Black Holes - the two points of view.

[Dale]

If the labels are changed and now t is spacelike then the logical thing is to put t instead of r in the functions, isn't it?

No, r is still the radial coordinate and still has the usual meaning in terms of the surface area of the sphere, even though it is timelike.

 

[DrGreg]

I have found no proof of the theorem that interprets t as spacelike.
That interpretation relies on comparing an inside metric with an outside one, but the proofs derive the metric as a unique and independent entity.

Once you've got the equation<br /> ds^2 = - \left( 1 - \frac{2m}{r} \right) dt^2 + \frac{dr^2}{1 - 2m / r} + r^2 \left( d\theta^2 + \sin^2 \theta \, d\phi^2 \right)<br />then t is automatically spacelike whenever r &lt; 2m, almost directly from the definition of "spacelike". No comparison required.

If the labels are changed and now t is spacelike then the logical thing is to put t instead of r in the functions, isn't it?

If you wanted to, you could rewrite the metric inside the horizon as<br /> ds^2 = - \left( 1 - \frac{2m}{T} \right) dR^2 + \frac{dT^2}{1 - 2m / T} + T^2 \left( d\theta^2 + \sin^2 \theta \, d\phi^2 \right)<br />and that might remove the confusion that it seems to be causing for you, but it has the disadvantage that you then have to repeat the same argument twice instead of just using the same formula everywhere.

 

[Dale]

I'm not referring to the statements, all the proofs I've read show precisely how a general isotropic metric which has two functions depending on time and space, when those function are solved for Rab=0 you get that those functions are no longer depending on time.

Did you go back and look at the proof you are thinking of? Are you sure that "time" and "space" are not just "t" and "r" coordinates? Please confirm.

 

[Dale]

Once you've got the equation<br /> ds^2 = - \left( 1 - \frac{2m}{r} \right) dt^2 + \frac{dr^2}{1 - 2m / r} + r^2 \left( d\theta^2 + \sin^2 \theta \, d\phi^2 \right)<br />then t is automatically spacelike whenever r &lt; 2m, almost directly from the definition of "spacelike". No comparison required.

If you wanted to, you could rewrite the metric inside the horizon as<br /> ds^2 = - \left( 1 - \frac{2m}{T} \right) dR^2 + \frac{dT^2}{1 - 2m / T} + T^2 \left( d\theta^2 + \sin^2 \theta \, d\phi^2 \right)<br />and that might remove the confusion that it seems to be causing for you, but it has the disadvantage that you then have to repeat the same argument twice instead of just using the same formula everywhere.

The problem with this is that then your space is foliated by spheres of radius T and spherical symmetry implies that everything is a function of T only. Both of those are generally associated with the coordinate r. That, and you will confuse everyone else. Best to just stick to the standard notation and learn the quirks.

 

[PeterDonis]

That's a vacuum alright, the problem is that the Schwarzschild metric regardless the signature convention used is static.

This is not correct. See below.

I have found no proof of the theorem that interprets t as spacelike.

Yes, you have. I've given it to you.

That interpretation relies on comparing an inside metric with an outside one, but the proofs derive the metric as a unique and independent entity.

There is no "interpretation" involved. The proofs show that the metric is independent of the t coordinate, for all values of r (except r = 0 where there's a curvature singularity, and requiring special handling at r = 2m). You don't have to make any comparisons to show that; the proof for any given value of r goes through regardless of what happens at any other value of r. And as others have already noted, if r &lt; 2m then t is spacelike. So the proof is valid for t spacelike.

If the labels are changed and now t is spacelike then the logical thing is to put t instead of r in the functions, isn't it?

You can't just arbitrarily "put t instead of r in the functions". The r coordinate is defined by the fact that the area of a 2-sphere at r is 4 \pi r^2. That fixes how r occurs in the "functions" (I'm not sure what you mean by that term but I assume you're referring to things like the metric coefficients and Einstein tensor components). You can change the name of the r coordinate to something else, but that doesn't change its meaning.

Similarly, the "t" coordinate is the coordinate that, once you've completed the derivation of Birkhoff's theorem, the metric turns out to be independent of. You can change the label on it, but that doesn't change its meaning. And that meaning is the same for r &lt; 2m as for r &gt; 2m; it's the *same* coordinate in both regions.

Have you actually read the proof in MTW? If you don't have access to MTW, I have posted a similar proof on my PF blog here:

https://www.physicsforums.com/blog.php?b=4211

My version doesn't use exponentials to write the metric coefficients, as MTW does; I commented on that in a previous post. So my version makes it explicit that there is no restriction on the signs of g_{tt} and g_{rr}, and therefore there's no assumption that either the t or the r coordinate is timelike or spacelike; the proof is valid for t both timelike and spacelike, and for r both spacelike and timelike.

 

[Mike Holland]

PAllen, I'm surprised you haven't mentioned your trillion star black hole in this context. When the trillion stars merge sufficiently to form a black hole, the stars are still separated. So when does the interior become a vacuum?

Mike

 

[Dale]

Hi Mike, if you are back, I would like to know if you understand why the idea that existence is determined by the coordinate system is unworkable in GR, as discussed in 149.

https://www.physicsforums.com/showpost.php?p=4069859&postcount=149

 

[Austin0]

A question related to your trillion star illustration.

Assume a clock at the center of the sphere at a time before catastrophic collapse , where the density is somewhat diffuse.
This clock is transmitting time signals to a static observer far outside the sphere.

What would be the time evolution regarding relative rates as the contraction progresses to the point of the appearance of the horizon??

If the EH begins in the center then it would seem to imply that the dilation factor also has a maximal value there but of course intuition might be leading me astray.

 

[PAllen]

PAllen, I'm surprised you haven't mentioned your trillion star black hole in this context. When the trillion stars merge sufficiently to form a black hole, the stars are still separated. So when does the interior become a vacuum?

Mike

Formation of a black hole and the complete SC geometry are two different (but related) things. The SC geometry is vacuum everywhere - the Einstein tensor is zero everywhere which is the definition of the vacuum. Meanwhile, the interior of a spherically symmetric collapse become vacuum soon after the singularity forms, assuming nothing new falls in. Further, a realistic collapse is believed to deviate substantially from the idealized solutions in the interior region due magnification of any deviations from perfect symmetry. The singularity theorems guarantee some form of singularity, however spacetime near it is likely extremely complex and twisted (in a realistic scenario, assuming GR).

Birkhoff's theorem specifies that wherever there is spherically symmetric vacuum, it must be a piece of SC geometry (or a different unique geometry if the cosmological constant is not zero). Thus, as a spherically symmetric collapse occurs, you have SC geometry outside any 2-sphere containing all the matter.

 

[PAllen]

A question related to your trillion star illustration.

Assume a clock at the center of the sphere at a time before catastrophic collapse , where the density is somewhat diffuse.
This clock is transmitting time signals to a static observer far outside the sphere.

What would be the time evolution regarding relative rates as the contraction progresses to the point of the appearance of the horizon??

If the EH begins in the center then it would seem to imply that the dilation factor also has a maximal value there but of course intuition might be leading me astray.

I covered this in one of my posts, but briefly:

The last signal (time reading) from the center clock received by a distant, hovering observer will be from before the collapsing matter passed the SC radius, in the following sense: it will reach the SC radius for the cluster a whole just as the cluster collapses to this point; it will be trapped there as will any outgoing light ray. Since it reaches the SC radius just as the outermost mass does, it was emitted before this - it had to get there from the center. This last signal from the central clock is the time of formation (per that clock) of true horizon, beginning at the center.

As for the rate observed on this central clock, the closer it is to the last time that ever escapes, the slower it appears to be going to an external hovering observer. The final time reading is seen after infinite wait by the external observer.

 

[Meselwulf]

Such ambuity about such things. Really all your quibbles lye in the idea that space and time switch places... If the general idea holds, then there would not be a problem.

None of you realize the realization of space and time changing coordinates. If you lot did, there would be no such complications. If you want to know such things, just ask.

Half the problems raised, aren't even problems.

 

[Meselwulf]

Also the idea that some observer outside of the Horizon observes no time pass. Yet the observer actually see's himself fall in. That does not mean it doesn't happen. It does, it just doesn't mean it happens to an outside frame of reference. An observer does not see an falling observer, but that observer will see himself falling no matter what.

 

[Mike Holland]

Dalespam, I have been lurking, interested in the discussion taking place here, but I feel I do not talk the same language. You and others (PAllen) say that changing coordinate systems cannot bring things into existence. However, simply changing one's position on the time stream of one coordinate system can! There are things which exist today that did not exist yesterday.

But you guys seem to have a different understanding of "existence", that if anything exists somewhere on a coordinate system, then it exists period! There is no becoming, no yesterday, today and tomorrow. If I see a supermassive star collapsing, then you say the black hole exists, where I would say it is forming and is going to exist. The fact that a guy falling in will experience the BH in a short time makes no difference, because I can see him hovering there, inching closer and closer, while his clock ticks over microsecinds.

Similarly, I cannot accept PeterDonis' idea of "now". He uses the word to cover the whole area between my past light cone and my future light cone. So a guy could live his whole life "now" on a planet 100 LY from me. He is born now, he is dead now. He is conceived now. This doesn't make sense to me. I would choose a simple method of drawing a line vertical to my world line in my space-time diagram, and say that defines my "now", while my past light cone defines an effective "now" because that is what I see now. A blind person might have a different view based on the speed of sound, but would agree with my geometric method.

The latest thing I cannot accept is PAllen sugggesting that time dilation is just a coordinate thing. If I hover near an Event Horizon and then return here, my clock will be retarded relative to yours. We can set our clocks side by side, and anyone in any coordinate system who can see them will agree that yours is ahead of mine. Then there is the experiment reported on Wiki :
"The Hafele–Keating experiment was a test of the theory of relativity. In October 1971, Joseph C. Hafele, a physicist, and Richard E. Keating, an astronomer, took four cesium-beam atomic clocks aboard commercial airliners. They flew twice around the world, first eastward, then westward, and compared the clocks against others that remained at the United States Naval Observatory. When reunited, the three sets of clocks were found to disagree with one another, and their differences were consistent with the predictions of special and general relativity."
How would these clock differences occur if time dilation is just a coordinate thing?

You lot seem so lost in your abstract theories that you forget there is a real world where things come into existence (like the flowers in my garden - it is Spring in Sydney) and then disappear, and where scientists actually measure time dilation and find that the facts agree with the theory.

Hope I haven't upset you with my "educated layman" perspective. I expect the three of you and others will jump on me.

Mike

Runs and ducks for cover!

 

[TrickyDicky]

There is no "interpretation" involved. The proofs show that the metric is independent of the t coordinate, for all values of r (except r = 0 where there's a curvature singularity, and requiring special handling at r = 2m). You don't have to make any comparisons to show that; the proof for any given value of r goes through regardless of what happens at any other value of r. And as others have already noted, if r &lt; 2m then t is spacelike. So the proof is valid for t spacelike.





Similarly, the "t" coordinate is the coordinate that, once you've completed the derivation of Birkhoff's theorem, the metric turns out to be independent of. You can change the label on it, but that doesn't change its meaning. And that meaning is the same for r &lt; 2m as for r &gt; 2m; it's the *same* coordinate in both regions.

Have you actually read the proof in MTW? If you don't have access to MTW, I have posted a similar proof on my PF blog here:

https://www.physicsforums.com/blog.php?b=4211

My version doesn't use exponentials to write the metric coefficients, as MTW does; I commented on that in a previous post. So my version makes it explicit that there is no restriction on the signs of g_{tt} and g_{rr}, and therefore there's no assumption that either the t or the r coordinate is timelike or spacelike; the proof is valid for t both timelike and spacelike, and for r both spacelike and timelike.

Peter, the exponentials for the coefficients are used precisely to avoid that kind of dimensional swapping,IOW to distiguish clearlybetween the spacelike and the timelike coordinates, whose nature is determined a priori, not based on an arbitrary relabeling.

Try using the Schwarzschild metric in isotropic cordinates and you'll see the fact r is < or > than 2GM doesn't change the sign ofthe coefficients.

 

[DrGreg]

Peter, the exponentials for the coefficients are used precisely to avoid that kind of dimensional swapping,IOW to distiguish clearlybetween the spacelike and the timelike coordinates, whose nature is determined a priori, not based on an arbitrary relabeling.

If you've already made up your mind "a priori", then you're not open to accepting anything that contradicts the arbitrary (and, it turns out, incorrect) assumption you've made.

Try using the Schwarzschild metric in isotropic cordinates and you'll see the fact r is < or > than 2GM doesn't change the sign ofthe coefficients.

So what? The fact that the Schwarzschild t coordinate is spacelike inside the horizon is a fact about Schwarzschild coordinates, it's not a fact about isotropic coordinates.

Kruskal sketch diagram depicting Schwarzschild coordinate grid (not to scale).

 

[stevendaryl]

The latest thing I cannot accept is PAllen sugggesting that time dilation is just a coordinate thing. If I hover near an Event Horizon and then return here, my clock will be retarded relative to yours. We can set our clocks side by side, and anyone in any coordinate system who can see them will agree that yours is ahead of mine. Then there is the experiment reported on Wiki :

In thinking about the "reality" of an in-falling observer crossing the event horizon, I think it's instructive to look at an analogous situation, which is the case of an accelerated observer in flat spacetime.

If you have a rocket ship accelerating at rate g in a straight line, then you can set up a coordinate system X, T using on-board clocks and rulers that are related to the usual Minkowsky coordinates x,t through

x = X cosh(gT)
t = X sinh(gT)

(For simplicity, I'm only considering one spatial coordinate. Also, I'm ignoring factors of c to make the equations simpler to write; it's easy enough to put them back in.)

In terms of the coordinate system X, T, things behave as follows:

1. Clocks that are "higher up" (larger value of X) tick faster.
2. Clocks that are "lower" (smaller value of X) tick slower.
3. There is an "event horizon" at X = 0 such that clocks at this location have rate 0; they are slowed to a stop.
4. An object in freefall will drift closer to the event horizon as T → ∞, but never reach it.

Note that effects 1-4 are exactly analogous to the event horizon of a black hole. Yet, in this case, we know that the conclusion that an observer can never cross the "event horizon" is baloney. The horizon at X=0 in the accelerated coordinates of the rocket corresponds to the points x=ct in the usual Minkowsky coordinates. "Above" the horizon are those points where x&gt;ct, and "below" the horizon are those points where x &lt; ct. Obviously, if an object is just sitting around, at rest in the x,t coordinate system, eventually it will "cross the horizon" so that x &lt; ct.

So of course it's possible to cross the event horizon. But the event of the object crossing the event horizon doesn't show up in the X,T coordinate system, unless you allow T = ∞. The existence of the event horizon, and the fact that apparently nothing can cross it, is an artifact of the accelerated coordinate system of the rocket; that coordinate system is only good for describing events "above" the horizon, where X &gt; 0, or alternatively, where x &gt; ct. The region at and below the horizon is just not adequately described by the coordinate system X,T.

 

[TrickyDicky]

If you've already made up your mind "a priori", then you're not open to accepting anything that contradicts the arbitrary (and, it turns out, incorrect) assumption you've made.

This kind of remark can be applied to your position as well.
But you can check the reason for using the exponential form in references as far back as Tolman's 1935 book on relativity.

So what? The fact that the Schwarzschild t coordinate is spacelike inside the horizon is a fact about Schwarzschild coordinates, it's not a fact about isotropic coordinates.

Well, if it is a purely coordinate fact, why do you derive geometric consequences(so physically bizarre) from it?
If the isotropic coordinates can be used in region II and t is still timelike, why use the coordinates that may produce far-fetched consequences?

 

[Dale]

If the isotropic coordinates can be used in region II and t is still timelike, why use the coordinates that may produce far-fetched consequences?

The usual isotropic coordinates cannot be used in region 2; r would be imaginary.

http://en.wikipedia.org/wiki/Schwar...c.29_formulations_of_the_Schwarzschild_metric

 

[DrGreg]

By the way, here's a Minkowski diagram to illustrate the coordinates in post #256.

 

[TrickyDicky]

The usual isotropic coordinates cannot be used in region 2; r would be imaginary.

Where did you get that?

Written in terms of isotropic coordinates (r, t), the Schwarzschild metric as usually given is static both outside and inside the Schwarzschild radius, i.e., admits a timelike Killing vector for all values of r and t. This has been considered a deficiency of the coordinates to be remedied because it goes against the conventional view about the inside region, but I think it is another example of the arbitrariness of all this.

 

[Dale]

Where did you get that?

The link I posted. Plug in any non-isotropic 0<r<2M and the isotropic r is imaginary.

Written in terms of isotropic coordinates (r, t), the Schwarzschild metric as usually given is static both outside and inside the Schwarzschild radius

Can you post a reference for isotropic coordinates which cover the interior? The usual ones that I am familiar with don't cover it.

 

[PAllen]

The latest thing I cannot accept is PAllen sugggesting that time dilation is just a coordinate thing. If I hover near an Event Horizon and then return here, my clock will be retarded relative to yours. We can set our clocks side by side, and anyone in any coordinate system who can see them will agree that yours is ahead of mine.
How would these clock differences occur if time dilation is just a coordinate thing?

Please don't radically distort what I said in responding to it. The post to which you refer described three different things:

1) Doppler, or more generally, the process of clock/observer sending timing signals to another clock/observer. This is an invariant physical measurement that is determined by two world lines in some geometry, producing a function parametrized by proper time on the one designated 'receiver'.

2) Differential aging: comparing two clocks that take different two different spacetime paths between different evetns. This is an invariant function of the two world lines.

3) Time dilation is a convention, typically realized in a coordinate system, to facilitate calculation of (1) or (2); it generally makes it easier to compute (1) only for a particular class of world lines. Specifically, the concept of 'gravitational time dilation' requires that there exist a global family (congruence in the formal terminology) of world lines for which the Doppler relation between them is constant, and has a particular behavior on exchange of which is considered emitter and receiver. In general spacetimes, this is not possible at all, and it is never necessary to isolate the concept of gravitational time dilation to make a physical prediction. One, simple, uniform computation predicts the result of an experiment like (1), given two world lines, a metric, and the assumption that communication between them is via light through a vacuum.

 

[TrickyDicky]

To make more precise what I was saying in #254, whether a coordinate is temporal or spatial is predefined for a given line element, once that decision that usually follows a convention is made, it doesn't matter at all what specific letter is used to name them, it makes sense to call a temporal coordinate t to avoid confusions but it makes no different as long as one is consistent about it.

When ambiguities can affect the results it is better to use the coefficients in exponential form, when MTW says that in the general case there is no such constraint on sign, it means exactly that, the general case, not the Schwarzschild's case, otherwise they could skip using the exponential form at all.

It is obvious that the change of sign of g00 and grr in the inside region when the exponential form is not used is a consequence of the particular coordinates used and the particular algebraic form of the coordinate functions.
Given the weird consequences of taking seriously what looks like a purely coordinate artifact for r<2GM , if one doesn't want to use the coefficients in exponential form, the sensible thing to do is use a different set of coordinates, like the isotropic ones and check that with them there is no coordinate temporal or spatial nature swap.
Of course if one is convinced beforehand that inside the Schwarzschild radius there is no timelike KVF, there is no way around it. But it looks rather arbitrary given the fact that the only reason to claim that is precisely the anomalous behaviour of the Schwarzschild coordinates for r<2GM when no constraint on sign is applied.

 

[TrickyDicky]

The link I posted. Plug in any non-isotropic 0<r<2M and the isotropic r is imaginary.

Can you post a reference for isotropic coordinates which cover the interior? The usual ones that I am familiar with don't cover it.

There is something you must be doing wrong, I took my answer precisely from a reference of the link you posted (note 14) by Buchdahl from 1985 who took the time to look for a general metric that avoided this because it didn't conform to the conventional view that the inside region must be non-static.

 

[TrickyDicky]

I find the attempt to derive something physical or geometrical from the Schwarzschild coordinate anomaly for r<2GM as misguided as trying to derive physical or geometrical consequences from the purely coordinate singularity at r=2GM. Guys, there is no real singularity there!

 

[PAllen]

Of course if one is convinced beforehand that inside the Schwarzschild radius there is no timelike KVF, there is no way around it. But it looks rather arbitrary given the fact that the only reason to claim that is precisely the anomalous behaviour of the Schwarzschild coordinates for r<2GM when no constraint on sign is applied.

You seem to be implying that there could exist a vector field that is timelike in one coordinate system and spacelike in another. However, whether a vector is timelike or spacelike in invariant. Thus, this is impossible.

If instead, you mean that in addition to the spacelike killing field associated with t in SC interior coordinate patch, there is some other timelike killing field which could be discovered in some other coordinates, you need to demonstrate this or provide a reference. I have seen numerous references in the literature to the mathematical fact that such a thing cannot exist (thought I admit I have never worked through a proof of this specific fact). For example, in the paper: http://arxiv.org/abs/1108.0449 , it is discussed as well known that:

1) isotopic coordinates as usually given cannot cover the SC geometry inside the event horizon: "The interior region R < 2GM of the Schwarzschild black hole is not
covered by metric (3)."

2) there cannot exist any timelike killing vector fields in the interior region
"In the interior
region of the Schwarzschild black hole, all Killing vectors are spacelike [6] and this implies
that the interior is nonstationary"

Interestingly, this paper does present a way to construct isotropic interior coordinates. However, consistent with (2), which implies the spacetime is not stationary, interior isotropic coordinates require that all diagonal metric components are functions of both a radial and a time coordinate (the off diags are zero, of course).

 

[TrickyDicky]

You seem to be implying that there could exist a vector field that is timelike in one coordinate system and spacelike in another. However, whether a vector is timelike or spacelike in invariant. Thus, this is impossible.

I'm not implying that at all. Rather the opposite.

 

[Dale]

There is something you must be doing wrong, I took my answer precisely from a reference of the link you posted (note 14) by Buchdahl from 1985 who took the time to look for a general metric that avoided this because it didn't conform to the conventional view that the inside region must be non-static.

I don't have access to that one. How do his isotropic coordinates differ from the ones on Wikipedia? The Wikipedia isotropic coordinates are not only imaginary for the interior region, but the restriction to the exterior region is explicitly given.

Unless Buchdahl is using different isotropic coordinates the isotropic coordinates simply do not cover the interior. For example, with c=G=M=1 the EH is located at r=2 which transforms to r'=1/2. If you try to obtain a point inside the EH by picking a smaller r', say r'=1/4, then you wind up with a location outside the EH, r=2.25 in this case.

 

[Dale]

When ambiguities can affect the results it is better to use the coefficients in exponential form,

I am fine with that, but you need to always be aware of what region of the spacetime is covered by a given coordinate chart. In this case, the exponential form restricts the chart to the exterior region. So you do avoid any ambiguities, but at the cost of coverage. You would need a new set of coordinates to cover the interior, which is OK, but you certainly cannot use the exponential form to make any claims about the interior region.

It is obvious that the change of sign of g00 and grr in the inside region when the exponential form is not used is a consequence of the particular coordinates used and the particular algebraic form of the coordinate functions.

Agreed.

Given the weird consequences of taking seriously what looks like a purely coordinate artifact for r<2GM

What weird consequences are you concerned about here.

if one doesn't want to use the coefficients in exponential form, the sensible thing to do is use a different set of coordinates, like the isotropic ones and check that with them there is no coordinate temporal or spatial nature swap

I agree that it would be better to use different coordinates for the interior region, but it cannot be the usual isotropic coordinates since those do not cover the interior region.

Of course if one is convinced beforehand that inside the Schwarzschild radius there is no timelike KVF, there is no way around it. But it looks rather arbitrary given the fact that the only reason to claim that is precisely the anomalous behaviour of the Schwarzschild coordinates for r<2GM when no constraint on sign is applied.

This seems to be a complete mischaracterization. There is no need to be convinced beforehand of anything, simply take the KVFs and compute whether they are timelike or spacelike. Personally, I suspected the opposite, that one of the four KVFs would be timelike, or that a linear combination of the spacelike KVFs would be timelike. So I took the KVFs and checked and found that they were all spacelike, contrary to my expectation beforehand. This is something that you can determine afterwards, without any preconceptions.

 

[TrickyDicky]

I don't have access to that one. How do his isotropic coordinates differ from the ones on Wikipedia? The Wikipedia isotropic coordinates are not only imaginary for the interior region, but the restriction to the exterior region is explicitly given.

Unless Buchdahl is using different isotropic coordinates the isotropic coordinates simply do not cover the interior. For example, with c=G=M=1 the EH is located at r=2 which transforms to r'=1/2. If you try to obtain a point inside the EH by picking a smaller r', say r'=1/4, then you wind up with a location outside the EH, r=2.25 in this case.

I only read the abstract where it says it admits any t and r, and by inspection it seemed the coordinates didn't have the problem with sign that the Schwarzschild have.