Black Holes - the two points of view.

[TrickyDicky]

One thing that confuses me and is not often mentioned in discussions about BHs and EHs, is the spacelike nature of the infaller observer's worldline, when it crosses the EH, for most people this seems to be something natural, but I thought it was generally assumed that physical observers are always timelike, their worldlines can never become null-like, unless they are massless like photons and much less become spacelike. Now since the EH is not considered a true singularity, one would tend to think physics at the interior side of the EH should not be that radically different than physics at the outside.

Since this is a physics forum rather than just a mathematical one I thought this might be relevant in a physics discussion. Maybe it is not.

 

[Dale]

One thing that confuses me and is not often mentioned in discussions about BHs and EHs, is the spacelike nature of the infaller observer's worldline, when it crosses the EH

An in falling observers worldline is timelike at all points. Remember, a free faller follows a geodesic, and a geodesic parallel transports its own tangent vector, and parallel transport preserves the norm. So if the tangent vector is timelike outside the EH then it remains timelike inside the EH.

 

[Mike Holland]

Mike, do you insist that seeing an Einstein ring means stars must be considered smeared into a ring by an observer that sees this? The effect BH on freezing light is physically the same phenomenon - just gravity bending or freezing light. There is no physical basis at all for interpreting this frozen light as 'reality' at all, let alone the only plausible reality for distant observers.

Not true. In addition to the red shift "freezing " of light, there is time dilation. Time dilation is real. We have measured the effect on clocks in high-flying aircraft and in orbit. If I go and hang out near a Black Hole for a while, and then return, I will have aged less than you, and my clock will be retarded. These effects are real, even if they are different for different observers. They have nothing to do with light paths. They have everything to do with the passage of time in a gravity field. I accept that there are light delay effects superimposed on this in some situations, and you are welcome to call these effects illusions.

Reality depends on your point of view. But that doesn't make it any less real. Points of view is all we have.


Mike

EDIT: Sorry, let me rephrase that. All we have is points of view and Einsteins theories to try and make sense of them.

 

[PAllen]

Not true. In addition to the red shift "freezing " of light, there is time dilation. Time dilation is real. We have measured the effect on clocks in high-flying aircraft and in orbit. If I go and hang out near a Black Hole for a while, and then return, I will have aged less than you, and my clock will be retarded. These effects are real, even if they are different for different observers. They have nothing to do with light paths. They have everything to do with the passage of time in a gravity field. I accept that there are light delay effects superimposed on this in some situations, and you are welcome to call these effects illusions.

Reality depends on your point of view. But that doesn't make it any less real. Points of view is all we have.


Mike

EDIT: Sorry, let me rephrase that. All we have is points of view and Einsteins theories to try and make sense of them.

But the time dilation you refer to is not a function of position, but a function of path. If, instead of comparing a distant observer with minimal proper acceleration with one who experiences enormous proper acceleration, we compare them with one in free fall toward the EH, starting from the distant observer, there is no significant time dilation between the near horizon free faller and the distant stationary observer. This near horizon observer sees their clock and the distant clock going at essentially the same rate. Who appointed you arbiter of which near horizon observers define reality?

[edit: In a real sense, GR does not have much 'relativity' in it - it is theory of invariants. The geometry of the manifold is the invariant 'reality'. Different coordinate systems on it are conventions. You want to elevate one coordinate system as the definition of reality rather than looking at the geometry of the manifold in a coordinate independent way.]

 

[TrickyDicky]

An in falling observers worldline is timelike at all points. Remember, a free faller follows a geodesic, and a geodesic parallel transports its own tangent vector, and parallel transport preserves the norm. So if the tangent vector is timelike outside the EH then it remains timelike inside the EH.

That seems the logical thing to expect, but there is a switch from timelike geodesics to spacelike ones at the EH, at least that is stated in many descriptions of black holes.

 

[PAllen]

That seems the logical thing to expect, but there is a switch from timelike geodesics to spacelike ones at the EH, at least that is stated in many descriptions of black holes.

No, this is incorrect. Perhaps you are confusing that the t coordinate in SC coordinates changes from timelike to spacelike. However, that is just an artifact of a coordinate choice. It doesn't happen for some other coordinates.

 

[PeterDonis]

That seems the logical thing to expect, but there is a switch from timelike geodesics to spacelike ones at the EH, at least that is stated in many descriptions of black holes.

No, there isn't. The tangent vector to a given geodesic must be of the same causal nature--timelike, spacelike, or null--everywhere on the geodesic. That's an elementary consequence of the geodesic equation.

What happens at the EH is that a particular vector *field*, the "time translation" Killing vector field, changes from timelike (outside the EH) to null (on the EH) to spacelike (inside the EH). But first, the integral curves of that vector field are not geodesics, and second, as you move from outside the EH to on the EH to inside the EH, you are moving to *different* integral curves of the vector field.

(In Schwarzschild coordinates, the "radial" vector field, defined by \partial_r, also changes from spacelike outside the EH, to null on the EH, to timelike inside the EH. But that is only true in that particular chart; in other charts, such as Painleve, the "radial" vector field stays spacelike all the way down to r = 0. The statement I made above about the time translation vector field is a coordinate-independent statement.)

 

[Dale]

That seems the logical thing to expect, but there is a switch from timelike geodesics to spacelike ones at the EH, at least that is stated in many descriptions of black holes.

There is no such switch. A path which switches from timelike to spacelike can not be a geodesic.

Could you provide a reference to such a description?

 

[DrGreg]

That seems the logical thing to expect, but there is a switch from timelike geodesics to spacelike ones at the EH, at least that is stated in many descriptions of black holes.

No, that is a mis-stated effect. In external Schwarzschild coordinates (i.e. outside the horizon), the t_e coordinate is timelike. In internal Schwarzschild coordinates (i.e. inside the horizon) -- a different coordinate system -- the t_i coordinate is spacelike & the r_i coordinate is timelike. That's just a matter of labelling coordinates.

Outside the event horizon its possible for a massive object to maintain a constant r_e, impossible to maintain a constant t_e coordinate.

Inside the event horizon its possible for a massive object to maintain a constant t_i, impossible to maintain a constant r_i coordinate.

 

[PeterDonis]

OK, I see how you are using the word "now", to include ALL events that COULD be "now" for us, depending on our chosen coordinate system. I would have called them "indeterminate".

That's a matter of terminology, not physics. However, the real point I was making is that there is *no* reasonable definition of "now" such that "now" includes events inside your past light cone.

just as our past light cone is distorted by the event horizon, so is our future one. The edge of my future light cone will approach the EH asymptotically (MY time frame, remember) and only reach it at t = infinity.

This doesn't change the fact (which you have agreed with) that a portion of the BH is in your future light cone. It also doesn't change the fact that the proper time elapsed for an infalling object, from your radius to the horizon, is finite. See below.

So none of these reference frames contradict calculations based on Schwarzschild coords.

They don't contradict them, but they do show limitations in them. See below.

The only question is, what time coordinates is my wristwatch based on? V = root(r/2GM - 1)? Or those of the falling raindrop?

That's the only question if we're concerned about proper time elapsed along *your* worldline, yes. But it's *not* the only question if we're concerned about proper time elapsed along some *other* worldline, spatially separated from yours.

If you think something can fall through the event horizon in a finite time, how does it avoid the time dilation?

It doesn't have to. "Time dilation" is relative; it depends on your worldline.

Since you're so insistent on doing calculations in Schwarzschild coordinates, try this one: write down the equation defining the proper time of an object freely falling radially inward from a finite radius r = R > 2M, to radius r = 2M. Write it so that the proper time is a function of r only (this is straightforward because it's easy to derive an equation relating r and the Schwarzschild coordinate time t, so you can eliminate t from the equation). This equation will be a definite integral of some function of r, from r = R to r = 2M. Evaluate the integral; you will see that it gives a finite answer. Therefore, the proper time elapsed for an infalling object is finite, even according to Schwarzschild coordinates.

 

[Dale]

Mike Holland, I don't know if you are avoiding my post 117 or simply have not had time, but I would like a response, in particular, to the question I posed there:

Do you believe that changing coordinate systems can make something start or stop existing?

 

[TrickyDicky]

No, that is a mis-stated effect. In external Schwarzschild coordinates (i.e. outside the horizon), the t_e coordinate is timelike. In internal Schwarzschild coordinates (i.e. inside the horizon) -- a different coordinate system -- the t_i coordinate is spacelike & the r_i coordinate is timelike. That's just a matter of labelling coordinates.

Just a matter of labelling coordinates? That was my initial idea but see for instance this quote by PeterDonis from a simultaneous similar thread:"The Killing vector field is not "swapped" by a coordinate transformation; it is a feature of the underlying geometry, independent of the choice of coordinates. So is the timelike, spacelike, or null nature of the Killing vector field at a particular event or within a particular region of the spacetime."
So then for instance the null-like condition of the EH is just a matter of labels or intrinsic to the geometry? And similarly what happens to the timelike or spacelike nature of geodesics outside and inside the EH, an arbitrary coordinate labelling issue? or a feature of the underlying geometry?

Inside the event horizon its possible for a massive object to maintain a constant t_i, impossible to maintain a constant r_i coordinate.

This doesn't seem a physical property for a timelike observer.

 

[DrGreg]

Just a matter of labelling coordinates? That was my initial idea but see for instance this quote by PeterDonis from a simultaneous similar thread:"The Killing vector field is not "swapped" by a coordinate transformation; it is a feature of the underlying geometry, independent of the choice of coordinates. So is the timelike, spacelike, or null nature of the Killing vector field at a particular event or within a particular region of the spacetime."
So then for instance the null-like condition of the EH is just a matter of labels or intrinsic to the geometry? And similarly what happens to the timelike or spacelike nature of geodesics outside and inside the EH, an arbitrary coordinate labelling issue? or a feature of the underlying geometry?

Whether a worldline is timelike, spacelike or null at a particular event is intrinsic to the geometry, and not dependent on a choice of coordinates. The scalar quantity given by the tensor expression g_{ab}U^aU^b, where U is a tangent vector to the worldline, is invariant, i.e. the same value no matter what coordinate system you use, and will be either positive, negative or zero. That determines whether the worldline is timelike, spacelike or null.

Particles with non-zero mass always travel along timelike worldlines (whether they are geodesics (in free fall) or not (acted on by a force)), everywhere, inside and outside the horizon. Particles with zero mass (e.g. photons) always travel along null worldlines.

 

[TrickyDicky]

The tangent vector to a given geodesic must be of the same causal nature--timelike, spacelike, or null--everywhere on the geodesic. That's an elementary consequence of the geodesic equation.

This is my understanding too.

What happens at the EH is that a particular vector *field*, the "time translation" Killing vector field, changes from timelike (outside the EH) to null (on the EH) to spacelike (inside the EH). But first, the integral curves of that vector field are not geodesics, and second, as you move from outside the EH to on the EH to inside the EH, you are moving to *different* integral curves of the vector field.

Are you sure?
I thought integral curves (flows) generated by Kiling vector fields in pseudoRiemannian manifolds were called geodesic flows because they were geodesic.

(In Schwarzschild coordinates, the "radial" vector field, defined by \partial_r, also changes from spacelike outside the EH, to null on the EH, to timelike inside the EH. But that is only true in that particular chart;

Didn't we agree that SC are valid only outside the EH?

The statement I made above about the time translation vector field is a coordinate-independent statement.)

And if so it must affect the geodesic flow consequently.

 

[TrickyDicky]

Whether a worldline is timelike, spacelike or null at a particular event is intrinsic to the geometry, and not dependent on a choice of coordinates. The scalar quantity given by the tensor expression g_{ab}U^aU^b, where U is a tangent vector to the worldline, is invariant, i.e. the same value no matter what coordinate system you use, and will be either positive, negative or zero. That determines whether the worldline is timelike, spacelike or null.

But that's what I said in my previous post and you replied that it was a matter of just labelling coordinates.

Particles with non-zero mass always travel along timelike worldlines (whether they are geodesics (in free fall) or not (acted on by a force)), everywhere, inside and outside the horizon. Particles with zero mass (e.g. photons) always travel along null worldlines.

This contradicts the quote from PeterDonis and your paragraph above, if you assert geometry determines whether a worldline is timelike, spacelike or null, you cannot say massive particles worldlines are always timelike independently of the intrinsic geometry.

 

[TrickyDicky]

There is no such switch. A path which switches from timelike to spacelike can not be a geodesic.

So I claim.

 

[TrickyDicky]

No, this is incorrect. Perhaps you are confusing that the t coordinate in SC coordinates changes from timelike to spacelike. However, that is just an artifact of a coordinate choice. It doesn't happen for some other coordinates.

I'm not talking about coordinates, I'm talking about Killing vector fields and the geodesic flows they generate in pseudoRiemannian manifolds.

 

[PeterDonis]

Are you sure?

Certainly. "Hovering" observers in Schwarzschild spacetime (observers who stay at a constant r > 2M for all time) travel along integral curves of the "time translation" Killing vector field. Those observers' worldlines certainly aren't geodesics; the observers have nonzero proper acceleration.

I thought integral curves (flows) generated by Kiling vector fields in pseudoRiemannian manifolds were called geodesic flows because they were geodesic.

They aren't. Where did you get the idea they were? [Edit: At least, they aren't in general. In some special cases, such as Minkowski spacetime, there are Killing vector fields that generate geodesic integral curves. I can't think of a case offhand of a curved spacetime where that's true, though.]

Didn't we agree that SC are valid only outside the EH?

No. Schwarzschild coordinates can be used inside the horizon, but that coordinate patch is disconnected from the patch that covers the exterior region (because of the coordinate singularity at r = 2M).

This contradicts the quote from PeterDonis and your paragraph above, if you assert geometry determines whether a worldline is timelike, spacelike or null, you cannot say massive particles worldlines are always timelike independently of the intrinsic geometry.

Yes, you can, because the law that "massive particle worldlines are always timelike" assumes that you already know the underlying geometry, so you know *which* curves are timelike. The law then just says that only those curves, the ones you already know are timelike, can be the worldlines of massive particles. (Similarly for massless particles like photons, with "timelike" replaced by "null".) Note that this goes for *any* curves, not just geodesics.

 

[Dale]

So I claim.

OK, so it sounds like you and I agree that an object which free falls across the EH must have a timelike worldline both inside and outside the EH. Then I guess it is just a mistake of one of the sources you have read that states otherwise. Could you please provide a link?

 

[TrickyDicky]

Certainly. "Hovering" observers in Schwarzschild spacetime (observers who stay at a constant r > 2M for all time) travel along integral curves of the "time translation" Killing vector field. Those observers' worldlines certainly aren't geodesics; the observers have nonzero proper acceleration.

These observers do not cross the EH so are not relevant to what I'm trying to get right.

They aren't. Where did you get the idea they were? [Edit: At least, they aren't in general. In some special cases, such as Minkowski spacetime, there are Killing vector fields that generate geodesic integral curves. I can't think of a case offhand of a curved spacetime where that's true, though.]

I think it is true of any spherically symmetric pseudoriemannian manifold. But let me find a good reference.

No. Schwarzschild coordinates can be used inside the horizon, but that coordinate patch is disconnected from the patch that covers the exterior region (because of the coordinate singularity at r = 2M).

Ah, ok, you just meant this.

Yes, you can, because the law that "massive particle worldlines are always timelike" assumes that you already know the underlying geometry, so you know *which* curves are timelike. The law then just says that only those curves, the ones you already know are timelike, can be the worldlines of massive particles. (Similarly for massless particles like photons, with "timelike" replaced by "null".) Note that this goes for *any* curves, not just geodesics.

Ok, so you are confirming that that law is subordinated to the geometry and therefore it should not contradict it.
So when DrGreg said that worldlines must obey the intrinsic geoemetry I uderstand that if the geoemetry says a worldline (geodesic or not) must be spacelike that means it cannot be followed by a massive observer, right?

 

[TrickyDicky]

OK, so it sounds like you and I agree that an object which free falls across the EH must have a timelike worldline both inside and outside the EH.

No, I agreed with what I quoted. I'm still trying to decipher how your statement here that "an object which free falls across the EH must have a timelike worldline both inside and outside the EH" does not contradict the intrinsic geometry of the K-S space.

 

[PeterDonis]

No, I agreed with what I quoted. I'm still trying to decipher how your statement here that "an object which free falls across the EH must have a timelike worldline both inside and outside the EH" does not contradict the intrinsic geometry of the K-S space.

Look at a Kruskal diagram, such as the one on the Wikipedia page:

http://en.wikipedia.org/wiki/Kruskal–Szekeres_coordinates

Any curve which is inclined to the vertical by less than 45 degrees on this diagram is timelike. There are many such curves that go from Region I to Region II; any such curve is a possible worldline for an infalling object. And of those curves, there are many that are geodesics.

If you want a mathematical description of such an infalling geodesic, it can be written in K-S coordinates, but it's kind of messy; it's easier to write it in ingoing Painleve coordinates for an ingoing timelike curve, or ingoing Eddington-Finkelstein coordinates for an ingoing null curve. For the timelike case, in Painleve coordinates, the ingoing geodesics are integral curves of the vector field \partial_T - \sqrt{2M / r} \partial_r, where T is the Painleve time coordinate. It is easy to show that this vector field is timelike everywhere.

 

[PeterDonis]

I think it is true of any spherically symmetric pseudoriemannian manifold. But let me find a good reference.

Please do, because Schwarzschild spacetime is a spherically symmetric pseudoriemannian manifold, and I gave an explicit example of a Killing vector field in it whose integral curves are not geodesics. So I'd be extremely surprised to find any reputable reference that claimed that the integral curves of a Killing vector field always *are* geodesics.

So when DrGreg said that worldlines must obey the intrinsic geoemetry I uderstand that if the geoemetry says a worldline (geodesic or not) must be spacelike that means it cannot be followed by a massive observer, right?

I would say "curve" instead of "worldline" for that precise reason; yes, a spacelike (or null) curve cannot be the worldline of any massive observer.

 

[Dale]

No, I agreed with what I quoted. I'm still trying to decipher how your statement here that "an object which free falls across the EH must have a timelike worldline both inside and outside the EH" does not contradict the intrinsic geometry of the K-S space.

What do you mean by "intrinsic geometry of the KS space"? I assume by KS you mean Kruskal–Szekeres coordinates, but that is a coordinate chart for a Schwarzschild spacetime and doesn't have any intrinsic geometry. The intrinsic geometry comes from the pseudo-Riemannian manifold representing the spacetime, not the coordinates.

In GR spacetime is represented by a pseudo-Riemannian manifold. The path of a massive particle is represented by a timelike curve in that manifold. So the worldline of any massive particle is always timelike regardless of whether it is geodesic or not and regardless of if it is inside or outside of an event horizon.

 

[Mike Holland]

Since you're so insistent on doing calculations in Schwarzschild coordinates, try this one: write down the equation defining the proper time of an object freely falling radially inward from a finite radius r = R > 2M, to radius r = 2M. Write it so that the proper time is a function of r only (this is straightforward because it's easy to derive an equation relating r and the Schwarzschild coordinate time t, so you can eliminate t from the equation). This equation will be a definite integral of some function of r, from r = R to r = 2M. Evaluate the integral; you will see that it gives a finite answer. Therefore, the proper time elapsed for an infalling object is finite, even according to Schwarzschild coordinates.

Peter, as I ubnderstand it, that is what Painleve did with his "raindrop" coordinates - provide a cioordinate frame that covers the object right through the falling process and beyond.

But I am at a loss to understand why there is an argument here. All along I have claimed that a falling observer will pass through the event horizon in his proper time, but that he won't in my proper time. This is what I understand from the calculations based on Schwarzschlild coordinates. So as long a I sit here in my armchair, I will see him hovering near the Evenyt Horizon forever edging closer and closer. As far as I am concerned, he hasn't fallen through it. But from his point of view in his spaceship, he just flies through and into the singularity. If the Black Hole is large enough, he won't even notice that he has gone through.

It is easy to fall into a Black Hole in a finite time, but not according to MY clock.

Mike

 

[Mike Holland]

Mike Holland, I don't know if you are avoiding my post 117 or simply have not had time, but I would like a response, in particular, to the question I posed there:
Do you believe that changing coordinate systems can make something start or stop existing?

My apologies, DaleSpam, I have been very busy. Also, your posts are usually the most difficult ones to answer. I need time to think.

If two coordinate systems do not overlap, then things that exist in one will not exist in the other. Similarly, if two systems have common space coordinates but no overlap in time coordinate, then you cannot convert x,y,z,t in one system to x,y,z,t in the other, so you cannot have the same event in both systems. But there can be cases of partial overlap to complicate things.

In the case of an Event Horizon, we can have a common x,y,z, but t does not overlap between my coordinates and the local ones. Any time coordinate at the event horizon corresponds to t = infinity in my coords, so no event there can be mapped into an x,y,z,t in my system. Events there do not exist for me. I can change coordinate system by jumping into the forming Black Hole, and when I get there it will come into existence for me.

Does that answer your question?
Mike

Edit: I've been thinking a bit further. What I said about Event Horizons still applies, but my ideas don't cover an event that did occur in my past, or will occur in my future. Such events exist on the timeline, but don't exist NOW. Things come into existence as I travel along my timeline, and others cease to exist as they move into my past. But in another sense of the word "exist", they do have an existence in my timeframe which Black Holes don't.

 

[TrickyDicky]

What do you mean by "intrinsic geometry of the KS space"? I assume by KS you mean Kruskal–Szekeres coordinates, but that is a coordinate chart for a Schwarzschild spacetime and doesn't have any intrinsic geometry. The intrinsic geometry comes from the pseudo-Riemannian manifold representing the spacetime, not the coordinates.

I was referring to the maximally extended Schwarzschild vacuum spacetime solution. KS coordinates happen to cover the whole spacetime.

In GR spacetime is represented by a pseudo-Riemannian manifold. The path of a massive particle is represented by a timelike curve in that manifold. So the worldline of any massive particle is always timelike regardless of whether it is geodesic or not and regardless of if it is inside or outside of an event horizon.

This is my understanding too, but IMO this contradicts the geometrically intrinsic definition of EH that PeterDonis gave.
See my answer to his posts below.

 

[TrickyDicky]

Look at a Kruskal diagram, such as the one on the Wikipedia page:

http://en.wikipedia.org/wiki/Kruskal–Szekeres_coordinates

Any curve which is inclined to the vertical by less than 45 degrees on this diagram is timelike. There are many such curves that go from Region I to Region II; any such curve is a possible worldline for an infalling object. And of those curves, there are many that are geodesics.

If you want a mathematical description of such an infalling geodesic, it can be written in K-S coordinates, but it's kind of messy; it's easier to write it in ingoing Painleve coordinates for an ingoing timelike curve, or ingoing Eddington-Finkelstein coordinates for an ingoing null curve. For the timelike case, in Painleve coordinates, the ingoing geodesics are integral curves of the vector field \partial_T - \sqrt{2M / r} \partial_r, where T is the Painleve time coordinate. It is easy to show that this vector field is timelike everywhere.

Yes, and it's a Killing vector field too, as long as Birkhoff's theorem holds.

Please do, because Schwarzschild spacetime is a spherically symmetric pseudoriemannian manifold, and I gave an explicit example of a Killing vector field in it whose integral curves are not geodesics. So I'd be extremely surprised to find any reputable reference that claimed that the integral curves of a Killing vector field always *are* geodesics.

I didn't say always, we are talking about spherically symmetric vacuums only.
According to Birkhoff's theorem a spherically symmetric vacuum has a timelike ∂t Killing vector field, in this context, the integral curves of this KVF are timelike geodesics.
Yor example is of an ingoing geodesic, so I don't understand what you mean by "are not geodesics".

Let me try and clarify where I see a problem here.
You said in the other thread " the event horizon is the boundary of the region in which the Killing vector field of the "time translation" isometry of Schwarzschild spacetime is timelike--i.e., the EH is the Killing horizon associated with the "time translation" isometry."
But then how is this compatible with the fact that spherically symmetric vacuums have timelike Killing vector fields (or to use Carroll's words in his Notes on GR:"We have therefore proven a crucial result: any spherically symmetric vacuum metric possesses a timelike Killing vector"), the logical question here would be then, is the region inside the EH (region II) a vacuum or not? If it is it must have ∂t KVF and therefore your EH definition would not be valid.

 

[Dale]

My apologies, DaleSpam, I have been very busy. Also, your posts are usually the most difficult ones to answer. I need time to think.

No problem, I understand that it can be difficult to keep up with a multiple-poster thread like this.

Does that answer your question?

Yes. The short answer is that you do believe that changing coordinate systems can make something start or stop existing.

Unfortunately, that position becomes ridiculous pretty quickly. In GR you are free to choose your coordinates in almost any way you like, you don't even need three for space and one for time, they don't have to be orthogonal or normal nor cover the whole spacetime. So, I could choose coordinates that do not cover anything to my right. Then, according to you, that light to my right, from which I am receiving a signal, does not exist. A change in coordinates and suddenly it exists.

Carried to the extreme, I do not even need to use coordinates which include me. So I can make myself not exist. Which begs the question, if I don't exist when I choose coordinates that don't include me, then how can something which doesn't exist choose coordinates?

Hopefully you can now understand why the idea of existence being determined by coordinate systems is repugnant to many people who understand the flexibility of coordinate systems in GR.

 

[Dale]

I was referring to the maximally extended Schwarzschild vacuum spacetime solution. KS coordinates happen to cover the whole spacetime.

OK, thanks for the clarification.

This is my understanding too, but IMO this contradicts the geometrically intrinsic definition of EH that PeterDonis gave.
See my answer to his posts below.

Since you and I agree then I will stop arguing . I didn't see any conflict with PeterDonis' comments (particularly since Killing vectors don't generally represent worldlines nor geodesics), but rather than try to explain what I think he meant I will let him explain directly. Perhaps you picked up on something I missed.