Removing the Schwarzschild coordinate singularity

[DrGreg]

I'm creating this thread to discuss some issues raised by kev in the Understanding maximally extended Schwarzschild solution thread, to avoid diverting that thread from its original question.

As any fule kno, the problem with Schwarzschild coordinates is their coordinate singularity at the event horizon. In fact, really there are two sets of Schwarzschild coordinates, those strictly inside and strictly outside the event horizon, which happen to share the same metric equation, but which are otherwise independent. For example, in the outside coordinates, t represents warped time (approaching infinity for any particle approaching the horizon), whereas in the inside coordinates, t represents a warped distance. The two ts are different entities and there's no real connection between them.

(For the avoidance of doubt, when I say the "inside coords", of course I refer to the vacuum solution inside the event horizon, not the "interior Schwarzschild solution" below the surface of a spherical mass.)

There are other coordinates, e.g. Kruskal-Szekeres or Eddington-Finkelstein, which remove the coordinate singularity. They do this by transforming from the each of the two Schwarzschild coordinate systems inside and outside, then "tweaking the initial conditions" to get the two solutions to match approaching the horizon from either side. The solution can then be extended to the horizon itself as the unambiguous, finite limit, and it turns out that a single metric equation describes the whole solution.

I have no problem with any of this, but the impression I get is that kev does have a problem. It is somewhat unsatisfactory in that we have to stitch two solutions together and add extra events (the event horizon) to bridge the gap. (The gap was there only because of a defect in the original coordinates (like the defect of the undefined longitude of the North Pole). There's no physical (i.e. spacetime-geometrical) gap. That last sentence seems to be a bone of contention with kev.)

All the technical expositions of (non-rotating, symmetric, uncharged, asymptotically flat) black hole metrics I have ever seen begin with a derivation of Schwarzschild coords, and then derive other coords (Kruskal-Szekeres, Eddington-Finkelstein etc) from them.

I wonder if there are any published works that do it the other way round, by deriving e.g. Kruskal-Szekeres coords "from first principles" without reference to Schwarzschild coords or event-horizon singularities?

 

[yuiop]

I'm creating this thread to discuss some issues raised by kev in the Understanding maximally extended Schwarzschild solution thread, to avoid diverting that thread from its original question.

As any fule kno, the problem with Schwarzschild coordinates is their coordinate singularity at the event horizon. In fact, really there are two sets of Schwarzschild coordinates, those strictly inside and strictly outside the event horizon, which happen to share the same metric equation, but which are otherwise independent. For example, in the outside coordinates, t represents warped time (approaching infinity for any particle approaching the horizon), whereas in the inside coordinates, t represents a warped distance. The two ts are different entities and there's no real connection between them.

I have no problem with any of this, but the impression I get is that kev does have a problem.

Thanks for starting a thread to address my concerns.

The main thing I have a problem with is that the conventional physical interpretation results in a singularity of infinite density at the centre of a black hole. I understand the textbook interpretation that the two ts are different entities, but also I claim I can show an alternative physical interpretation of the GR equations that removes the problem of infinite mass density (which I think most people would agree is a serious problem). However, I can not outline my unconventional interpretation here, that is in effect a counter example to the textbook interpretation, without risking getting banned. If anyone is interested, I have posted some items on my alternative approach in my blog and website linked from there. That is probably the best place to continue this discussion and I hope to soon put together a presentation suitable to submit to the independent research section of this forum.

 

[Sam Park]

The main thing I have a problem with is that the conventional physical interpretation results in a singularity of infinite density at the centre of a black hole... I claim I can show an alternative physical interpretation of the GR equations that removes the problem of infinite mass density (which I think most people would agree is a serious problem).

Your comments are based on several fundamental misconceptions, some of which I'll mention below, but before even addressing those, your claim is self-evidently self-contradictory, because you object to the manifold singularity at r=0, but your "solution" is to propose an even more pathological manifold singularity at r=2M, where of course the manifold is perfectly smooth and well-behaved according to general relativity. So, even if your "new metric" made sense (which it doesn't), you would still be promoting a self-evidently inane claim. (By the way, I'm a new member here, and the rule I was required to read said that users are not supposed to use this as a forum to promote their crackpot web sites. It seems to me this is precisely what you're doing.)

Now, as to your "new interpretation" of general relativity, what you've described is utterly incompatible with general relativity, as several others have already explained to you (without you seeming to notice). The field equations of general relativity are based on a conceptual foundation which can be interpreted in terms of the differential geometry of a pseudo-Riemannian manifold. The fundamental basis of this conceptual framework - the thing that makes it a coherent theory - is the existence of invariants, i.e., quantities that are the same, regardless of what arbitrary system of labels (coordinates) we choose to apply. The most fundamental invariant of all is the absolute interval, (d tau)2, where tau denotes the proper time along any timelike path. It's obvious from your comments that you totally reject this conceptual basis, so your ideas really have nothing at all to do with general relativity, nor even with any metrical theory of any kind.

Needless to say, no one is obligated to espouse metrical theories, but when someone claims to be espousing a metrical theory (as you do), and then immediately begins spouting the kind of nonsense that you've put on your web site, that is frankly just intellectually dishonest. And the worst part is not the deception of others, who can easily see for themselves how non-sensical your ideas are. The worst part is the deception of yourself. Since you reject the conceptual basis of general relativity, it really makes no sense at all for you to base your brand new original theory of the universe on grade-school algebraic manipulations of a particular equation (the Schwarzschild line element) that is derived from the very conceptual basis that you've rejected. Can you see how senseless that is?

One more point: You say you "understand" the two different "t" coordinates, but it is not at all evident from your comments (let alone your web site) that you do. It takes about 2 seconds to do a google search on "schwarzschild coordinate time" and find some excellent explanations of the issues involves, so there's really no excuse for you to be promoting the kind of silliness that you've presented in these threads (and on your web site).

 

[yuiop]

To Sam: I see that you have made a total of 3 posts in this forum. In this thread you have accused me of being "intellectually dishonest" and in an unrelated thread ( https://www.physicsforums.com/showpost.php?p=2032154&postcount=44 ) you accuse Tam Hunt of being "personally dishonest". Do you see a common theme there? That leaves only one post where you have not accused someone of being dishonest.

I agree with Tam Hunt's response and I quote "Whoa whoa whoa. Sam, check the attitude." I am not sure if it is in the rules or not, but by tradition, the physicsforums members are usually civil and polite to each other in their discussions.

...because you object to the manifold singularity at r=0, but your "solution" is to propose an even more pathological manifold singularity at r=2M, where of course the manifold is perfectly smooth and well-behaved according to general relativity.

With the singularity at r=0, all the mass of the black hole is contained in a volume of exactly zero (therefore infinite density) and most people agree the laws of physics break down at that point. Perhaps you could elaborate in your own words (but without the usual aggressive attitude) exactly how a singularity at r=2m is "even more pathological" than that?

 

[Sam Park]

To Sam: I see that you have made a total of 3 posts in this forum. In this thread you have accused me of being "intellectually dishonest" and in an unrelated thread ( https://www.physicsforums.com/showpost.php?p=2032154&postcount=44 ) you accuse Tam Hunt of being "personally dishonest". Do you see a common theme there?

Yes I do. The common theme is that in both messages I pointed out blatent intellectual dishonesty.

I agree with Tam Hunt's response and I quote "Whoa whoa whoa. Sam, check the attitude."

That wasn't a substantive response when Tam posted it, and it isn't a substantive response when you post it. In both cases the dishonesty was carefully explained in detail, and in both cases no substantive rebuttal was forthcoming. In both cases my comments were entirely accurate.

I am not sure if it is in the rules or not, but by tradition, the physicsforums members are usually civil and polite to each other in their discussions.

As I mentioned in my previous message, one of the rules here is that members will not use this as a forum to promote their crackpot web sites... And this is precisely what you've been doing (and what you continue to do).

My messages have not been uncivil or impolite, but they have been blunt and accurate. If something is nonsense, it should be permissible to call it nonsense. If these forums had a rule against calling nonsense nonsense, they wouldn't be worth much.

You mustn't mistake having your ideas called nonsense for impoliteness. There are reasons other than impoliteness that might prompt someone to call your ideas nonsense. For example, your ideas may be nonsense. If they are, then surely you wouldn't hold it against me for calling them nonsense, would you?

With the singularity at r=0, all the mass of the black hole is contained in a volume of exactly zero (therefore infinite density) and most people agree the laws of physics break down at that point. Perhaps you could elaborate ... exactly how a singularity at r=2m is "even more pathological" than that?

Sure. The volume of the spherical surface at r=2M is exactly zero, and your claim is that all matter "falls" to this surface (from inside as well as outside), so it has infinite density. And of course the spacetime manifold (according to you) is explicitly discontinuous at this surface, which consists of infinitely many infinitely dense points (rather than just the single point r=0), where the laws of physics break down and your proper time changes discontinuously from real to imaginary. How you can imagine that this proposal (even if it made sense) would represents a reduction in the discontinuities of general relativity is a mystery. Remember, when people talk about the laws of physics breaking down, they mean the field equations of general relativity don't apply... but in your proposal the field equations are blatently violated at infinitely many points. Your proposal, even without discussing the detailed technical errors on which it is based, is simply nonsense. I don't say this to be rude, I'm just telling you the truth.

 

[Dale]

Hi Kev,

You might like this nice little arXiv paper: http://arxiv.org/abs/gr-qc/0311038v2"

 

[Hurkyl]

Yes I do. The common theme is that in both messages I pointed out blatent intellectual dishonesty.

Intellectual dishonesty entails knowing better.

To Sam: I see that you have made a total of 3 posts in this forum...

That's completely irrelevant to whether or not his criticisms are valid.

 

[Mentz114]

I always thought dishonesty was telling lies, deception and so on. Whatever the merits of the subject matter, I don't see any dishonesty.

Sam Park sounds like an outraged parent, he scores high on the self-righteous pomposity meter. His catch phrase must be "How dare you !"

Kev is inexperienced and dove into the deep end of a boiling pool. He got it wrong. Lots of people get things wrong. Why the outrage ?

 

[JustinLevy]

Look,
I've been in situations where I've danced around my real question because just blurting it out instead of making people first understand the reason why I'm having trouble with a concept ... well, people would dismiss my question as 'stupid' and move on. Or, in a similar vein, at times I've hit upon an idea that I realize contradicts mainstream but I can't figure out why it is wrong ... learning why for such situations is very helpful (this is the very reason so many SR books contain all the standard 'paradoxes', because it helps people learn). (For example, I still haven't figured this 'paradox' out: https://www.physicsforums.com/showthread.php?t=159157 )

kev collects his thoughts in one place, and you interpret this in the worst possible way. We see many crackpots come through here all the time. kev is much more willing to learn than that.


Regardless of our individual opinions of posters here, please let's return to the original question.

I wonder if there are any published works that do it the other way round, by deriving e.g. Kruskal-Szekeres coords "from first principles" without reference to Schwarzschild coords or event-horizon singularities?

I don't know of any. Hopefully someone here does, as I would be interested in reading it.

 

[DrGreg]

I'd like to get this thread back on to a technical discussion instead of name-calling.

I created this thread in the hope that someone could answer my original question which would persuade kev that his interpretation was wrong. Unfortunately nobody seems to have an answer. One person did send me a private message in reply, but that reply gave no published answer but instead gave hints on how I could work out an answer for myself. That wasn't really what I wanted, and by sending me a response in private did not help in my objective of persuading kev in public.

I know kev is well aware of Rindler coordinates, those of an observer in a Born-rigid accelerating rocket in flat spacetime (i.e. in the absence of gravity). They are related to Minkowski coordinates (t,x) (in 2 dimensions for simplicity) by

$$ct = X \sinh \frac {aT}{c}$$
$$x = X \cosh \frac {aT}{c}$$​

Relative to the Rindler observer undergoing constant proper acceleration a at $X = c^2 / a$, there is an event horizon as $X \rightarrow 0$, corresponding to the photon worldline $x = ct$. The Rindler observer dropping an apple from his spaceship will see it drop towards the horizon at a distance of $c^2 / a$ below him, but it will take an infinite Rindler time to get there. (When I say "see", in this case it applies equally to optical observation or Rindler coordinate calculation.) However, from the point of view of any inertial observer, (including the apple itself) the apple reaches $x = ct$ in a finite time, crosses without impediment and continues. There is no accumulation of matter at the event horizon, no switch from real to imaginary time, no singularity in any physical sense, just a singularity in Rindler coordinates.

This demonstrates the general principle of how a coordinate singularity at an event horizon can be removed (but in flat instead of curved spacetime).

(Note that performing the change of variable

$$R = \frac{1}{2} \left( \frac {aX^2}{c^2} + \frac{c^2}{a} \right)$$​

makes the parallel between this and the Schwarzschild to Kruskal-Szekeres transformation even clearer. See this old post, rescaled here to give R the units-dimension of distance. The thread of that post gives further information about Rindler coordinates for anyone not familiar with them.)

 

[yuiop]

... your proper time changes discontinuously from real to imaginary...

You should not be so frightened of imaginary time. Steven Hawking is very fond of it and used it in his no boundary proposal and in explaining how virtual particles with imaginary mass/energy just outside the event horizon become particles with real (but negative) mass energy when they fall into an black hole, thus subtracting from the total mass of the black hole while its partner escapes as Hawking radiation.

Maybe your issue is the "discontinuous" part. Well if you have studied these things you would know that the equation describing the path of a particle falling into a black hole in coordinate time changes from real to complex as it crosses the event horizon. To "fix" the problem, an arbitrary imaginary constant of integration of i*pi is added to make the path real everywhere. The justification for adding the arbitrary constant to the equation fo r<2m but not for r>2m is precisely because the there is a discontinuity in time at the event horizon.

This quote is from mathpages

"This occurs because, in such cases, we are integrating from a = 0 where r = R (which is greater than 2m) to a value corresponding to r less than 2m, and hence we must perform a complex integration around the singularity at r = 2m, offsetting the result by ±pi (assuming the path of integration doesn’t make any complete loops around the singularity). This is not surprising, because the t coordinates are discontinuous at r = 2m, so we cannot unambiguously “carry over” the labelling of the t coordinates in the region r > 2m to the region r < 2m." http://www.mathpages.com/rr/s6-04/6-04.htm

To my way of thinking, if you require two different equations to describe the motion of a particle then there has to be a physical reason to justify it. For example imagine a very deep mine shaft is drilled into the surface of the Earth. An object dropped from high up would accelerate proportional to 1/R^2 but once it is below the surface of the Earth the acceleration changes to 1/R. Two separate equations are required to describe the falling particle's motion because there is a real physical change. If the acceleration equation changed from 1/R^2 to 1/R at precisely 1 mile above the Earth's surface we would want a physical justification as to why that happens. If there is no physical boundary then the motion should be able to be described by a single equation.

The entropy of a black hole is proportional to the surface area of its event horizon. To me, this hints at the physical nature of the event horizon.

Now if we go back to the equations of a particle falling into a black hole as given by mathpages and set the apogee of the trajectory to r<2m then we find that while the path can be described in real positive proper time there is no corresponding path in coordinate time in real or imaginary terms unless you count infinite coordinate velocity as a real path. In coordinate terms the particle travels finite distance in zero coordinate time or if you reverse time and distance it travels forward in time and remains spatially stationary while traveling from r=2m to r=0 implying there is no distance between r=2m and r=0. Can anyone rationalise that?

Is it not reasonable to assume that if the path of a particle can be described in one coordinate system, that it should still exist in all other valid coordinate systems?

... The Rindler observer dropping an apple from his spaceship will see it drop towards the horizon at a distance of $c^2 / a$ below him, but it will take an infinite Rindler time to get there. (When I say "see", in this case it applies equally to optical observation or Rindler coordinate calculation.) However, from the point of view of any inertial observer, (including the apple itself) the apple reaches $x = ct$ in a finite time, crosses without impediment and continues. There is no accumulation of matter at the event horizon, no switch from real to imaginary time, no singularity in any physical sense, just a singularity in Rindler coordinates.

This is indeed true, but are Rindler coordinates (acceleration in flat spacetime) exactly equivalent to curved spacetime?

There are many reasons to think they are not exactly equivalent and we have explored some of the ways they differ in previous threads. For example I can equally use Rindler coordinates to "prove" that there is no central singularity in a black hole in the same way that you "proven" there is no singularity at the event horizon. If you draw a straight line vertically on a Minkowski chart (representing an inertial path) it clearly shows that the falling particle never arrives at a final destination that is equivalent to a central singularity or a singularity anywhere else for that matter. Clearly singularities are a feature of curved space which does not show up in flat space, so using flat space to justify an assertion about curved space is not rigorous Maybe someone can tell us if distinction between the strong equivalence principle and the weak equivalence principle is relevant here?

In fact if you look at the graph you posted in the Rindler thread

, it can be claimed that the Rindler chart has nothing to say about the region below the Rindler horizon and that is why there no worldlines in the blue and yellow regions of your Minkowski diagram, so to me it, it is difficult to justify that a particle “continues” past the Rindler horizon using Rindler coordinates, because the Rindler coordinates do not cover that region. The black curve your Rindler spacetime chart is the path of an inertial observer. It terminates at the Rindler horizon. Can you plot how the black curve of the inertial observer extends beyond the Rindler horizon on the Rindler chart using real values?
(Note: The black line in the first Minkowski chart is an accelerating observer).

 

[JesseM]

To my way of thinking, if you require two different equations to describe the motion of a particle then there has to be a physical reason to justify it.

That's just absurd, since for any dividing line in space it's possible to invent some arbitrary coordinate system where you need two different equations to describe the motion of the same particle on different sides of the dividing line. For example, it would be possible to come up with a coordinate system on a globe such that, for a particle moving at constant speed (in ordinary 3D cartesian coordinates) along a line of longitude from one pole to another, you'd need a different equation to describe its path after it crossed the equator than the equation you used before it crossed the equator. Do you disagree?

You told me earlier that you understand that GR is normally interpreted as a geometric theory where coordinate systems have no significance in themselves, and only geometric quantities that are the same in all coordinate systems are seen as genuinely physical. And yet you continue to act as though Schwarzschild coordinates have some sort of privileged place--are you trying to dispute the theory of GR itself, and put in its place a non-geometric theory, or do you believe that there is some purely geometric, coordinate-independent sense in which there's a singularity at the horizon?

For example imagine a very deep mine shaft is drilled into the surface of the Earth. An object dropped from high up would accelerate proportional to 1/R^2 but once it is below the surface of the Earth the acceleration changes to 1/R. Two separate equations are required to describe the falling particle's motion because there is a real physical change.

In this case I think there'd be actual local coordinate-independent facts that would change as a freefalling observer passed the surface, like the way tidal forces between the top and bottom of the chamber he's falling in would be changing as a function of his own proper time. Can you point to any local coordinate-independent facts that change when an observer crosses the horizon?

The entropy of a black hole is proportional to the surface area of its event horizon. To me, this hints at the physical nature of the event horizon.

No one has said the horizon isn't physical, it represents the boundary between the region where events can send signals that escape to infinity and the region where everything in the future light cone of an event will hit a singularity. But this is a global definition involving the entire spacetime, in GR nothing locally weird is happening at the horizon.

Incidentally, the idea of entropy being proportional to surface area is thought to be part of a general "holographic principle" that applies to cases other than black holes, see this article. The article mentions a broad generalization which applies to any possible 2D surface in space:

In 1999 Raphael Bousso, then at Stanford, proposed a modified holographic bound, which has since been found to work even in situations where the bounds we discussed earlier cannot be applied. Bousso's formulation starts with any suitable 2-D surface; it may be closed like a sphere or open like a sheet of paper. One then imagines a brief burst of light issuing simultaneously and perpendicularly from all over one side of the surface. The only demand is that the imaginary light rays are converging to start with. Light emitted from the inner surface of a spherical shell, for instance, satisfies that requirement. One then considers the entropy of the matter and radiation that these imaginary rays traverse, up to the points where they start crossing. Bousso conjectured that this entropy cannot exceed the entropy represented by the initial surface--one quarter of its area, measured in Planck areas. This is a different way of tallying up the entropy than that used in the original holographic bound. Bousso's bound refers not to the entropy of a region at one time but rather to the sum of entropies of locales at a variety of times: those that are "illuminated" by the light burst from the surface.

Bousso's bound subsumes other entropy bounds while avoiding their limitations. Both the universal entropy bound and the 't Hooft-Susskind form of the holographic bound can be deduced from Bousso's for any isolated system that is not evolving rapidly and whose gravitational field is not strong. When these conditions are overstepped--as for a collapsing sphere of matter already inside a black hole--these bounds eventually fail, whereas Bousso's bound continues to hold. Bousso has also shown that his strategy can be used to locate the 2-D surfaces on which holograms of the world can be set up.

Now if we go back to the equations of a particle falling into a black hole as given by mathpages and set the apogee of the trajectory to r<2m then we find that while the path can be described in real positive proper time there is no corresponding path in coordinate time in real or imaginary terms unless you count infinite coordinate velocity as a real path. In coordinate terms the particle travels finite distance in zero coordinate time or if you reverse time and distance it travels forward in time and remains spatially stationary while traveling from r=2m to r=0 implying there is no distance between r=2m and r=0. Can anyone rationalise that?

The only problems you are pointing to are about how weird this path looks in Schwarzschild coordinates, not that there is any coordinate-independent physical sense in which it's weird. Again, do you somehow disagree with the idea that only coordinate-independent facts are truly physical? You can draw a path that never leaves the horizon in Kruskal-Szekeres coordinates and it doesn't seem particularly problematic, just look at the path E-E'-E'' in the Kruskal-Szekeres diagram on p. 835 of MTW which I've included as an attachment.

Is it not reasonable to assume that if the path of a particle can be described in one coordinate system, that it should still exist in all other valid coordinate systems?

Half of the path "exists" in Schwarzschild coordinates (the whole thing doesn't exist because these coordinates don't cover the whole maximally extended spacetime), you can see it drawn as the line E'-E'' on the Schwarzschild diagram on p. 835. The fact that this path seems to involve infinite velocity in these coordinates is just a feature of the coordinate system, it doesn't suggest a physical problem--indeed, for any path, even a timelike worldline in Minkowski coordinates, it's not hard to come up with a coordinate system where that path has infinite coordinate velocity.

This is indeed true, but are Rindler coordinates (acceleration in flat spacetime) exactly equivalent to curved spacetime?

Of course they're not equivalent, DrGreg was just showing that Rindler coordinates have some features that are analogous to Schwarzschild coordinates. Curvature is a real geometric aspect of a given spacetime, so there's a real physical distinction between a spacetime that's curved like the Swarzschild spacetime and flat Minkowski spacetime which isn't curved, you can't erase this difference with your choice of coordinate systems.

it can be claimed that the Rindler chart has nothing to say about the region below the Rindler horizon and that is why there no worldlines in the blue and yellow regions of your Minkowski diagram, so to me it, it is difficult to justify that a particle “continues” past the Rindler horizon using Rindler coordinates, because the Rindler coordinates do not cover that region

Uh, what? If we create a coordinate system on a globe that only covers the region of the globe North of the equator, do you somehow believe that an object traveling South would cease to exist when it crosses the equator? Again you must distinguish between the geometry of a surface and facts about a coordinate system drawn on this geometric surface (which may fail to cover the entire surface, as with a coordinate system on a globe that only covers the region North of the equator, as with Rindler coordinates which fail to cover the yellow and blue regions in Minkowski coordinates which do cover the entire Minkowski spacetime, and as with Schwarzschild coordinates that fail to cover regions III and IV in Kruskal-Szekeres coordinates which do cover the entire maximally extended Schwarzschild spacetime)

 

[Dale]

Sam, I don't think it is appropriate to quote something that old and from a different forum here. First, I am sure kev's thinking has advanced somewhat from more than 7 years ago. Second, that forum had different rules on the content of posts. Regardless of his "agenda", in my experience kev has always been careful to follow the PF rules here and quoting him from a forum where the rules are different makes it difficult for him to respond both in the context of the other forum and within the rules of this forum.

If you have a problem with what he said on that forum then you should respond on that forum, not this one.