On the nature of the infinite fall toward the EH

[PeterDonis]

Except possibly as a brief, initial speculation, corrected almost immediately, I never claimed monotonic was possible.

Yes, I agree, you didn't. I was only trying to make the point that, even though the singularity is spacelike, there *is* a possible monotonic "time ordering" of events on the singularity, which matches the time ordering of events on Lucky's worldline. That's kind of counterintuitive for a spacelike surface.

 

[Mike Holland]

Mike, I think what PAllen was referring to is that this "ideal coordinate frame" of yours is only valid if all the objects involved are at rest relative to one another, since gravitational time dilation can only be defined in a system that is static.

OK, but I was thinking of observers taking into account any motion relative to the black hole or any local gravity fields, and using SR and GR to calculate its effects on their observations. I thought they would be left with a time dilation which would be the same for all distant observers observing the same clock near the same supermassive object. They should see what O-S calculated for their ideal case.

PAllen, I obviously need to read up on close orbiting neutron stars, but what does SOL mean?

 

[PeterDonis]

OK, but I was thinking of observers taking into account any motion relative to the black hole or any local gravity fields, and using SR and GR to calculate its effects on their observations.

But any such "effects" will be frame dependent. There are no invariants corresponding to "gravitational time dilation" for objects that are falling into the black hole.

I thought they would be left with a time dilation which would be the same for all distant observers observing the same clock near the same supermassive object.

For a static clock, yes, you can meaningfully define a "time dilation" relative to distant observers. But you can't for an infalling clock.

They should see what O-S calculated for their ideal case.

O-S calculated the *proper* time along an infalling worldline. That's not the same as calculating a time dilation; they didn't do that for an infalling object, because it can't be done. There is no invariant relationship between the proper time O-S calculated for an infalling object and any sort of "time dilation".

 

[harrylin]

Yes it would. Unfortunately, the correct answer is not known. Somewhere in this thread I posted links to a 2007 paper by Krauss et.al. that argues one position; and a paper by Padmanabhan et.al. from 2009 that claims to refute the former. My belief is that the 2009 paper represents the 'majority view' (and I can't find any response to it from the 2007 authors), but it is far from 'settled physics'. Without responding to the 2009 paper, there are certainly new papers written in the framework of the 2007 paper. It appears to me that both string theory and LQG are more consistent with the framework of the 2009 paper, as is Hawking's proposal for resolving the information paradox.

The 2007 paper gives a clear conclusion about pure GR in a separate GR discussion; however the 2009 paper doesn't as clearly separate GR from GR+QM, and so I did not spot or understand what error in the "classical" GR calculation was supposedly demonstrated in the 2009 paper - or even if they claim that they did.
Please clarify what the claimed error is according to you - I'm sure that you understand it much better than I do.

 

[PAllen]

The 2007 paper gives a clear conclusion about pure GR in a separate GR discussion; however the 2009 paper doesn't as clearly separate GR from GR+QM, and so I did not spot or understand what error in the "classical" GR calculation was supposedly demonstrated in the 2009 paper - or even if they claim that they did.
Please clarify what the claimed error is according to you - I'm sure that you understand it much better than I do.

Except that your interpretation of an alleged classical result is at odds with how every expert here reads the paper, how every expert here reads the press releases, and how other authors refer to the 2007 paper (it is considered new only insofar as the quantum result).
You are entitled to your interpretation, but it is important to note that it is considered incorrect by every expert here.

 

[PAllen]

PAllen, I obviously need to read up on close orbiting neutron stars, but what does SOL mean?

SOL is a vernacular abbreviation I don't think I can render here. Google it. First urban dictionary meaning.

 

[stevendaryl]

Except that your interpretation of an alleged classical result is at odds with how every expert here reads the paper, how every expert here reads the press releases, and how other authors refer to the 2007 paper (it is considered new only insofar as the quantum result).
You are entitled to your interpretation, but it is important to note that it is considered incorrect by every expert here.

In an early paper by Padmanabhan and Narlikar, the authors do challenge the classical notion of the Schwarzschild solution. The paper mentions black hole radiation, but most of the paper is really concentrating on classical black holes.

http://www.academia.edu/2120610/The..._and_Padmanabhan_Foundations_of_Physics_1988_

 

[PAllen]

In an early paper by Padmanabhan and Narlikar, the authors do challenge the classical notion of the Schwarzschild solution. The paper mentions black hole radiation, but most of the paper is really concentrating on classical black holes.

http://www.academia.edu/2120610/The..._and_Padmanabhan_Foundations_of_Physics_1988_

Actually, I don't see any fundamental challenges to established understanding. The observation the information about the black hole is never in the past light cone of an external observer is standard. Stating that if new physics preventing actual horizons and BH in the real universe, physical theory would be 'in better shape' is also not a new idea or particularly controversial. In their conclusion, they also note that new physics is required to avoid the singularity as prediction of GR - there is no solution in classical GR. I also notice they don't address the singularity theorems at all, which is an unfortunate omission. They do mention negative energy as a way to avoid BH formation, which sidesteps the assumptions of the singularity theorems, but most would call that new physics - even within established quantum theory, inequalities governing negative energy imply it can't be a solution to the singularity problem of GR.

Despite the above caveats, it is also worth mentioning that this paper is very early in Padmanabhan's career (1987), and is published in a journal which at the time was a dumping ground for work unpublishable in mainstream journals.

 

[harrylin]

Except that your interpretation of an alleged classical result is at odds with how [..] other authors refer to the 2007 paper (it is considered new only insofar as the quantum result). [..]

You are ducking my question. Please clarify what the claimed error is according to you.

 

[PAllen]

You are ducking my question. Please clarify what the claimed error is according to you.

I don't believe the authors dispute the mainstream interpretation of BH formation except in light of the quantum result which allows a physical justification for saying part of the classical solution is not part of the universe - which goes beyond saying it is not observable by an external observer. Every expert here who has commented on the paper and press release interprets it this way, not as claiming any new classical interpretation.

 

[Dale]

You are ducking my question.

see post 221.

Pot kettle

 

[harrylin]

see post 221.

Pot kettle

No problem, although not exactly. I had to become selective as I received here an exponentially growing number of replies with requests for answers instead of answers; and that one I decided not to reply, as you were beginning to argue against my lack of appreciation of an argument that you presented and which I had no desire to discuss after having discussed practically the same recently in other threads. In contrast, I asked PAllen a simple question about the meaning of his argument that he presented, and the second time he clarified that he referred to the 2009 paper as it appears to refute the QM solution of the 2007 paper.

 

[PAllen]

No problem, although not exactly. I had to become selective as I received here an exponentially growing number of replies with requests for answers instead of answers; and that one I decided not to reply, as you were beginning to argue against my lack of appreciation of an argument that you presented and which I had no desire to discuss after having discussed practically the same recently in other threads. In contrast, I asked PAllen a simple question about the meaning of his argument that he presented, and the second time he clarified that he advanced the 2009 paper as it appears to refute the QM solution of the 2007 paper.

A question I thought I had answered 6 times already (in other threads), and immediately answered yet again.

 

[harrylin]

A question I thought I had answered 6 times already (in other threads), and immediately answered yet again.

I'm sorry that I can't afford to study every reply in every thread - even your last reply needed reading between the lines in order to extract your answer from it. As a reminder, this is the "classical GR" solution (based on related calculations) of Vachaspati et al that apparently is not (yet) disputed:

"we find that Schwarzschild coordinates are sufficient to answer the very specific set of questions we ask from the asymptotic observer’s viewpoint. [..]
the standard result [is] that the formation of an event horizon takes an infinite (Schwarzschild) time if we consider classical collapse.
[calculations ...]
R(t) = R_S only as t → ∞. This result is similar to the well-known result (for example, see [5]) that it takes an infinite time for objects to fall into a pre-existing black hole as viewed by an asymptotic observer [6]. In our case there is no pre-existing horizon, which is itself taking an infinite amount of time to form during collapse." -http://arxiv.org/abs/gr-qc/0609024

Their result disagrees with Kevin Brown's rather convincing sounding argument that "the "frozen star" interpretation [is not] logically consistent", because "there is nothing to prevent an event horizon from forming and expanding, and this implies that the value of m inside the horizon increases in finite coordinate time" - http://www.mathpages.com/rr/s7-02/7-02.htm

 

[Dale]

No problem, although not exactly. I had to become selective as I received here an exponentially growing number of replies with requests for answers instead of answers; and that one I decided not to reply, ... In contrast, I asked PAllen a simple question about the meaning of his argument that he presented, and the second time he clarified that he referred to the 2009 paper as it appears to refute the QM solution of the 2007 paper.

Understood, and I wasn't going to call you on it since it has been a fast moving thread. I just think it is funny of you to demand responses from PAllen. Particularly since it doesn't seem that your question to him is any more simple or important than mine to you.

 

[Dale]

you were beginning to argue against my lack of appreciation of an argument that you presented and which I had no desire to discuss after having discussed practically the same recently in other threads

Btw, despite your assertions to the contrary, I take this as clear evidence that you do not understand the mathematics and mechanics of the manifold/chart concept as presented by Carroll and as used in standard GR. I think that this is the single key issue for your GR-related education, if you desire.

 

[PAllen]

I'm sorry that I can't afford to study every reply in every thread - even your last reply needed reading between the lines in order to extract your answer from it. As a reminder, this is the "classical" solution (based on related calculations) of Vachaspati et al that apparently is not (yet) disputed:

"we find that Schwarzschild coordinates are sufficient to answer the very specific set of questions we ask from the asymptotic observer’s viewpoint. [..]
the standard result [is] that the formation of an event horizon takes an infinite (Schwarzschild) time if we consider classical collapse.
[calculations ...]
R(t) = R_S only as t → ∞. This result is similar to the well-known result (for example, see [5]) that it takes an infinite time for objects to fall into a pre-existing black hole as viewed by an asymptotic observer [6]. In our case there is no pre-existing horizon, which is itself taking an infinite amount of time to form during collapse." -http://arxiv.org/abs/gr-qc/0609024

Their result disagrees with Kevin Brown's rather convincing sounding argument that "the "frozen star" interpretation [is not] logically consistent", because "there is nothing to prevent an event horizon from forming and expanding, and this implies that the value of m inside the horizon increases in finite coordinate time" - http://www.mathpages.com/rr/s7-02/7-02.htm

And I already answered many times, as have others here, that you are misinterpreting this. Note the 'as viewed' , 'specific questions', 'Schwarzschild time'. Other advisers here (not just me) read this in contact with established usage of these qualifying terms. Nowhere is there a statement that a shell observer's view is invalid classically. Nowhere is there a claim that prior mainstream understandings are questioned. This classical analysis is presented to lay the basis for the quantum calculation. Then, and only then, is there a physical basis to consider the classical interior and shell observers irrelevant - because the quantum result is claimed to modify their history and physics from classical expectations 'quite early' according to their clocks [that is, their physics diverges from the classical prediction earlier than the shell observer clock time the classical prediction assigns to event horizon formation and crossing].

This has been debated ad nauseum. It seems clear you will hold onto your interpretation of the paper. However, if you do, I need to keep pointing out that multiple people here think you are misinterpreting the paper, claiming a result its authors do not claim. Thus, for us, the 2009 paper focuses only on quantum issues because there is no other new content to the 2007 paper. That is, without the quantum argument, there is nothing left to the 2007 paper.

 

[harrylin]

Understood, and I wasn't going to call you on it since it has been a fast moving thread. I just think it is funny of you to demand responses from PAllen. Particularly since it doesn't seem that your question to him is any more simple or important than mine to you.

Sorry that I can't afford the time to explain the difference to you; I would like to have infinite time but I already used half of it.

 

[harrylin]

And I already answered many times, as have others here, that you are misinterpreting this. [..]

Try to look what I wrote:
- no metaphysical interpretation; your tirade is completely misdirected.
- I did compare that with mathpages. That is relevant for rjbeery (the OP of this thread), as it is related to an earlier apparently erroneous answer to him some two years ago. I first checked it with you as I didn't understand the 2009 paper which you apparently do understand; your reply encouraged me to give this information. If he asks for more clarification then I'll give my 2cts.

 

[PAllen]

Try to look what I wrote:
- no metaphysical interpretation; your tirade is completely misdirected.
- I did compare that with mathpages. That is relevant for rjbeery (the OP of this thread), as it is related to an earlier apparently erroneous answer to him some two years ago. I first checked it with you as I didn't understand the 2009 paper which you apparently do understand; your reply encouraged me to give this information. If he asks for more clarification then I'll give my 2cts.

And again: I claim, along with others here, that there is no classical claim in the 2007 paper inconsistent with mathpages. This is based on understanding the math and background. Instead the 2007 paper deviates from mathpages (which just presents the mainstream understandings of GR since about 1960, on this issue) only by virtue of the quantum result.

As long as you make a false claim about a paper, it must be corrected on these pages to preserve their pedagogical value.

 

[harrylin]

And again: I claim, along with others here, that there is no classical claim in the 2007 paper inconsistent with mathpages. This is based on understanding the math and background. Instead the 2007 paper deviates from mathpages (which just presents the mainstream understandings of GR since about 1960) only by virtue of the quantum result.

As long as you make a false claim about a paper, it must be corrected on these pages to preserve their pedagogical value.

I can't make anything else of it; but everyone can make mistakes. So, if anyone else here expresses any doubts that Vachaspati found as "classical GR" solution an infinite Schwarzschild coordinate time for black hole forming whereas (according to me as well as this forum in 2010) Brown argues on his pages that this is impossible, I will ask Vachaspati to clarify this point in view of the pedagogical value.

 

[PAllen]

I can't make anything else of it; but everyone can make mistakes. So, if anyone else here expresses any doubts that Vachaspati found infinite Schwarzschild coordinate time for black hole forming whereas (according to me as well as this forum in 2010) Brown argues on his pages that this is impossible, I will ask Vachaspati to clarify this point in view of the pedagogical value.

This encapsulates several errors. Everyone agrees on infinite Schwarzschild coordinate time for black hole formation. Brown, and mainstream GR since 1960 supplements this statement with the understanding that this coordinate time has a limited meaning, and that if you ask what is predicted for the infalling matter you must conclude BH formation in finite clock time of the infalling clocks. And that there are many way besides SC coordinate time by which these events can be correlated with external events. This keeps circling back to the same misunderstanding explored with you in several threads and hundreds of posts here. Except in this context, you project your misunderstanding onto others.

 

[harrylin]

[..] Everyone agrees on infinite Schwarzschild coordinate time for black hole formation. [..].

I'm sure that that is very interesting for the OP, in view of the answers I saw that he received in 2010; the rest of what you commented is entirely unrelated to my remark about Schwarzschild coordinate time; that is your misunderstanding of what I said. I leave any further comments to him, thanks for the feedback.

 

[stevendaryl]

This encapsulates several errors. Everyone agrees on infinite Schwarzschild coordinate time for black hole formation. Brown, and mainstream GR since 1960 supplements this statement with the understanding that this coordinate time has a limited meaning, and that if you ask what is predicted for the infalling matter you must conclude BH formation in finite clock time of the infalling clocks. And that there are many way besides SC coordinate time by which these events can be correlated with external events. This keeps circling back to the same misunderstanding explored with you in several threads and hundreds of posts here. Except in this context, you project your misunderstanding onto others.

Is it really true that "Everyone agrees on infinite Schwarzschild coordinate time for black hole formation"? It sure seems that Brown is arguing otherwise.

Consider a black hole of mass m. The event horizon has radius r = 2m in Schwarzschild coordinates. Now suppose a large concentric spherical dust cloud of total mass m surrounds the black hole is slowly pulled to within a shell of radius, say, 2.1m. The mass of the combined system is 2m, giving it a gravitational radius of r = 4m, and all the matter is now within r = 4m, so there must be, according to the unique spherically symmetrical solution of the field equations, an event horizon at r = 4m. Evidently the dust has somehow gotten inside the event horizon. We might think that although the event horizon has expanded to 4m, maybe the dust is being held "frozen" just outside the horizon at, say, 4.1m. But that can't be true because then there would be only 1m of mass inside the 4m radius, and the horizon would collapse.

http://www.mathpages.com/rr/s7-02/7-02.htm

 

[PAllen]

Is it really true that "Everyone agrees on infinite Schwarzschild coordinate time for black hole formation"? It sure seems that Brown is arguing otherwise. http://www.mathpages.com/rr/s7-02/7-02.htm

I just read that whole link and I see nothing contradicting the statement that it takes infinite Schwarzschild coordinate time for a black hole to form. He goes to great lengths to explain exactly what this does and doesn't mean, physically, but never states anything different. He describes this as a mysterious fact that warrants explanation in light of other facts. People may be over-interpreting the following:

"Nevertheless, if mass accumulates near the exterior of a black hole's event horizon the gravitational radius of the combined system must eventually increase far enough to encompass the accumulated mass, leading unavoidably to the conclusion that matter from the outside must reach the interior, and it must do so in a way that is perceptible in finite coordinate time for a distant observer, which seems to directly conflict with Item 2 (and certainly seems inconsistent with the "frozen star" interpretation)."

However, note that he doesn't use Schwarzschild here, and calls this a paradox to be resolved. Throughout the later text, I see no contradiction to the statement that infinite Schwarzschild coordinate time is required. Instead, the text explains very clearly why this means much less than many people think.

Now, he also makes another statement which I consider false, and contradicted by many sources:

"(1) An event horizon can grow only if the mass contained inside the horizon increases."

On a quick read I don't quite see where he denies or fully explains the issue with this statement. However, for a collapsing shell, the horizon grows from the center with no mass at all inside it. Thus the explanation of this statement is simply that it is false in GR.

 

[PAllen]

"My finger never ends because the foobar coordinate goes to infinity on approach to my finger tip".

That is the sum total of so many debates here. Yes, foobar length of my finger is infinite. Yes, my finger is not very long; foobar length has a limited meaning. These are compatible, not contradictory statements.

 

[rjbeery]

"My finger never ends because the foobar coordinate goes to infinity on approach to my finger tip".

That is the sum total of so many debates here. Yes, foobar length of my finger is infinite. Yes, my finger is not very long; foobar length has a limited meaning. These are compatible, not contradictory statements.

OK it's rare that I literally laugh out loud at my computer screen...

 

[Dale]

So, if anyone else here expresses any doubts that Vachaspati found as "classical GR" solution an infinite Schwarzschild coordinate time for black hole forming whereas (according to me as well as this forum in 2010) Brown argues on his pages that this is impossible, I will ask Vachaspati to clarify this point in view of the pedagogical value.

I have doubts.

 

[PAllen]

I have doubts.

About which part?

 

[Austin0]

Everyone agrees on infinite Schwarzschild coordinate time for black hole formation. Brown, and mainstream GR since 1960 supplements this statement with the understanding that this coordinate time has a limited meaning, and that if you ask what is predicted for the infalling matter you must conclude BH formation in finite clock time of the infalling clocks. And that there are many way besides SC coordinate time by which these events can be correlated with external events. This keeps circling back to the same misunderstanding explored with you in several threads and hundreds of posts here. Except in this context, you project your misunderstanding onto others.

So everyone agrees on infinite coordinate time on a static clock at infinity for BH formation or infalling approach to EH but you all keep reiterating that this coordinate time has no physical meaning and that any of us who think there are questions here are attributing incorrect meaning to this evaluation.
That the proper time of the falling clock is a finite value.

But are you not attributing equal physical meaning to the subjective time of the infaller??
Time dilation is inherently a relative evaluation. What difference does it make what the elapsed time on the falling clock is.

These were addressed to Mike Holland in the original thread but I would like your response as he didn't address them.

you say the falling observers clock is never stopped in either frame because the distant observers clock never reaches infinity.
I agree. but you seem to ignore the fact that this is only true in the region where the faller has NOT reached the singularity (The EH.)
you then want to magically have the faller PASS the horizon without ever having reached it.

It appears you interpret time dilation in a way that creates alternate contradictory realities.
If your premise that reaching the horizon requires infinite coordinate time for the distant observer is correct, that means that at all points in that interval the times at the two locations will be related by the SC metric. Both observers will agree on these relative elapsed times and both observers will agree that the faller has not reached the horizon.

the fact that the time subjectively passes normally for the faller does not affect this relationship.

Would you disagree with this?

Austin0

An analogous scenario:
As system passes that is accelerating from the distant past that now has a gamma factor of 10²⁰ At this point we "observe" a passenger starting to walk from one end to the other. A stroll requiring 10 sec of ship time.
We jump ahead an interval on the order of the age of the universe 10¹⁰ Earth years. A future observer would see the passenger in virtually the same point in the walk with an elapsed time on his watch of 0.0018 secs.
Ahead another 10¹⁰ years etc.etc.

In fact the 10 seconds on the ship would equate to approx 5.5 x 10¹³yrs.
even without factoring in the increased gamma from the acceleration over this time.

SO for the next 3,500 ages of the universe both frames will agree the passenger has not reached the far end of the ship. The fact that time appears to be passing normally for the passenger does not mean that he will ever complete his trip in the real universe.

Which is what you are suggesting here . One universe where the passenger never completes his walk (reaches the horizon) and another where he finishes his walk and moves on (reaches the horizon and moves past it)

 

[PAllen]

So everyone agrees on infinite coordinate time on a static clock at infinity for BH formation or infalling approach to EH but you all keep reiterating that this coordinate time has no physical meaning and that any of us who think there are questions here are attributing incorrect meaning to this evaluation.
That the proper time of the falling clock is a finite value.

But are you not attributing equal physical meaning to the subjective time of the infaller??

It is proper time (= time on clocks; progress of physical processes) that is the observable quantity. Coordinate time and time dilation are not measurements or observables at all.

GR gives no preference to any observer or clock. It makes the specific predictions:

- distant observer never sees anything cross 'cross' a computed radius called the event horizon. If isolated, there is a black surface infinitesimally larger than this radius.

- infalling observer crosses horizon in and reaches singularity in finite time time on their clock.

These are the unambiguous, physical predictions. They are not contradictory.

Time dilation is inherently a relative evaluation. What difference does it make what the elapsed time on the falling clock is.

No, you confuse time dilation which is not an observable at all, with the most direct physical prediction of GR: proper time = progress of any physical process that evolves. Proper time is invariant not relative. Time dilation, the non-observable quantitiy, is what varies by observer and even convention.

These were addressed to Mike Holland in the original thread but I would like your response as he didn't address them.


Would you disagree with this?

Austin0

Obviously, I disagree with almost all of it. It is just wrong.

 

[PeterDonis]

But are you not attributing equal physical meaning to the subjective time of the infaller??

No, we are attributing physical meaning to the directly observable proper time on the infaller's clock. That is not "subjective", except in the trivial sense that it's that particular observer who directly observes it. But that directly observable number is an invariant; anyone can calculate it using any coordinate chart they like that covers the appropriate portion of the infaller's worldline, and they will get the same answer.

Furthermore, the proper time on the infaller's clock is only being used to make assertions about what happens along the infaller's worldline, i.e., along the worldline where that proper time is directly observable. The coordinate time is being used, by those who make assertions about what it "means", to make assertions about what happens *elsewhere* than on the worldline of an observer "at infinity", for whom coordinate time = proper time. It's the fact that something that can only be observed on one particular worldline (and on an idealized one at that, since it's the worldline of the observer "at infinity") is being used to make assertions about the entire spacetime, that creates the problem.

Time dilation is inherently a relative evaluation. What difference does it make what the elapsed time on the falling clock is.

The assertion that's being made is not about "time dilation". It's not relative. It's an assertion that the infaller's worldline continues all the way down to the singularity, because the infaller's proper time is finite and the spacetime curvature in the infaller's vicinity is finite all the way down to the singularity. Those are physical invariants--direct observations that the infaller can make. For the claim not to be true, physics along the infaller's worldline would suddenly have to start working differently at the horizon, for no apparent reason. That's why it makes a difference what the elapsed time on the falling clock is.

Would you disagree with this?

The relationship between the elapsed time on the infaller's clock and the coordinate time is fine for the portion of the infaller's trajectory that is above the horizon. And yes, both observers will agree that the infaller has not yet reached the horizon, *on that portion of his trajectory*.

But when the infaller reaches the horizon, he "disappears" from the distant observer's coordinates, and from his "line of sight", since light rays emitted at or inside the horizon can't get back out to the distant observer. The problem arises when people try to translate "the infaller disappears from the distant observer's sight at the horizon" into "the infaller never reaches the horizon, period". That's not a valid translation.

 

[Nugatory]

But are you not attributing equal physical meaning to the subjective time of the infaller??
Time dilation is inherently a relative evaluation. What difference does it make what the elapsed time on the falling clock is?

Use a sample of radioactive material, steadily decaying according to whatever its half-life is, as your clock. Now it's easier to see that the "reading" on this clock, namely the fraction of the original material that has decayed, has real physical significance; it's not in the least bit subjective. That's proper time.

Time dilation is the ratio of proper time on a given world line to proper time on some other world line. Neither of these proper times are subjective or relative, but the ratio between them depends on which "some other world line" you choose to calculate this ratio.

 

[Dale]

But are you not attributing equal physical meaning to the subjective time of the infaller??

No. GR attributes MORE physical meaning to proper time than to coordinate time. Proper time is an invariant and objectively measurable quantity, coordinate time is a frame variant mathematical convention. They are not given equal meaning.

 

[PeterDonis]

the ratio between them depends on which "some other world line" you choose to calculate this ratio.

And also on what simultaneity convention you adopt, so that you can pick out "corresponding" events on each worldline between which you are going to calculate the proper time elapsed, for comparison.