Singularity one-dimensional?

A Brief Natural History of Singularities

I'm assuming you mean a "curvature singularity", as treated in various spacetime models in gtr (or similar theories). The short answer is that consistently assigning a "dimension" to such a locus can be very difficult, particularly in Lorentzian spacetimes. The reasons are various and technical, but one point which might help a bit is to recall that singularities are not part of the spacetime; they are places where the manifold structure breaks down badly.

The kinds of "geometric singularities" (i.e. loci where strange things happen independently of choice of coordinate chart) which can arise in gtr are much varied than most newbies probably appreciate. Interestingly enough, there is a rough classification by severity of singularity, or rather a bunch of classifications. A review is long overdue. In lieu of that, I offer an informal description of some examples:

1. The curvature singularity in the future interior of the Schwarzschild vacuum solution is a strong spacelike scalar curvature singularity. Here, "scalar singularity" means that the scalar curvature invariant [itex]R_{abcd} \, R^{abcd}[/itex], which is sometimes called the Kretschmann scalar, blows up. "Strong" (more precisely "crushing and destructive") means that every body whose world sheet runs into this singularity is crushed and destroyed.

2. "The" curvature singularity in the deep interior of the Kerr vacuum is a strong timelike scalar curvature singularity. Just to add to the confusion, in the maximal extension of this model (not physically realistic!) there are infinitely many different singular loci!

3. A "Big Bang" type curvature singularity in a typical cosmological model is generally a strong spacelike scalar curvature singularity. A famous conjecture, the BKL conjecture (after Belinsky-Khalatnikov-Lifschitz, the same Lifschitz who comes after Landau and whose name I am, some would say, mistransliterating, in order to avoid autobleeping ), says very very roughly that "generic" curvature singularities should have this nature, in some (not yet precisely known) sense of the term "generic". At various times, researchers have announced the solution to this conjecture, but its been around a long time and to paraphrase Piet Hein, when you poke a hard problem, it hits back!

4. Many pp-wave solutions (these are generalizations of plane waves in flat spacetime) possess lightlike nonscalar curvature singularities. Curvature singularities in pp-wave solution can never be "scalar curvature singularities" because all the scalar invariants of these spacetimes vanish identically. Some of these lightlike nonscalar curvature singularities are strong, as in the so-called waves of death. These are plane wave solutions (a special type of pp-wave--- the terminology is confusing but standard!) which propagate through the universe, destroying spacetime itself as they go. A directed beam with similar effects is called a thunderbolt.

Even more intriguing, some of these singularities are weak lightlight nonscalar curvature singularities, which means they are possibly survivable. In the latter case, interestingly enough, the "weakness" is revealed by looking at the expansion tensor of timelike geodesic congruences, and finding that these do not diverge, even though the tidal forces may diverge!

One way to think about this is that the tidal forces blow so quickly that small objects have no time to respond by being crushed or torn apart before they have already passed the singular locus! An interesting consequence in such cases is that gtr cannot predict what happens after they pass through such a singularity.

5. Many colliding plane wave (CPW) models feature strong spacelike curvature singularities which develop in the interaction zone after two plane waves collide. In some cases, these singularities are weaker than expected, which is of great interest since these models sometimes turn out to be locally isometric to the "shallow interior" of certain black hole models, and the weak singularity corresponds to the event horizon, i.e. this locus is not remarkable in terms of curvature; it's significance arises at the level of conformal or causal structure. In addition to these, CPW models generally feature lightlike but non-curvature singularities called fold singularities, which occur "ahead" of the departing waves. In fact, you can think of these (but probably shouldn't) as propagating backwards in time from the curvature singularities! Describing the geometry of a typical CPW model more accurately would take a lot of work, unfortunately.

6. The family of Weyl vacuum solutions, which can be written down in terms of choice of axisymmetric harmonic function, give all static axisymmetric vacuum solutions in gtr, so they correspond in Newtonian gravitation to axisymmetric gravitational potentials, i.e. axisymmetric harmonic functions. Which makes sense, except that the correspondence is quite tricky! Anyway, many of these turn out to have geometric singularities on the axis of symmetry which are often called struts and which have unrealistic features making them behave a bit like rigid rods which make no contribution to the gravitational field but can hold apart massive objects. They are generally regarded as artifacts due to inappropriate choice of boundary conditions.

7. Similarly, the Robinson-Trautman null dusts are a family of exact solutions which can be regarded as Schwarzschild holes perturbed by massless radiation. It turns out that these generally feature singularites sometimes called pipes, which have unrealistic properties and are again generally regarded as artifacts due to inappropriate choice of boundary conditions.

8. Many spacetime models also feature conical singularities which are analogous to the vertex of a paper cone. These loci are places where angular deficits (or angular excesses) are concentrated.

For those who didn't understand very much of what I just said: I can't and don't really expect anyone who doesn't already know most of this stuff to understand very much. I just wanted to try to vaguely popularize the idea that it is not at all easy to single out characteristics common to all singularities; their taxonomy is simply too diverse.

 

Thanks, Chris!
that's a fascinating list of various types of curvature singularity that occur in various models of spacetime----mainly I guess in the GTR (general theory of relativity) context.

several I hadn't heard of, which were fun to imagine

From my perspective (and I wonder if you would agree) it is important to stress that these singularities occur in man-made models
and that doesn't automatically imply they ever occur in nature

the original poster (O.P.) who asked the question may not be clear about this---many people aren't---and may be thinking of singularities as *real things that happen in nature*.

So if you agree (you being the local GTR expert) I would like to add that singularities are places in an artificial model where that model breaks down and fails to compute reasonable numbers---say it starts giving infinities for the curvature if we are talking about GTR.

the way you deal with singularities is you fix the model (if you can see a way to do that) so that it does not break down---sometimes *quantizing* a model will fix its singularities (it has been known to happen) and then you have to test the new model experimentally to check that it's better in other ways as well.

I think the O.P. was asking about the dimensionality of singularities: are they one dimensional or two dimensional or what?
Clearly from the examples you gave we should expect there to be singularities of all different dimensionality and physical extent.

Sometimes people have the idea that the *big bang singularity* is pointlike. Actually as far as I know among professional cosmologists (please correct me if I am wrong) the most common picture is of an infinitely extending 3D hypersurface.
People seem to get the idea that the singularity is pointlike because the word "singularity" sounds like "single" and the word "single" suggests a point.

So I'd hasten to assure the O.P. that there is no one type of geometry that singularities must have---they can have various different dimensionality, and shape, and size. They can extend spatially off to infinity, or they can be spatially bounded.
they are artificial loci where a model fails, and they can be as various as the models that give rise to them.

and considerable research these days is devoted to getting rid of singularities (by replacing the model with one that doesnt break down). there was that 3-week workshop at Santa Barbara about it earlier this year. maybe the O.P. would like to check out the videos of some of the talks

 

Easier question first...

"Big Bang theory" is something of a misnomer. I was trying to explain, among other things, that "big bang type cosmological singularities" are strong, spacelike, scalar curvature singularities, the most destructive and the least avoidable, if you will. But none of these singularities really have "dimensions" in the sense you mean. It is true that you can readily embed the FRW dust with S^3 hyperslices orthogonal to the world lines of the dust particles in [itex]E^(1,4)[/itex], and then--- after supressing two dimensions so that you have two dimensional manifold embedded in [itex]E^(1,2)[/itex] --- it looks like an American style "football" with Big Bang and Big Crunch singularities corresponding to the two "tips". But this picture, while vivid, is misleading if it leads you to think of the singularities as pointlike. Remember, the embedding is artificial and introduces distracting irrelevancies.

 

Serves me right for attempting a nonmathematical sketch...

All in gtr, but closely related theories will have similar aspects regarding plane waves and so on.

Wow, I am sure glad you asked because you must have partially misunderstood. I was trying to say that many exact solutions studied in gtr turn out to possess unphysical features such as struts or pipes. Careful authors (IMO) deprecate these, but many authors slur over their implausibility, which (IMO) can lead to seriously misleading attempted inferences about more realistic scenarios. I was trying to suggest that these features are highly suspect within the context of gtr itself.

Contrast the strong spacelike scalar singularities which generally make perfect sense within gtr, and thus should be regarded as genuine predictions of this theory.

Theoretical considerations external to gtr, e.g. musings on a possible quantum theory of gravity, tend to cast doubt upon whether gtr can give an accurate picture of spacetime at the Planck scale, but such curvatures are well beyond anything astronomers are likely to be able to observe in the foreseeable future!

I am glad you asked, because at the level of classical physics, they should regard these as real features of gtr.

Now, you said "which happen in nature", but I'd say that physics is about constructing theories which describe what specific measurements will show in various situations, or if you like about understanding "how things behave" rather than "what things really are".

Do electrons "really exist in Nature"? I don't even know what that would mean. Does "the Sun" really exist in Nature? If your answer is "yes", do you think it has a well defined radius "in reality"? I'd say that "the Sun" is a convenient fiction, adding that this is nothing to worry too much about. What matters is that we have a good theory in which "the electron" is well-defined. We have good theories in which we can construct good models of idealized "stars" without worrying overmuch about what it might mean to say that stars "really exist".

Does Homo sapiens "really exist"? Wise biologists would say that "species" are also a convenient fiction (Ernst Mayr wrote a book on this topic). Science is full of idealizations or abstractions which may not correspond to "reality" but which help one to construct and think about mathematical models which allow us to predict what will happen in a given situation, even to engineer devices which work.

Perhaps I don't know what you mean by "artificial". Or would you say that all mathematical models in all physical theories are "artificial"?

Be this as it may, I was trying to explain, among other things, that when an object encounters a weak nonscalar singularity, the tidal forces measured by an observer riding on this object probably diverge, but so quickly that the expansion tensor of the congruence of world lines of bits of matter in the object doesn't diverge, i.e. the object doesn't have time to respond by being destroyed before the "bad place" is in the past.

Its crucial to understand that in the context of gtr, curvature singularities cannot be avoided or fixed up. For that matter, I don't think I agree that singularities in field theories generally can usually be "fixed up", e.g. point mass potential in Newtonian gravitation.

Agreed.

No! I was not very clear about this because I lack the energy to try to explain any of the technicalities, which are formidable, but I was trying to state (not explain) that it is best not to try to assign any "dimension" to curvature singularities in Lorentzian manifolds.

You're probably thinking of pictures in Weinberg, The First Three Minutes. Those are good pictures, and indeed the singular locus appears as a coordinate plane, but that locus does not belong to the manifold and you shouldn't think of it as having a dimension. As matter of fact, while I deprecated "pointlike", if you simply must think of it as having a dimension, pointlike would be infinitely better than sheetlike!

Oh dear, oh dear, oh dear! I didn't mean to illustrate the fact I so often decry, that it simply isn't possible to discuss subtle theories in nonmathematical terms, but I am beginning to think it was a mistake to have tried to offer a nonmathematical sketch...

There are few things in there which I vaguely recognize as bearing some resemblance to the intuition I was trying to convey, but unfortunately, for the most part I feel that is directly contrary to what I was trying to say!

Huh?

Oh well, I'm glad you liked my post, Marcus, even though I guess it's back to the drawing board for me as a populizer and you as a consumer of PF-style popsci

Perhaps literally, in my case--- all this was happening while I was grabbing some stuff I somehow neglected to previously install on this machine, so that I could draw some conformal diagrams to illustrate my post above, but I can see that these figures would probably also be seriously misunderstood, unless someone first writes a PF tutorial on interpreting conformal diagrams!

 

In case you want to look up the videos of this conference about the latest work on resolving singularities, here is the URL
http://online.kitp.ucsb.edu/online/singular_m07/

"kitp" is the Kavli Institute for Theoretical Physics at UC Santa Barbara

the people who gave those talks are one way or another involved with the question of what to replace GTR with so as to get rid of the singularities.
especially the big bang singularity.

we know GTR is wrong because it breaks down at a certain places and has these unnatural glitches called singularities. so the question is, what do you replace GTR with so that it will be just as good as GTR where GTR is a success
but not have these unnatural failure glitches.

what you see in that list of talks is kind of the frontline leading edge research in various approaches to replacing GTR with something that is not subject to singularities. It is hard to do. Also the proponents of the different approaches argue a lot amongst themselves.

I don't recommend watching these hourlong videos, it would take too much time and be incomprehensible. Just realize they are there.
We have no scientific evidence that singularties exist in nature. World class people are working on developing a spacetime model that won't have a bigbang singularity.

 

Marcus, I feel you have seriously misunderstood the scientific rationale for seeking a quantum theory of gravity. Many people seem to hate event horizons and the notion of a beginning or an end on religious or philosphical grounds. It is good to remember that such prejudices can blind you to how Nature behaves. This might also explain how you so badly misunderstood what I wrote. I advise you not to rush to attribute to either myself or to the researchers you mention motivations which we might not share, or even understand.

 

Chris - Thank you for your posts here, there has been a consensus of opinion on these Forums that when GR meets QT in the singularity of a BH or BB it will be GR that breaks down and the presence of singularities in GR proves that in these regimes GR breaks down and the singularities are 'unphysical'. I am glad you are shooting for the other side.

Why don't you write something on interpreting conformal diagrams and illustrate you ideas here?

Garth

 

Conformal diagram tutorial and clarification

Hi, Garth, thanks for the encouragement since I was well and truly pole-axed by what Marcus wrote in his last two posts! Pig-stuck. Thrown for a loop. Whatever Athapaskan verb means "caused to wonder whether the entire world has gone utterly mad". You get the picture

Let me try again to briefly sketch my take on (1) event horizons and curvature singularities and (2) the people who loathe them (I mean deep-down felt-in-the-gut fear and loathing):

1. These are are certainly real predictions of gtr, and thus should be expected to "occur in Nature" (apparently this is a phrase which can be more badly understood than I had recalled!) in regimes where gtr is accurate.

2. To judge from their own writings, the people who loathe the very notion of event horizons or destructive curvature singularities do so because they fearfully believe these notions, if valid, would "scientifically disprove" the core beliefs of their world view. I believe that Fear of Event Horizons and Fear of Singularities are irrational fears which arise from a more plausible human fear: fear of isolation and fear of death. People simply need not to confuse issues in theoretical physics with the issue of their own mortality, a psychological (spiritual? existential?) problem for which I can offer no assistance. Ironically, in my view, it seems clear that according to current mainstream belief, regions of strong curvature can kill humans, but there are many more likely ways to be Reaped. IOW, people who loathe black holes should probably loathe the family car. Similarly for people who fear the prospect that one day humans will become extinct. In a way, I feel they may take gtr (and science generally) much too seriously by reacting to scientific theories which employ (often very abstract) notions which seem contrary to their world view as if these notions might "really exist in Nature" in some "absolute" sense, which I feel is naive. In fact, scientific knowledge is far more supple and adaptable thing. Science loathers suffer at once from too much imagination--- and too little!

Marcus, you probably feel from the above that I have misunderstood your motivations and beliefs--- and if so you're probably right. To prevent further misunderstanding, I emphasize that the above comments are based on my prior experience with dozens of persons who seem to express strong negative feelings about black holes.

I hope that my psychological speculations will smooth the suddenly troubled waters in this thread, rather than fanning the flames of controversy! If that isn't mixing metaphors. I'm trying to advise the singularity loathers: lighten up! No pun intended

 

dont know these many people or how they seem to you. I was not attributing motivations to you...certainly not analyzing your motives, Chris

I just think that General Relativity fails as a theory (has a limited domain of applicability) and will eventually be replaced by some better theory which does not break down at some of these singularities

instead of singularities I would prefer to call them "limits to applicability where the model blows up"

In the past other theories have had singularities---and this has been recognized as a flaw, or sign that the theory was of limited usefulness---and they have been replaced by better theories. I suppose my attitude is conservative---I expect that Gen Rel is no exception and that science will proceed as usual and Gen Rel will be replaced by something a bit more rugged that doesn't suffer from the same problems.

 

I don't see the consensus you refer to---it doesnt include me. Nor do I see a combat between two sides shooting at each other.

It seems to me that both conventional quantum theory and Gen Rel have problems.

It strikes me as simplistic or naive to suppose that, when they are joined, one or the other will be a "winner".

I don't see the current state of physics theory as a contest between quantum theory and GR. I see something more constructive than that going on.

I don't see it as appropriate to argue about which theory is going to "win" and to think of opposing sides rooting or shooting for one or the other. My guess would be that merging quantum theory and spacetime theory is a creative effort that will involve learning how to modify both.

 

Now you are sounding reasonable again to me, Marcus! Just drop the bit about insisting that whatever the first workable quantum theory of gravity looks like, it will neccessarily "exorcise" singularities or whatever, because that is not at all clear and it may well turn out not be true.

 

I am glad that you find me reasonable, Chris. I'm not aware of having shifted my basic position---but I can't always account for how you take what I say.

One of the enjoyments of observing the progress of scientific research is that one really cannot predict how things will go (at least I feel that I cannot.) One can have *expectations* however.

I see that a considerable number of smart people consider the old (1915) Gen Rel to be flawed because it suffers from singularities (such as the BB and BH, in particular) and a growing number of people are searching for a theory of spacetime and matter to replace Gen Rel---duplicating its impressive success where it does work and extending coverage to situations where Gen Rel breaks down.

If this search succeeds, which I expect it to, it will in a certain sense replace the singularities with a deeper understanding of what goes on in, and possibly also beyond, them.

the conventional meaning of a singularity is where a physical theory breaks down.

So you could say that the singularity is removed or resolved when you get a new theory which does not break down there.

But if you prefer, when that happens I suppose you could use the word in a slightly different way and say that *the singularity is still there, we just understand better what goes on there*

Some people call what replaces the former BB singularity in their models by the name "the Planck regime"-----I don't pretend to understand what is meant by that----allegedly in certain cases the model cranks along smoothly thru the former singularity, but usual ideas of space and time momentarily cease to apply.

 

Clarifying the Mainstream Viewpoint on Singularities in GTR

Unfortunately, I am back to being flummoxed by something you just wrote which I consider to be potentially seriously misleading.

You have consistently written statements in which I agree with the second half but not with the first half! So let me reverse the order of those statements:

The second halves of these statements are correct summaries of the current mainstream:

I agree entirely! Furthermore, I think we all agree that the search for a new theory of gravitation is a thoroughly mainstream activity. (In this context, it is amusing to note that the mathematician John Baez, author of the semi-humorous Crackpot Index, has contributed to this effort.)

But the first halves of those statements are seriously misleading:

My objection is that in statements like this you suggest the misleading conclusion that the object of the mainstream effort is to exorcise black holes and the Hot Big Bang Theory from astrophysics and cosmology. This is quite untrue. As I thought everyone knew, the object of the mainstream effort is to

1. reconcile quantum mechanics with a classical field theory, general relativity,

2. elucidate some fascinating connections between the notion of black holes and notions of thermodynamics.

An important point regarding (2) is that it could well be that the next "gold standard theory of gravitation" might be more "thermodynamical" than "quantum". One of the fascinating trends in physics in the past few decades has involved growing recognition that mathematical techniques developed in the context of classical or quantum physics turn out to apply to the other arena. In addition, in the past decade there has been a good deal of work on nongravitational analogs of black holes which suggests that this notion may be best understood via thermodynamics.

Regarding the current mainstream view on the major technical issues within gtr itself, including dealing with various kinds of geometric singularities, I could give many citations, but one short book which I particularly like is the Chandrasekhar memorial volume edited by Robert Wald, Black Holes and Relativistic Stars, University of Chicago Press, 1998. I'd highly recommend this to anyone who wants to know more about current mainstream views on theoretical issues in gtr and the search for a "better theory".

Before I say anything else, I need to stress something: your personal objections to the notion that black holes "really exist in Nature" seem to be based upon the prediction in gtr that curvature singularities exist inside the horizons. It's important that newbies understand that historically, mainstream objections to the notion of black holes (pre 1975 or so) have really been objections to the notion of "event horizon", which should be thought of as the defining characteristic of "black hole"; as my list above should make clear, many exact solutions in gtr which are nothing like black holes, including plane waves, exhibit curvature singularities. Furthermore, many exact solutions which can be regarded as cosmological models (but certainly not as models which resmemble the Universe in which we live), such as the Goedel dust, contain no curvature singularities. So the existence of curvature singularities is certainly not a defining characteristic of either black holes or cosmological models!

The above mentioned objections quickly moved to the fringe with the discovery of objects which can (according to the current mainstream viewpoint) only be interpreted as black holes in the sense of gtr. Vaguely similar objections are still promoted on the web, sometimes including PF; these should be regarded as incorrect crank opinions which are greatly at variance with the current scientific mainstream.

OK, back to the book: the chapters are based upon talks delivered in 1996, subsequently revised by the various authors, but despite a major development in cosmology (the cosmological constant thing), the mainstream has not budged on the points which are most relevant here. Some particularly relevant chapters:

1. Martin Rees, "Astrophysical Evidence for Black Holes": Martin Rees recounts how and why it suddenly became universally accepted that black holes (think: event horizons) "really exist in Nature", modulo my comments above about the nature of physics. Rees certainly does not say that mainstream researchers consider "Gen Rel to be flawed because it suffers from singularities (such as the BB and BH", or anything even close to that statement. Your statement does, however, somewhat resemble the early objections (c. 1960) to the Hot Big Bang Theory back when Continuous Creation was still regarded as viable, and to early objections to Black Holes (think "event horizons", not "curvature singularities") back when (c. 1975) the suggestion that black holes are common objects in our Universe was considered highly speculative and dubious by most physicists.

2. Roger Penrose, "The Question of Cosmic Censorship": Penrose discusses some theoretical issues involving Cauchy horizons and geometric singularities in gtr, which have not yet been resolved within gtr (or by going beyond it). In particular, he discusses "thunderbolts". You will search in vain for any assertions that gtr is unworkable because of the mere existence of singularities, rather, Penrose and many others have put great effort into understanding the nature of generic solutions of the EFE within the context of gtr, effort predicated on the assumption that the theory, while clearly difficult, is not fundamentally flawed simply because singularities exist. You will search in vain for any statements to the effect that Penrose himself or other researchers consider "Gen Rel to be flawed because it suffers from singularities (such as the BB and BH)".

3. Werner Israel, "The Internal Structure of Black Holes": Israel discusses "mass inflation" and the question of what "generic" black hole models look like in gtr. That is, the maximal extension of the Schwarzschild and Kerr solutions are regarded as "physically unrealistic" due to simple but unrealistic choice of boundary conditions. Specifically, the black holes which apparently exist in nature are thought to have been formed by gravitational collapse and therefore have very different causal structure inside the horizon. Penrose, Israel, and others, working with gtr itself, have discussed theoretical considerations suggesting that the interior of black holes "as they really exist in Nature" is quite different from what the Schwarzschild and Kerr vacuum solutions would suggest. One line of attack on elucidating this issue involves the remarkable local isometry discovered by Chandrasekhar between the "shallow interior" of the Kerr vacuum and a certain CPW model, the Chandrasekhar-Xanthopolous vacuum. By perturbing the two incoming waves of this model one obtains exact solutions which are locally isometric to a perturbation of the shallow interior of the Kerr vacuum. Again, one will search in vain for any statements to the effect that Israel himself or other researchers consider "Gen Rel to be flawed because it suffers from singularities (such as the BB and BH)".

In addition to these chapters, in Part I of the book, the entirety of Part II is devoted to survey articles on the motivations for the search for a new theory of gravitation. These survey articles can be readily supplemented by others available at the arXiv, including papers by Rovelli on the motivation for the seach for quantum gravity and the original paper by Jacobson on an important reinterpretation of the Einstein field equation. If anyone, after consulting these resources, is unconvinced that I am correctly describing the motivations for mainstream efforts working toward a quantum theory of gravity, I'd suggest posting a query in sci.physics.research specifically asking for responses from John Baez, Steve Carlip and Ted Jacobson, all of whom read that newsgroup at least sometimes and all of whom have contributed to the search under discussion. (To prevent further misunderstanding, I'd request that anyone following this suggestion include the URL of this PF thread.)

Regarding this search, I feel your statements require some further qualifications. You used the phrase "replacement" and "better theory" in your posts. These are weasel words which could easily mislead students and the general public if left unaccompanied by suitable qualification.

You could say that Newtonian gravitation was "replaced" by gtr in 1919, when the first solar system test decisively agreed* with gtr and disagreed with Newtonian theory. But it is important for students and the general public to understand that Newtonian gravitation is alive and well, and for good reason: it's much simpler to work with, so much so that it makes good sense to use it whenever you can get away with this. In particular, vacuum solutions in Newtonian theory are governed by harmonic functions, which are rather well understood mathematically. Contrast the solution space of the vacuum EFE, which after 90 odd years is still not well understood mathematically. (See again the articles by Penrose and Israel.) Or contrast the way in which Newtonian gravitation has been employed for many decades to study statistically the evolution of stellar clusters and note that relativistic elaborations have recently become popular topics of research. For all these reasons, I prefer to say that Newtonian gravitation is known to break down under certain circumstances. We know how to tell when we should work with gtr instead, and we have some theoretical arguments suggesting an upper bound for the curvatures/energies at which we think that gtr too must break down.

(*Modulo later assertions that Eddington's data analysis was flawed--- let's not get into that; suffice it to say that gtr has been tested very thoroughly and has held up very well indeed. There is no doubt that the four classical solar system tests, and some even more impressive tests as well, give results in excellent agreement with gtr.)

You could say that a quantum theory of gravitation will be a "better theory" than gtr, simply because gtr is a classical field theory, yet nothing has been better confirmed by twentieth century physics than the fact that Nature adores the quantum. This theoretical conflict at the very heart of physics is aesthetically objectionable, as I think almost everyone would agree. But it is important to stress that ultimately, the true test of which of two theories is "better" is which agrees better with observation and experiment. Here, we have a problem, because it is not yet clear that experimental tests of the long sought theory of quantum gravity which could decisively confirm the expected breakdown of gtr under certain conditions can be conducted in the forseeable future. This leads to discussion of some philosphical issues which arise from the prediction of event horizons in gtr, and the search for a self-consistent quantum theory of gravitation, issues which seem to challenge the Baconian notion of the scientific method. However, discussion of these issues should probably move to the philosophy subforum.

There's more to it than that, I think! Context is everything. You are probably thinking of the broad usage described in such sources as http://mathworld.wolfram.com/Singularity.html (which is discussing how the term is used in mathematics generally, especially analysis, including applied mathematics, including physics). For a previous discussion at PF, see https://www.physicsforums.com/showthread.php?t=124016

I agree that removing the coordinate singularity in the Schwarzschild exterior chart by passing to a new chart, such as the ingoing Eddington chart, is analogous to removing a removeable singularity when studying some holomorphic function in complex analysis. I might even agree that geometric singularites in gtr are somewhat analogous to non-removeable singularities of holomorphic functions. One has to be very careful not to try to push this analogy too far, however. In particular, the natural smoothness requirement in gtr is [itex]C^\infty[/itex] or less , depending on context. As has been hinted at above, the maximal real analytic extensions of the exterior Schwarzschild and Kerr vacuums are considered to be unrealistic; to obtain reasonable boundary conditions you must drop the assumption of analyticity. The reason is that analytic functions are much too "rigid"; knowledge of the derivatives at some point determines the function in an entire neighborhood. To deal with radiation and avoid undesirable asymptotic properties we generally need to work with functions built out of "bump functions", which are not analytic.

ADDENDUM: thanks for the link the Kavli Institute conference, Marcus!

The very first slide I examined, Slide 04 from the talk by Beverly Berger, obviously illustrates a piece of the issue I alluded to above, that the interior of astrophysical black holes is currently believed to be somewhat similar to the future interior of the Schwarzschild vacuum but utterly unlike the RN electrovacuum or Kerr vacuum. As I noted, even discussing this issue, while natural within the context of gtr, appears to raise some startling philosophical challenges to the Baconian model of the scientific method!

Slide 06 refers refers obliquely models (particularly the mixmaster model) with which I am familiar. In another recent post I wrote out the Bianchi II analog for the classical (Bianchi IX) mixmaster model. These are homogeneous but anisotropic exact dust solutions, expressed in terms of a certain ODE (a different one for each of the different Bianchi types) which feature a "Big Bang type" strong spacelike scalar curvature singularity. The BKL conjecture originally arose the context of asserting that the approach to a generic curvature singularity in gtr would resemble the behavior of the mixmaster model.

Slide 11 illustrates the time evolution of the Kasner exponents for the vacuum limit of the Bianchi I model (aka Kasner dust). I have investigated this in detail for all the models. Slide 13 shows the result of computing the Kretschmann scalar; this confirms what I just said, that the singularity is a scalar curvature singularity.

And this is cool! Slide 16 illustrates the vacuum limit of the very Bianchi II model I just mentioned. See my Post # 4 in the thread https://www.physicsforums.com/showthread.php?t=168995 The investigation I just mentioned showed that the Bianchi II model is similar to the Bianchi IX model, but the others can exhibit rather different behavior. As she says, the fascinating thing about the Mixmaster model is the infinite sequence of "Kasner epochs", with transitions being governed (Slide 22) by the expansion of a simple continued fraction! Since continued fractions came up in my diss (on generalizations of Penrose tilings!) I have always found that fascinating!

Slide 30 is related to something I obliquely alluded to and have discussed elsewhere at much greater length--- adding a massless scalar field or massless radiation to a CPW model can drastically change the nature of the curvature singualarity. Recall that such models can be locally isometric to models of black hole interiors (at least, roughly speaking "the outer half").

Slide 65 is worth bookmarking as a good illustration of current thinking on the topic of the article by Werner Israel cited above http://online.kitp.ucsb.edu/online/singular_m07/berger/oh/65.html The null singularity is thought to be weak and possibly survivable, in which case it would also function as a Cauchy Horizon (CH; no relation).

69 slides, wow--- how long did this talk last?! But a great set of slides, nonetheless. But unfortunately, there seems to be something wrong with the links to many of the slides; many of them seem to be duplicates, as if someone made a goof when uploading the slides by hand I hope she turns these slides into a proper survey article.

The term Planck regime generally refers to sectional curvatures (which have the same units as energy density in relativistic units) associated with energies approaching the Planck energy, which is generally regarded as the upper bound of the region where, in theory, gtr might be valid. This regime lies far, far beyond the limits of the regime where observation and experiment have determined that gtr is valid within current error bars. It is currently believed that gtr should ultimately turn out to be useful, as a fundamental theory of gravitation, well beyond the curvatures expected near the exterior/interior of stellar mass black holes, but also generally acknowleged that good models in the context of gtr might require appeal some "effective field theory" taking account of quantum effects (see also the semiclassical approximation for the exterior), and also that gtr might break down at smaller energies than the Planck energy.

I have offered above some citations which I feel will help interested lurkers to better understand the current mainstream viewpoint concerning both theoretical problems in gtr and the motivations for the "next generation gravitation theory". Since I've gone to considerable effort to try to clarify these issues, I hope that at least some lurkers with a serious interest in modern astrophysics will follow up by studying these citations. I stress that I hope and believe that the articles in the above cited book will make at least some sense to those lacking mathematical or physical background! I also feel that the horrified reaction in some segments of the general population to black holes and to modern cosmology are based largely upon serious misunderstandings of what these notions, and science generally, really concern.

 

I disagree with that notion. Most of the GR tests are weak field tests. No single test has been made that would indicate a singularity exists in nature.

While GR is a wonderful theory the usability has been mostly exaggerated. Apart from a set of "Mickey Mouse" solutions not even a simple two body situation can be modeled without great difficulties.

Some people fall in love with a theory, sometimes they have invested a lifetime of work into it, and then feel a need to defend it to the teeth, they would only "allow" changes that extend and not invalidate earlier work. Emotions can run pretty high, even for different views within the same theory. We only have to look at Eddington's quite appalling behavior towards Chandrasekhar in trying to discredit him.

As in the case of Newton's theory also Einstein's theory will be surpassed. And that could mean a complete paradigm shift, not just some adjustments.

 

Oh dear, my post grew too long! I just wanted to add, after "best understood via thermodynamics":

It is true that it is widely recognized that one possible "side benefit" of a successful next generation gravitation theory is that it may well clarify the physical nature of the putative curvature singularities and other geometric singularities of gtr. But this is not guaranteed. In particular, it is widely recognized that the next generation theory may not in any sense "exorcise" the features which lead (in gtr considered as some kind of limiting case of this yet unknown next generation thory) to the appearance of curvature singularities.

However, I would emphasize that the most troubling kind of "singularity" which arises in gtr is probably weak singularities which function as Cauchy horizons, as in the slide from Beverly Berger's talk which I archived above, because gtr refuses to predict what happens after an object passes through the locus. This really should be regarded, I think, as a serious theoretical defect of gtr. To anthropomorphize, one can hope that the next generation theory will not say in such situations, "and after this, something interesting might well happen, but if so I haven't a clue what that might be!" But unfortunately, at this point it seems that nothing is guaranteed.

OK, I hope we are all converging on agreement regarding all the fundamental points now!

 

something of value I want to keep from this thread.
no flattery or exaggeration intended.
I didn't reply earlier because I wished not to cover it up (often people only read the ends of threads and don't delve back)

Agreement is a nice bonus when it happens. But I don't necessarily expect it, and can also approve situations where several acceptable viewpoints on fundamental issues are recognized.

 

See a new thread I just started in the "Special and General Relativity" subforum titled "Roy Kerr on the Kerr vacuum", in which I attempt to clarify some remarks in a new eprint by Roy Kerr himself regarding the singularities lurking inside his famous solution. This eprint appeared just after the above posts.

 

Planck scale cutoff?

why does there have to be a Planck scale cutoff, singularities (geodesics no longer descriptive), and therefore the assumed solution: quantization of C_R? As an alternative, consider the abstraction of manifold (continuum, inbetweenness) for still finer scale of <C_p, which is matched, vis-a-vis homeomorphism (bicontinuous), to the rational set. That is using Cantor's terney set wherein the middle third is removed iteratively. So does nature abhor initial/final conditions, and special conditions like cutoffs? So have the mathematicians already been there? That is, they start with abstract concept of manifold, and then add constructs like differentiation, metric as more elaboration 'as they metaphorically rise above the Planck scale?'