What is the Relationship Between Proper Time and Coordinate Time in Black Holes?

[Naty1]

Can you expand a little on the difference between relative and absolute horizons?

Not as well as I would like as this source is the only one where I've seen it explained. Here are a few insights from Kip Thorne:

The areas of absolute horizons will almost always increase and can never decrease

This quote implies to me Hawking and Beckenstein's entropy work and area increase theorem are based on ABSOLUTE horizons. But relative horizons should provide the same experimental results...see below. ...if one can get through more obscure mathematics...

Penrose's relative horizon is "the outermost location where photons trying to escape the hole get pulled inward by gravity...Hawking (later) realzied this old definition of the horizon...was an intellectual blind alley...he gave it a slightly contemptenous name...which would stick...the apparent horizon...Hawking's new definition was absolute (the same in all reference frames)...the boundary in spacetime between events (outside the horizon) that can send signals to the distant universe and those inside the horizon that cannot...when a hole eats another hole or collides with another hole or does anything at hole iots absolute horizon changes shape and size in a smooth, continuous way instead of a jumping way...

(parenthetical expressions are from the quoted text.)

I'm not sure I understand the fine points of his difference in definition, but Hawking's work relies on the new absolute horizon.

Hawking was well aware the choice of definition of horizon, absolute or apparent, could not influence in any way any predictions for the outcomes of experiments...however the choice of definition could influence the ease with which theoretical physicsts deduce from Einsteins equations the properties and behaviors of black holes.

I assume this means that both definitions result in the same "no hair" result. And should provide the same area increase/thermodynamic/entropy insights. It's unclear to me whether any experimental evidence could ever distinguish between absolute and apparent even horizons...I'm guessing not.

When matter starts to fall into a black hole the absolute horizon starts to grow before matter reaches it...(effect before cause) ..Hawking and James Hartle were able to develop a set of elegant equations that describe... the smooth and continuous growth of the absolute horizon...

The seeming paradox, effect before cause, has a simple origin. The very definition of the absolute horizon depends on what will happen in the future: on whether or not signals will ultimately escape to the universe. It is a teleological definition...and it forces the horizon's evolution to be teleological.

I don't get that!

My next reading assignment:
http://en.wikipedia.org/wiki/Absolute_horizon

 

[DrGreg]

Compare the uniformly accelerating rocket

rjbeery, I think it would be well worthwhile for you to consider in more detail the scenario suggested by George in post #4[/color].

Look at the more detailed diagram attached to about the Rindler metric - post #9[/color].

This takes place in flat space-time, i.e. no gravity, and we compare the "Minkowski" coordinate system (t, x) of an inertial observer with the "Rindler" coordinate system (T, X) of a uniformly accelerating (Born-rigid) rocket.

This scenario is remarkably similar, in many (but not all) ways, to a black hole. If you are in the rocket experiencing a proper acceleration of a and you drop a clock out of the rocket, you'll see the clock falling underneath you, but as it approaches a distance of c²/a, it slows down and never quite gets there. The time you can see on the clock slows down and never quite reaches a value of c/a since you dropped it.

In the left hand diagram, the black line is the accelerating observer, each red line is at a constant distance from that observer, in his coordinates, and each green line is a line of simultaneity for that observer. From the point of view of the accelerating observer, events in the purple region of the diagram never occur -- he will never see those events and cannot assign a time coordinate to them. In particular, the events along the diagonal line from (0,-10) through (20,10) have a time coordinate of T = \infty.

There is more about the mathematics of this in the thread in which that diagram appears.

 

[Naty1]

From
http://en.wikipedia.org/wiki/Apparent_horizon

Differences from the (Absolute) Event Horizon
In the context of black holes, the term event horizon refers almost exclusively to the notion of the absolute horizon. Much confusion seems to arise concerning the differences between an apparent horizon (AH) and an event horizon (EH). In general, the two need not be the same. For example, in the case of a perturbed black hole, the EH and the AH will generally not coincide as long as either horizon is fluctuating.

As I suspected...

 

[rjbeery]

Just in case the event horizon formed in xantox's bedroom , you may try http://motls.blogspot.com/2008/11/wh...nto-black.html .

Oh my, "may try" is the operative phrase! The link obviously pertains to this thread but that "explanation" is so obfuscated I think I pulled a brain muscle. Does it make sense to you, atyy? I'll give it another try after my morning coffee...

Consider a polar system of coordinates LaTeX Code: (u,\\theta) , where LaTeX Code: u=1/r. This system has an infinite value of LaTeX Code: u at the origin. Clearly there is a coordinate transformation that undoes the infinite coordinate at the origin and makes it finite. How is this any more difficult to understand?

But isn't this because moving from polar coordinates to cartesian is a bijective function? (1/inf, theta) is a discrete point and maps to (0,0), while a distant observer of black holes would claim that all of them form "at the same time" (i.e. never).

Melia's section 1.4 has a discussion of when black holes form in terms of cosmic time http://arxiv.org/abs/0705.1537.

Melia's paper was interesting, but the analysis presumed that the black hole formation begins at the time the Schwarzschild radius is reached, which is the very presumption that I am questioning.

xantox: You are describing the relationship between the finite proper time of black hole creation as seen from the free-falling body to its infinite coordinate time. The infinite observer will see all black hole formation time calculations as infinite, I believe.

Although their motives were different (they were struggling with the potential "information loss" paradox associated with black holes) some researchers at Case Western Reserve in Ohio are suggesting the same thing I am:
http://www.ccnmag.com/article/astronomers_may_have_solved_information_loss_paradox_to_find_black_holes_do_not_form"

http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=PRVDAQ000076000002024005000001&idtype=cvips&gifs=yes" is a link to their paper which was published in Physics Review D...

 

[DrGreg]

Oh, and by the way, the phenomenon I referred to in post #62 is an apparent horizon, but is not an absolute horizon.

 

[atyy]

Oh my, "may try" is the operative phrase! The link obviously pertains to this thread but that "explanation" is so obfuscated I think I pulled a brain muscle. Does it make sense to you, atyy? I'll give it another try after my morning coffee...

Try comparing the two lines in George Jones's picture in post #3 with the two lines in Motl's Penrose diagram. In both cases, the frame that is being used contains both observers. In special and general relativity, one shouldn't infer that things cannot happen by using particular frames that don't cover all of spacetime. In special relativity, the frame of an accelerated observer is a frame that doesn't cover all of spacetime - this is the frame of the guy below the red line in George Jones's picture. In general relativity there is the further possibility that no frame may cover all of spacetime, and you must patch your description together from partially overlapping frames (this is not the case in George Jones's and Motl's picture).

In general relativity, if you can determine all the data on some surface, then you can determine the spacetime you are in. However, if you can only determine some of the data, then you can only say the data you have is consistent with one of several spacetimes - it may be hard to list all the alternatives, so sometimes one goes with what one has as the present best guess. I don't know whether black holes are in the realm of determined completely by the available data, or just our best guess consistent with the available data.

 

[atyy]

"Observers cannot yet definitively confirm the form of the metric in the strong-gravity region, in order to prove that BHs are indeed described by the Kerr metric. The flow patterns close to the hole offer, in principle, a probe of the metric. Until X-ray interferometry is developed, actually ‘imaging’ the inner discs is beyond the capabilities of current instruments."
http://arxiv.org/abs/astro-ph/0701512

BTW, this is about whether the present data allows us to say a black hole exists for sure according to classical general relativity. Classical general relativity probably needs to be replaced by a yet unknown quantum theory of gravity, but here we are assuming that classical general relativity holds well enough.

 

[xantox]

You are describing the relationship between the finite proper time of black hole creation as seen from the free-falling body to its infinite coordinate time. The infinite observer will see all black hole formation time calculations as infinite, I believe.

If you agree on the first calculation giving a finite proper time for the horizon crossing by the infalling observer, then I don't see how you can say that black holes never actually form: if they form for the infalling observer, then they form. About the other observers, what does matter is the definition of time they use. In terms of the distant observer coordinate time or Schwarzschild time, the horizon forms at infinite t, for purely geometrical reasons. In terms of infalling proper time, conformal time or cosmological time then it forms at a finite t.

Although their motives were different (they were struggling with the potential "information loss" paradox associated with black holes) some researchers at Case Western Reserve in Ohio are suggesting the same thing I am

The very speculative proposal by Vachaspati et al. suggests that an horizon never forms not because it takes infinite time, but because some quantum effect would supposedly radiate away the mass quickly before the horizon forms – this is different from your argumentation in this thread, which seems based on the relation between infinite coordinate time and finite proper time – this relation is uncontroversial within general relativity.

 

[tiny-tim]

nothing happens until it happens

… if they form for the infalling observer, then they form.

But that's begging the question … how do we know that they form for the infalling observer?

Nothing happens until it happens, and the infalling observer doesn't find it forming until our-time is infinite, ie until our-never.

 

[xantox]

But that's begging the question … how do we know that they form for the infalling observer?

Nothing happens until it happens, and the infalling observer doesn't find it forming until our-time is infinite, ie until our-never.

We know that the infalling observer crosses the horizon in finite time simply because the theory predicts it, ie because we can calculate it. Or is it your question about how we could verify it experimentally?

 

[tiny-tim]

nothing happens until it happens

We know that the infalling observer crosses the horizon in finite time simply because the theory predicts it, ie because we can calculate it.

Exactly … predicts!

We can only predict that there will be a horizon! ​

 

[xantox]

Exactly … predicts!

We can only predict that there will be a horizon! ​

We can also postdict horizon formations in the distant past (eg primordial black holes, etc).

 

[rjbeery]

The very speculative proposal by Vachaspati et al. suggests that an horizon never forms not because it takes infinite time, but because some quantum effect would supposedly radiate away the mass quickly before the horizon forms – this is different from your argumentation in this thread, which seems based on the relation between infinite coordinate time and finite proper time – this relation is uncontroversial within general relativity.

Vachaspati's proposal is equivalent to mine because, as you say, quantum effects radiate the mass quickly before the horizon forms, but "quickly" is relative to the free-falling body's frame. In fact, the further one is from the apparent horizon, the more slowly this evaporation seems to occur, yet it always seems to occur before anybody is able to reach it (according to their calculations). Quickly or not, I've never seen a proposal suggesting that the evaporative process is infinite from any frame, and if that's true then the two arguments are equivalent in my opinion.

Let's make it simpler...the distant frame calculates the event horizon to be formed at t=infinity, while complete evaporation of the black hole due to Hawking radiation is calculated to be t = 5120*pi*G^2*M^3/(hc^4).
Isn't that sufficient to be able to say that the free falling body will be destroyed before it reaches the event horizon? Surely the ordering of local events cannot change (regardless of what kind of axes motls uses in his graph )

 

[xantox]

Vachaspati's proposal is equivalent to mine because, as you say, quantum effects radiate the mass quickly before the horizon forms, but "quickly" is relative to the free-falling body's frame.

To say that black hole horizons are not forming, some explanation is required. Vachaspati's speculative and controversial explanation is that they do not form because of some physical mechanism able to radiate away the mass during the extremely short time the black hole would take to collapse. Your explanation since post #1 seems to be another one, ie you seem to consider that the reason they do not form is because distant observers always register them forming in their infinite time and that you consider this to be inconsistent with a formation in finite time in the infalling frame.

Let's make it simpler...the distant frame calculates the event horizon to be formed at t=infinity, while complete evaporation of the black hole due to Hawking radiation is calculated to be t = 5120*pi*G^2*M^3/(hc^4).
Isn't that sufficient to be able to say that the free falling body will be destroyed before it reaches the event horizon? Surely the ordering of local events cannot change (regardless of what kind of axes motls uses in his graph )

When Hawking radiation is included in the picture, the distant observer does no longer calculate the event horizon to be forming at infinite coordinate time, but instead at a finite time corresponding with the end of the evaporation process. Nonetheless, the infalling body still had entered the black hole quickly, at the beginning of the incredibly long time it would take for an astronomical black hole to Hawking-evaporate.

 

[pesto]

Sorry to interject so late, and please let me know if my questions are unwelcome, but did I read correctly that an outside observer would not be able to see the creation of a black hole, but an observer inside the gravitational field would? This would be essentially the same situation as watching an object enter a black hole, and being the object?

Thank you.

 

[rjbeery]

during the 0.3 milliseconds a 10-solar mass black hole would take to collapse.

But this calculated .3 milliseconds is local to the formation! Why do you seem to assign such an absolute value to this time calculation when from most frames the collapse takes "forever"?

When Hawking radiation is included in the picture, the distant observer does no longer calculate the event horizon to be formed at infinite coordinate time, but instead precisely when the evaporation is completed.

Fair enough, but we both agree that relativity does not allow for a reordering of local events due to a transition in frames, correct? If the collapse happens after the evaporation is complete (or precisely as it completes) as calculated from the distant frame, then how can the local frame claim that the horizon was ever available to be crossed?

Sorry to interject so late, and please let me know if my questions are unwelcome, but did I read correctly that an outside observer would not be able to see the creation of a black hole, but an observer inside the gravitational field would? This would be essentially the same situation as watching an object enter a black hole, and being the object?

The outside observer (sitting at "an infinite distance" because it is mathematically simpler if he cannot feel any gravitational pull from the black hole) sees, or equivalently calculates, that the black hole never forms. Any body (someone called it a beer can) approaching the area where the event horizon supposedly resides would slow to a stop and dim into nothingness. The distant observer would calculate that the beer can will cross the event horizon at a time = infinity. Yet the beer can, from its perspective according to relativity, would cross just over just fine and quite quickly. I claim that the beer can's perspective is irrelevant because before "infinity" occurs other things happen such as the beer can and whole ball of wax evaporating into space over many billions of years (from the distant observer's perspective). Who knows what the beer can would actually experience? I'm sure it wouldn't be pleasant regardless...

 

[xantox]

But this calculated .3 milliseconds is local to the formation! Why do you seem to assign such an absolute value to this time calculation when from most frames the collapse takes "forever"?

Because the physical mechanism proposed by Vachaspati et al. must also act local to the formation, in order to radiate the local mass before it's locally too late and the horizon locally forms.

Fair enough, but we both agree that relativity does not allow for a reordering of local events due to a transition in frames, correct? If the collapse happens after the evaporation is complete (or precisely as it completes) as calculated from the distant frame, then how can the local frame claim that the horizon was ever available to be crossed?

Note that things happening in some place only happen in that place and nowhere else, and they do not ask permission to distant observers in order to happen. If an horizon was crossed there, then it was crossed – distant observers just receive or not receive some signals such as light rays carrying an information about that event. If the spacetime is too curved those light rays will get slowed down, and eventually remain trapped, then too bad for the distant observer who will get no photograph of the events, but in no way this means the events did not happen. Events are not getting reordered, as the evaporation happens behind the event horizon so it is less redshifted than signals at the horizon which can only get released when the horizon is no more.

Yet the beer can, from its perspective according to relativity, would cross just over just fine and quite quickly. I claim that the beer can's perspective is irrelevant because before "infinity" occurs other things happen such as the beer can and whole ball of wax evaporating into space over many billions of years (from the distant observer's perspective). Who knows what the beer can would actually experience? I'm sure it wouldn't be pleasant regardless...

The infalling body perspective is not irrelevant, to the contrary it's the only relevant one if we want to talk about what happens to the infalling body. You seem to suggest that during the few milliseconds of the body infall, by some trick infinite time has passed and these few milliseconds are just an illusion, but this is certainly not the case. What a distant observer calls "time" is not what the infalling observer calls "time", even if they share the same name – it is not possible to compare them more than it is possible to compare apples with oranges. What matters is that the free-falling observer has a watch, which is as good as any other in measuring proper time on his worldine, and under that worldline the proper time of the horizon crossing is short.

 

[rjbeery]

xantox: To me, if a black hole completes its evaporation precisely as the event horizon forms then at its final evaporative moment it is but a point, and the supposed event horizon is also a mere singularity which never came to be and therefore offers nothing to cross...

Anyway I've enjoyed this discussion, xantox, thank-you, this is how I learn. It seems we disagree on nothing but the interpretation of the facts and I think it's clear that our differences have boiled down to subjective opinion.

 

[qraal]

Just a thought, but if the horizon nevers forms there can be no Hawking radiation either. One follows from the other.

 

[xantox]

Just a thought, but if the horizon nevers forms there can be no Hawking radiation either. One follows from the other.

Correct. The two entirely different scenario commented above are:
a) Framework theory: classical general relativity. Horizons form at any cosmological time. They evolve. They can collide and merge. There are primordial black holes, and (in the semiclassical formalism) they eventually disappear by Hawking evaporation so there can be also young and old black holes, and some may form at cosmological times which are subsequent to the disappearance of prior ones. In case they form by gravitational collapse of a dense body, they form in a short proper time.
b) Framework theory: canonical quantum gravity. Vachaspati et al. speculate that horizons may never form, since the collapsing mass could be radiated by a quantum mechanism (not Hawking radiation) before it gets too dense.

 

[xantox]

xantox: To me, if a black hole completes its evaporation precisely as the event horizon forms then at its final evaporative moment it is but a point, and the supposed event horizon is also a mere singularity which never came to be and therefore offers nothing to cross...

Anyway I've enjoyed this discussion, xantox, thank-you, this is how I learn. It seems we disagree on nothing but the interpretation of the facts and I think it's clear that our differences have boiled down to subjective opinion.

In fact I failed to precisely grasp your argumentation. It seemed to me to be at moments an epistemological statement (such as "if a black hole doesn't form in my own frame then I may consider it doesn't exist", similar to "if nobody looks at the moon then the moon doesn't exist"), at moments a discussion within the general relativistic theory of black holes where you seemed to consider by comparing times measured on different frames that its conclusions about the formation of horizons are inconsistent and that horizons cannot form (this is however wrong), and at moments a statement of support of Vachaspati theory (where horizons cannot form for other reasons than relativistic time dilation and spacetime curvature). It is always nice to finish a discussion upon scientific agreement, however unless the above is clarified I cannot comment further.

 

[tiny-tim]

nothing happens until it happens

… If the spacetime is too curved those light rays will get slowed down, and eventually remain trapped, then too bad for the distant observer who will get no photograph of the events, but in no way this means the events did not happen. …

Yes it does …

in no way does this mean the events will not happen …

(though perhaps the most accurate way of describing it is to say: "the events will happen … never!")

but they certainly did not happen, in any sense of the word "did"!

Who knows what the beer can would actually experience? I'm sure it wouldn't be pleasant regardless...

From The Hitch-hiker's Guide to the Galaxy …

It would be unpleasantly like being drunk! ​

Just a thought, but if the horizon nevers forms there can be no Hawking radiation either. One follows from the other.

But wouldn't we see the same amount of radiation from the strong gravitational field outside an almost-black-hole? …

the "inward" half of the radiation, which with a black hole falls through the event horizon to vanish, will instead fall onto the surface of the almost-black-hole and vanish!

There would be no way to tell the difference.

 

[rjbeery]

This is true, I was fudging the math when I assumed that "pre"-Hawking radiation would occur at the same rate as Hawking radiation because I simply don't know that, and then I began referring to both generically as radiation or evaporation. This isn't a foundation of my point though, and my intuition (as well as tiny-tim's) is that the evaporation rate would be the same between a certified black hole and an "almost" black hole.

Xantox and I will have to agree to disagree because my interpretation is that t=infinity from our perspective means that black holes do not exist from our perspective. Is this different from saying that they don't exist at all? I don't know, but this thread has evolved and my ORIGINAL question was...

I'm pretty sure the SR time dilation math shows that the "outside world" clocks move to infinity during a black hole's formation, yet we seem to readily postulate that black holes currently exist...

Can someone that understands black holes please explain?

This question has been answered to my satisfaction, and the answer is that they do NOT currently exist in our frame. The rest is semantics and opinion.

 

[sloughter]

How do we know if a neutron star under collapse (going from closest packing configuration, being crushed, going to even more dense state), isn't capable of producing an event horizon where no light escapes? In other words, there is no need for a singularity.

To say that black hole horizons are not forming, some explanation is required. Vachaspati's speculative and controversial explanation is that they do not form because of some physical mechanism able to radiate away the mass during the extremely short time the black hole would take to collapse. Your explanation since post #1 seems to be another one, ie you seem to consider that the reason they do not form is because distant observers always register them forming in their infinite time and that you consider this to be inconsistent with a formation in finite time in the infalling frame.


When Hawking radiation is included in the picture, the distant observer does no longer calculate the event horizon to be forming at infinite coordinate time, but instead at a finite time corresponding with the end of the evaporation process. Nonetheless, the infalling body still had entered the black hole quickly, at the beginning of the incredibly long time it would take for an astronomical black hole to Hawking-evaporate.

 

[DrGreg]

...and the answer is that they do NOT currently exist in our frame.

That pre-supposes we all share the same frame. We don't. For those of us who choose to hover at a fixed distance above a collapsing black hole, we won't see it form. For those of us who choose to drop into the hole, we will see it form.

 

[George Jones]

How do we know if a neutron star under collapse (going from closest packing configuration, being crushed, going to even more dense state), isn't capable of producing an event horizon where no light escapes? In other words, there is no need for a singularity.

The Penrose-Hawking singularity theorems show that, under reasonable hypotheses, if there is a trapped surface, as there is inside an event horizon, then spacetime is singular. If you think spacetime isn't singular in this case, you need to show which of the hypotheses of the Penrose Hawking singularity theorems are violated.

 

[skeptic2]

I think the argument in this thread is not that spacetime is not singular inside the event horizon but that there is no inside to the event horizon.

 

[George Jones]

I think the argument in this thread is not that spacetime is not singular inside the event horizon but that there is no inside to the event horizon.

Not sure if you're referring to me, but my previous post was a response to sloughter's scenario of event horizon and no singularity.

 

[rjbeery]

Originally Posted by rjbeery View Post

...and the answer is that they do NOT currently exist in our frame.

That pre-supposes we all share the same frame. We don't. For those of us who choose to hover at a fixed distance above a collapsing black hole, we won't see it form. For those of us who choose to drop into the hole, we will see it form.

"Just when I thought I was out...they pull me back in!"
Tell me DrGreg, which among us "currently" chooses a frame dropping into a black hole? Someone at the LHC perhaps?

 

[xantox]

Yes it does …
in no way does this mean the events will not happen …
(though perhaps the most accurate way of describing it is to say: "the events will happen … never!")
but they certainly did not happen, in any sense of the word "did"!

No, as the sentence described the horizon crossing by an infalling body. For the body crossing the horizon, this did happen, and too bad for distant observers.

Xantox and I will have to agree to disagree because my interpretation is that t=infinity from our perspective means that black holes do not exist from our perspective. Is this different from saying that they don't exist at all? I don't know, but this thread has evolved and my ORIGINAL question was...This question has been answered to my satisfaction, and the answer is that they do NOT currently exist in our frame. The rest is semantics and opinion.

I guess the question is ill-posed. What "black holes do not exist from our perspective" may mean here?

• Either it means that we do not receive light signals from them (in that case, the statement is purely epistemological, and yes, this is different from saying that they don't exist at all, since at least some observers in the full manifold can receive light signals emitted at the horizon).
• Either it means that we describe the dynamics of their formation in terms of some "time" variable valid for us and find they don't form. I realize these two alternatives are what you called respectively "seeing" and "calculating" at the beginning of the thread. Unfortunately, this second approach is not unique, since there are many possible definitions of time. I emphasize that if we just say "time" without stating which definition is used, then the discussion is entirely devoid of meaning. Also, Newtonian or special relativistic notions of time cannot be used when dealing with black holes. In general relativity we can for example call time one of these:
  
  1. Coordinate time, which is a projection onto a global system of coordinates, allowing to map spacetime events just like longitude and latitude allow to map Earth locations. Here it is possible to locate a distant event in terms of a vector in the coordinate system. However, some coordinate systems, such as Schwarzschild coordinates, are singular at the horizon, so they give t=infinity. This simply means we need to use a different coordinate system which is not broken at the horizon and which gives a finite value there.
  2. Proper time measured on a worldline in spacetime. Here it is critical to understand that generally, each proper time is specific to its own worldline, and that it is not possible to express global dynamics as evolution in proper time, which is not spatially global, thus you cannot use this definition of time for the above argument. That is, the sentence "the event horizon of a black hole will happen at the distant observer's infinite proper time", eg where you attempt to define the dynamical evolution in terms of a generic distant observer's proper time, has absolutely no meaning whatsoever. However it is possible to say "light rays emitted by a body while crossing the event horizon of a black hole will be received from a distant observer after an infinite amount of his own proper time", here it is correct to measure the wordline of the observer until it "sees" a photon coming from the event horizon.
  3. Cosmological time, where the isotropic coordinates of comoving observers are singled out. And in general, we can single out some dynamical parameter such as the radius of the universe, so that evolution can be expressed in terms of that parameter. Here it is possible to locate the black hole formation in terms of such parameter (even if the whole manifold cannot be covered in general). When drawing the horizon on a conformal diagram using as vertical time coordinate such a time, eg a primordial black hole horizon segment will appear to begin at the bottom of the diagram eg in the young universe region, and not on the top, where is the infinite future.

So to resume, and using your terms of "seeing vs calculating":

• If you use coordinate time you will "calculate" that the horizon forms at a finite "t" upon choice of suitable coordinates. This "t" is just a coordinate.
• If you use cosmological time you will also "calculate" that the horizon forms at finite "t". This is a metric time obtained by operating a preferred slicing of the whole manifold.
• But, you cannot "calculate" anything about the horizon by using the proper time of a distant observer, as proper time on a specific worldline cannot be used to describe the evolution of the universe. You can merely say the distant observer will "see" a photon emitted at the horizon after infinite proper time on his worldline – which is not inconsistent with a horizon "calculated" as forming in finite proper time for the infalling body.