks down in these extreme regions.

I have seen the same thing in my reading of professional papers.

However, your conclusion doesn't follow. Most professional physicists I know of, definitely believe BH's are part of our universe and are mainstream physics. They would have many of the macroscopic properties predicted by GR, but differ in details. Among those differences is that there is no actual singularity at the 'center'. As I mentioned, there is a wider range of views about the likely 'real nature' of the horizon. I think (this is more of guess compared to singularity viewpoint) is that a majority think the efforts to visualize a horizon in galactic center BH's will come out positive. This simply means there is something macroscopically behaves a lot like a horizon. There are several speculative theories of BSM physics that have this feature macroscopic horizon-like behavior, while micrcoscopically, there is no horizon.

I wouldn't consider the singularity in a black hole a mere "detail" one can simply drop. Without singularity there is no theoretical justification for the black hole concept, you may have some other object or process that explains certain astrophysical observations but certainly wouldn't be a BH unless one defines it as "whatever there is up there that was formerly known and interpreted as a black hole".

 

[PAllen]

I wouldn't consider the singularity in a black hole a mere "detail" one can simply drop. Without singularity there is no theoretical justification for the black hole concept, you may have some other object or process that explains certain astrophysical observations but certainly wouldn't be a BH unless one defines it as "whatever there is up there that was formerly known and interpreted as a black hole".

The absence of the singularity is not detail - it is a signal of breakdown of GR. What differs only in possibly undetectable detail is the properties of an actual BH (modified by quantum corrections with no singularity), versus a classical GR black hole with a singularity as observed from afar.

 

[atyy]

I think a black hole is defined by the event horizon, not the singularity.

Concerning the view that singularities show that classical GR breaks down, is that true? I don't understand why one couldn't consider singularities real, and that we don't observe them because of cosmic censorship.

In quantum GR, yes the theory does break down mathematically at singularities (at least as far as we can see perturbatively).

 

[TrickyDicky]

I think a black hole is defined by the event horizon, not the singularity.

That is what wikipedia says, yes, but could you explain what the sense of having an EH without gravitational singularity is.
It is usually considered as the "mathematical" boundary that forms when gravitational collapse occurs, if there is no collapse since there's no gravitational singularity, it would seem no EH would form, there would be no boundary of no return for light.

 

[PAllen]

I think a black hole is defined by the event horizon, not the singularity.

Concerning the view that singularities show that classical GR breaks down, is that true? I don't understand why one couldn't consider singularities real, and that we don't observe them because of cosmic censorship.

In quantum GR, yes the theory does break down mathematically at singularities (at least as far as we can see perturbatively).

Generally, throughout history of physics, singular results have been taken to mean breakdown of the theory (as a model of reality, as opposed to a mathematical model). I don't see any difference between the classical infinities and quantum infinities as indicative of theory breakdown.

Whether we observe them or not, since it is possible to travel across the event horizon in a short time (for you) in classical GR, you must take seriously the singularity - or admit that you don't believe this prediction of GR [that you can travel across an event horizon in finite time].

Though, I guess you are a counter example to 'find one'.

Also, note that numerical simulations of ever greater sophistication, spanning 3 decades, have basically come down in favor of cosmic censorship being false as a prediction of GR.

 

[TrickyDicky]

The absence of the singularity is not detail - it is a signal of breakdown of GR. What differs only in possibly undetectable detail is the properties of an actual BH (modified by quantum corrections with no singularity), versus a classical GR black hole with a singularity as observed from afar.

Ok, but in that case I would see no reason to keep calling those objects black holes.

 

[PAllen]

Ok, but in that case I would see no reason to keep calling those objects black holes.

Fine, that is terminology. Most physicists prefer to say 'if it looks in almost every detectable way from afar like classical GR black hole with horizon', then we will continue calling it a black hole (even though a string theorist would tend to think its internal structure is a fuzzball). In BSM physics, people talk about quantum black holes rather than 'things that look like a black hole but are not'.

 

[atyy]

Generally, throughout history of physics, singular results have been taken to mean breakdown of the theory (as a model of reality, as opposed to a mathematical model). I don't see any difference between the classical infinities and quantum infinities as indicative of theory breakdown.

Whether we observe them or not, since it is possible to travel across the event horizon in a short time (for you) in classical GR, you must take seriously the singularity - or admit that you don't believe this prediction of GR [that you can travel across an event horizon in finite time].

Though, I guess you are a counter example to 'find one'.

Also, note that numerical simulations of ever greater sophistication, spanning 3 decades, have basically come down in favor of cosmic censorship being false as a prediction of GR.

Why can't I take the singularities seriously? Do they lead to mathematical contradictions?

Censorship is in general false, but we only need to hold in physical situations. Presumably we can even allow its falsity as a prediction of the theory, as long as the theory is not self-contradictory.

 

[DrGreg]

If Birkhoff's theorem is true, if a region of matter is smaller that its own Schwarzschild radius, an event horizon forms, regardless of the distribution of matter inside the horizon (provided it's spherically symmetric), and therefore regardless of whether the matter collapses to a singular point or not.

 

[PAllen]

If Birkhoff's theorem is true, if a region of matter is smaller that its own Schwarzschild radius, an event horizon forms, regardless of the distribution of matter inside the horizon (provided it's spherically symmetric), and therefore regardless of whether the matter collapses to a singular point or not.

True, but Birkhoff's theorem is only valid for perfect spherical symmetry, and no rotation. There are no perfect SC geometries in our universe. There is no analog of Birkhoff's theorem in the presence of rotation.

[edit: Also, note that Birkhoff's theorem assumes GR, which requires that an absolute singularity form in such a case, at least within the assumptions of the singularity theorems.

If you are suggesting that GR predicts that almost(?) all collapsed objects are surrounded by an event horizon, I agree and don't think many would disagree. I think there is also little disagreement that this implies that a successor theory including gravity, in order to match known predictions of GR, would predict something 'nearly like an event horizon', if not an actual event horizon. ]

 

[PAllen]

Why can't I take the singularities seriously? Do they lead to mathematical contradictions?

Censorship is in general false, but we only need to hold in physical situations. Presumably we can even allow its falsity as a prediction of the theory, as long as the theory is not self-contradictory.

You can if you want. You can be the exception that proves the rule. It is not a mathematical issue, nor is it in doubt that GR predicts them. The issue is what you think this says about GR as a physical theory. To me, and every physicist I know of who has written significantly on this, singularity is taken (by assumption, belief, not mathematical inconsistency) to represent the breakdown of the theory as a model of our universe, and the need for an alternative.

 

[DrGreg]

True, but Birkhoff's theorem is only valid for perfect spherical symmetry, and no rotation. There are no perfect SC geometries in our universe. There is no analog of Birkhoff's theorem in the presence of rotation.

Yes. A wholly valid point.

 

[atyy]

You can if you want. You can be the exception that proves the rule. It is not a mathematical issue, nor is it in doubt that GR predicts them. The issue is what you think this says about GR as a physical theory. To me, and every physicist I know of who has written significantly on this, singularity is taken (by assumption, belief, not mathematical inconsistency) to represent the breakdown of the theory as a model of our universe, and the need for an alternative.

Yes, that makes sense to me (actually, from the physics point of view, QM would kick in first, so this point should be mathematical, not physical:) BTW, is it clear that GR is mathematically consistent in the presence of singularities? I've been assuming that it is because we can make sense of eg. FRW or Schwarzschild solutions. But would a mathematician see it the same way?

 

[PAllen]

Yes, that makes sense to me (actually, from the physics point of view, QM would kick in first, so this point should be mathematical, not physical:) BTW, is it clear that GR is mathematically consistent in the presence of singularities? I've been assuming that it is because we can make sense of eg. FRW or Schwarzschild solutions. But would a mathematician see it the same way?

Sure, why not? Not that I'm a mathematical expert, but mathematicians study and classify singularities all the time. The proof of the Poincare Conjecture didn't involve banning singularities - just showing that they weren't of an unmanageable type.

 

[TrickyDicky]

BTW, is it clear that GR is mathematically consistent in the presence of singularities? ...would a mathematician see it the same way?

I started a thread that is now 10 pages long with those questions in mind.
I don't think a mathematician would object singularities per se, but I'd imagine they might object calling smooth manifold a space with singularities.

 

[atyy]

Sure, why not? Not that I'm a mathematical expert, but mathematicians study and classify singularities all the time. The proof of the Poincare Conjecture didn't involve banning singularities - just showing that they weren't of an unmanageable type.

I don't see why not, but I have zero understanding of anything from a rigourous point of view, so I wish to be handed the definitive answer from "on high"

 

[atyy]

I started a thread that is now 10 pages long with those questions in mind.
I don't think a mathematician would object singularities per se, but I'd imagine they might object calling smooth manifold a space with singularities.

I have no idea of rigour, but I've sometimes tried to read http://relativity.livingreviews.org/Articles/lrr-2005-6/ . There also seems to be something called the BKL conjecture, which is apparently well-defined enough to be a mathematical conjecture even though it's about singularities, eg. http://arxiv.org/abs/1102.3474 .

 

[Austin0]

Redshift is always calculated the same way: take two timelike worldlines, on one pick two nearby events, calculate the proper time between them, find a null geodesic from each event to an event on the other worldline, calculate the proper time between those, the redshift is the ratio.

This procedure works in flat spacetime, curved spacetime, with or without motion.

Just to clarify. Here you seem to be talking about relative dilation which requires two events in each worldline

I'm just saying that there is only one process for calculating redshift. You don't have to do two calculations and add or multiply them together.

If we are considering an EM emission from a free falling frame at a particular potential altitude to a receiver at infinity does it still hold? Or does it require calculation of the g dilation and the relativistic Doppler due to velocity?
I guess in a way I am just asking if there are two separate effects or only one, and if not two ; why not? Thanks

 

[PAllen]

Just to clarify. Here you seem to be talking about relative dilation which requires two events in each worldline
If we are considering an EM emission from a free falling frame at a particular potential altitude to a receiver at infinity does it still hold? Or does it require calculation of the g dilation and the relativistic Doppler due to velocity?
I guess in a way I am just asking if there are two separate effects or only one, and if not two ; why not? Thanks

It still holds exactly as Dalespam described it, in all cases he listed (SR pure doppler, GR any case, any space time, any world lines). One calculation, one core phenomenon. Any possibility of separation into 'gravitational redshift' versus doppler in GR depends on the special case of a sufficiently static metric. This is often done as a matter of convenience, but is undefinable if the geometry is far from static.

There is a less intuitive version of the same concept, due to J.L. Synge (1960) [in the cosmological framework, this approach was pushed in a well known paper by Bunn and Hogg (2008), but it was demonstrated in greater generality by Synge in 1960]. This approach also is true for every SR and GR case , one operation: parallel transport the 4 velocity of the emitter at moment of emission, along the null path the light follows to the receiver, then apply SR doppler formula using the transported emitter 4-velocity expressed in the local frame of the receiver, and the null path tangent also expressed in this local receiver frame. This will give the correct shift for every case. Synge took the view that there was really no such thing as gravitational or cosmological red shift per se, only relative velocity doppler corrected for the case of dynamic metric of GR (using the specified parallel transport). He especially railed against gravitational redshift because it is definable only for the special case of (sufficiently) static metric.

[Admittedly, Synge had somewhat icon iconoclastic views on GR. For example, he felt the principle of equivalence was mathematically false, and therefore of no value except historically.]

 

[Austin0]

It still holds exactly as Dalespam described it, in all cases he listed (SR pure doppler, GR any case, any space time, any world lines). One calculation, one core phenomenon. Any possibility of separation into 'gravitational redshift' versus doppler in GR depends on the special case of a sufficiently static metric. This is often done as a matter of convenience, but is undefinable if the geometry is far from static.

There is a less intuitive version of the same concept, due to J.L. Synge (1960) [in the cosmological framework, this approach was pushed in a well known paper by Bunn and Hogg (2008), but it was demonstrated in greater generality by Synge in 1960]. This approach also is true for every SR and GR case , one operation: parallel transport the 4 velocity of the emitter at moment of emission, along the null path the light follows to the receiver, then apply SR doppler formula using the transported emitter 4-velocity expressed in the local frame of the receiver, and the null path tangent also expressed in this local receiver frame. This will give the correct shift for every case. Synge took the view that there was really no such thing as gravitational or cosmological red shift per se, only relative velocity doppler corrected for the case of dynamic metric of GR (using the specified parallel transport). He especially railed against gravitational redshift because it is definable only for the special case of (sufficiently) static metric.

[Admittedly, Synge had somewhat icon iconoclastic views on GR. For example, he felt the principle of equivalence was mathematically false, and therefore of no value except historically.]

If I am understanding you correctly then in the case I outlined there is only one effect.
It sounds like the resulting shift is purely dependent on the relative velocity and in this case the received frequency would be the same whether the gravitating mass was there or not. Is this right??
If this is so then I don't understand what you mean when you say it can only be separated into gravitational and velocity components in a static metric. it sounds like there is no gravitational component to be separated in any metric.
If Synge doubted the g redshift to what did he attribute the observed relative dilation related to potential that occurs with static sources and receivers and remote clocks??
In what way is the EP not mathematically supported according to Synge??
Thanks

 

[PAllen]

If I am understanding you correctly then in the case I outlined there is only one effect.
It sounds like the resulting shift is purely dependent on the relative velocity and in this case the received frequency would be the same whether the gravitating mass was there or not. Is this right??
If this is so then I don't understand what you mean when you say it can only be separated into gravitational and velocity components in a static metric. it sounds like there is no gravitational component to be separated in any metric.
If Synge doubted the g redshift to what did he attribute the observed relative dilation related to potential that occurs with static sources and receivers and remote clocks??
In what way is the EP not mathematically supported according to Synge??
Thanks

You can factor it for a static metric because there is an identifiable class of static observers. Then you define redshift relations between these observers (computed e.g. with either Dalespam's approach or Synge's) as 'gravitational'. Then for, other observers, you figure total redshift, compare to instantly co-located static observers and call the difference kinematic. But for non-static metric, there is no class of static observers to perform this separation.

It is not true that mass makes no difference under this scheme. Stress-energy and geometry are interlinked, and parallel transport is affected by geometry (as are the way null paths connect world lines in Dalespam's approach). It is just that there is no need to factor it into separate effects, and in the general case, you can't.

Synge's position was that the spacetime was either curved or not, period. And that the difference is detectable mathematically in an arbitrarily small region; in the limit at a single point. Therefore he felt it was simply false to say gravity and acceleration in flat spacetime were locally equivalent. I don't agree this makes the principle useless - it just defines the bounds of its accuracy.

 

[Austin0]

You can factor it for a static metric because there is an identifiable class of static observers. Then you define redshift relations between these observers (computed e.g. with either Dalespam's approach or Synge's) as 'gravitational'. Then for, other observers, you figure total redshift, compare to instantly co-located static observers and call the difference kinematic. But for non-static metric, there is no class of static observers to perform this separation.

Forgive me if I am a little slow tonight. But let me see if I've got it right:
Given static observers S₁, S₂ with S₂at infinity and free falling observer FF with rel velocity v wrt S₂
S₁ and FF emit identical signals at the moment of co-location.
As received at S₂ the signal from FF will be equivalent to the signal from S₁ with the addition of a purely classical Doppler shift for relative velocity v.
Is this right?
If we consider another inertial frame I ,in flat spacetime with the same v relative to S₂ would there be any difference in received signals at S₂, between those from FF and I ?

It is not true that mass makes no difference under this scheme. Stress-energy and geometry are interlinked, and parallel transport is affected by geometry (as are the way null paths connect world lines in Dalespam's approach). It is just that there is no need to factor it into separate effects, and in the general case, you can't.

I thought that parallel transport of a vector along a geodesic left the vector unchanged. is this incorrect?
Thanks for your patience