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Atrium > Mathematics, Physics & Philosophy: Kerr metric toplogy

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  1.  
    • CommentRowNumber1.
    • CommentAuthorIan_Durham
    • CommentTimeMay 20th 2010
    • (edited May 20th 2010)
    The following is an honest question and I have no desire to start an argument. It's bugging me scientifically and you all have a very interesting perspective that I would appreciate on this. I apologize in advance for its length but it ultimately has to do with topology (I think - correct me if I am wrong).

    In the debate awhile back about spacetime, Urs mentioned that nothing actually rotates in the Kerr metric. I had always understood it differently and thought spacetime itself rotated which resulted in frame-dragging, but wasn't sure if it was just me. I hadn't had time to look into it, but then was redirected from the nLab to the Wikipedia entry on Kerr metrics. While there I found this quote (apologies for its length):

    A surface on which light can orbit a black hole is called a photon sphere. The Kerr solution has infinitely many photon spheres, lying between an inner one and an outer one. In the nonrotating, Schwarzschild solution, with a=0, the inner and outer photon spheres degenerate, so that all the photons sphere occur at the same radius. The greater the spin of the black hole is, the farther from each other the inner and outer photon spheres move. A beam of light travelling in a direction opposite to the spin of the black hole will circularly orbit the hole at the outer photon sphere. A beam of light travelling in the same direction as the black hole's spin will circularly orbit at the inner photon sphere. Orbiting geodesics with some angular momentum perpendicular to the axis of rotation of the black hole will orbit on photon spheres between these two extremes. Because the space-time is rotating, such orbits exhibit a precession, since there is a shift in the φ variable after completing one period in the θ variable.



    So Gravity Probe B was supposed to be looking for this frame-dragging effect. Last I heard, they hadn't been able to pull it out of the noise yet when analyzing the data. In any case, supposing they are able to verify it, then it would seem (I'm not sure?) it would verify the existence of the photon spheres. At any rate, whomever wrote the entry obviously had the same impression I did about stuff rotating (which may be wrong, who knows).

    But here's what I find interesting and what I'm wondering about. So, an empiricist would look at data supporting the existence of the photon spheres as supporting the fact that spacetime itself is rotating. To Urs it isn't rotating. Is this because Urs and/or you guys view the photon spheres topologically, meaning they are static topological aspects of the spacetime and the photons end up on one or the other simply based on the relative geometry of their (the photons') approach?

    And how would a mathematical physicist react if GP-B data can't improve the accuracy of the frame-dragging data? I ask because it appears they're having trouble getting the signal to match what they expected for this. In essence, I'm wondering if this could potentially confirm or refute one or the other interpretation of what's physically happening.
    • CommentRowNumber2.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 20th 2010
    • (edited May 22nd 2010)

    This may or may not help, but as far as I understand it, the stuff with horizons is all geometry: the Schwarzschild spacetime is, at its most complicated, topologically ℝ 4−{0}\mathbb{R}^4 - \{0\} (as far as I understand it - I’ve not studied topology in GR properly) (Edit: this is patently absurd, as Tim vB points out in #17 below. I meant (ℝ 3−{0})×ℝ(\mathbb{R}^3 - \{0\})\times \mathbb{R}, of course). The interesting stuff depends on the manifold/metric structure. A photon sphere is defined in terms of lightlike paths, but this is metric dependent, even though it defines a submanifold with mildly interesting topology (it is S 2×ℝS^2\times \mathbb{R} in simplest form). Interestingly, this is not too dissimilar to how Poincare discovered/introduced the idea of topology, by looking at stability of planetary orbits and if they were closed or not asymptotically in time. The size/shape of the orbit was not as important as the topological properties of the worldlines.

    Please also consider the difference between saying spacetime is rotating and saying space is rotating. Spacetime is (for argument’s sake) the whole 4-dimensional Einstein manifold. It doesn’t rotate, it just is. If however one thinks of what happens to test objects (which are only used to see what is happening in the metric) in a time-evolving spacelike slice, then one can observe rotational behaviour. Compare the statement: ’the worldline of the particle is orbiting the sun’. I’m sure you’d agree it is meaningless. The worldline of the particle is a static object, because there is not another ’external time’ by which to say it is doing anything dynamic. I know you clarify that your impression is perhaps not correct, so this is just to give you the explanation from my point of view of the ’static topological’ description.

    Anyway, I’ll let the real experts give you a proper answer.

    And how would a mathematical physicist react if GP-B data can’t improve the accuracy of the frame-dragging data?

    Easy - get the experimentalists to make better instruments :P The ’worry’ would be if the accuracy was say an order of magnitude higher than the predicted size of the effects, and no frame-dragging was observed.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMay 20th 2010
    • (edited May 20th 2010)

    Urs mentioned that nothing actually rotates in the Kerr metric.

    No.

    Here is the exchange we had in the spacetime thread, the first quote from you, the second from me:

    The example I gave there is the Kerr metric that explicitly has rotation built into the metric. Rotation does not necessarily have anything to do with gravity.

    You realize that there is no thing that rotates in a Kerr spacetime except the gravitational field itself? It’s a vacuum solution to Einstein’s equations.

    • CommentRowNumber4.
    • CommentAuthorIan_Durham
    • CommentTimeMay 20th 2010
    Thanks David. Yes, I would generally think of a geodesic as not rotating since the natural question to ask if it was would be "rotating with respect to what?" (Note that Eddington, among others, actually postulated a fifth macroscopic dimension to answer this question and some folks involved in GP-B, led by Paul Wesson, actually were looking for evidence of it in the GP-B data.)

    Urs, sorry for misquoting you. I think I (finally) see the miscommunication. What I meant by that wasn't that the geodesics were rotating, necessarily, just that the metric allowed for something to be rotating, i.e. the embedded massive object had a measurable angular momentum.

    So, in essence, you see it as just warping the spacetime in a different way, but the spacetime is still static.

    I think what has long-bugged empiricists about this is the behavior of the photons being different depending on which directions they go around the hole. I think they just have a hard time "seeing" it topologically and thus I think they actually see the geodesics as rotating (and have never paused to ask themselves "with respect to what?"). In other words, I think they have a hard time visualizing how a static spacetime could produce that effect.
    • CommentRowNumber5.
    • CommentAuthorIan_Durham
    • CommentTimeMay 20th 2010
    • (edited May 20th 2010)
    By the way, does this mean someone should change the Wikipedia article? It must have been written by an experimentalist...
    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMay 20th 2010

    the embedded massive object had a measurable angular momentum.

    No. There is no massive embedded object in a Kerr spacetime. It is a vacuum solution. It still has non-vanishing angular momentum. This is entirely carried by the gravitational field itself.

    • CommentRowNumber7.
    • CommentAuthorIan_Durham
    • CommentTimeMay 20th 2010
    • (edited May 20th 2010)

    No. There is no massive embedded object in a Kerr spacetime. It is a vacuum solution. It still has non-vanishing angular momentum. This is entirely carried by the gravitational field itself.



    Hmmm. So what causes the spacetime to curve in a Kerr metric? I think I'm more confused than ever so please pardon my apparent lug-headedness. So, if the angular momentum is carried by the gravitational field, does that mean it is separate from spacetime itself and thus while the field rotates, the spacetime manifold on which it is defined does not? Or are you saying that angular momentum here is a little like "spin" in QM - it acts like an angular momentum, but nothing is really moving, per se? I think I have spent waaaaay too much time studying Eddington's Fundamental Theory...
    • CommentRowNumber8.
    • CommentAuthorTim_van_Beek
    • CommentTimeMay 20th 2010

    So what causes the spacetime to curve in a Kerr metric?

    Let’s say we have a rotating massive star, then the Kerr metric is supposed to be a good approximation if you keep a certain distance from the star. The Kerr spacetime is the mathematically idealization of the situation that you let the radius of the star go to zero. If you yourself can move along timelike curves in the Kerr spacetime without disturbing it (neglecting your effect on the spacetime, that is), you won’t ever experience a collision of any sort.

    I don’t intend to put words in Urs’ mouth, but that would be the way I would explain “there is no massive object in Kerr spacetime”.

    So, if the angular momentum is carried by the gravitational field, does that mean it is separate from spacetime itself and thus while the field rotates, the spacetime manifold on which it is defined does not?

    There are some mistakes I repeat a hundred time before I get it :-) While I think I can guess what you mean by “the spacetime manifold rotates”, you will see, that the participants of this thread already agreed that this statement does not make any sense :-)

    Same goes for “nothing is really moving”…You can only move relative to the gravitational field, the gravitational field itself defines what is movement and what is “stationary”, ergo there cannot be a movement of the gravitational field. It’s like: define space to be ℛ 3\mathcal{R}^3, choose an orthonormal basis {e 1,e 2,e 3}\{e_1, e_2, e_3\}, question: is e 2e_2 moving?

    The Kerr spacetime is asymtotically Minkowskian, that is if we are far away we could play a game and say:

    Kerr spacetime, in the area where we are, = Minkowski spacetime + some small, strange effects.

    Kerr spacetime is a family of vacuum solutions with two parameters M and a, if you set M = a = 0 you get the Minkowski spacetime, so question is: can we use these small, strange effects to measure M and a? (I guess you already know the answer, what I intend to achieve is to provide context that serves as a common ground where we can meet :-)

    …it acts like an angular momentum, but nothing is really moving, per se…

    Well, maybe the electron does spin, who knows? :-) You won’t see it easily unless you paint a dot on it somewhere… But seriously: The spin of an electron is an intrinsic property of the electron that cannot be changed by any process (well, at least that’s one of the assumptions of QFT). While you cannot change spacetime (see above), you certainly can influence and change the gravitational field in your spatial environment…

    • CommentRowNumber9.
    • CommentAuthorTim_van_Beek
    • CommentTimeMay 20th 2010
    • (edited May 20th 2010)

    I may have mentioned it before: When I learned a bit about GR the book “Quantum Gravity” by Carlo Rovelli was very helpful. You don’t have to be interested in quantum gravity, the chapter 2 about GR will be very worthwhile to read by all means.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeMay 20th 2010
    • (edited May 20th 2010)

    Ian, you write:

    So what causes the spacetime to curve in a Kerr metric? […] So, if the angular momentum is carried by the gravitational field, does that mean it is separate from spacetime itself and thus while the field rotates, the spacetime manifold on which it is defined does not? Or are you saying that angular momentum here is a little like “spin” in QM - it acts like an angular momentum, but nothing is really moving, per se? I think I have spent waaaaay too much time studying Eddington’s Fundamental Theory…

    This is good: you are now asking questions and no longer making statements! This is the way to progress. Very good.

    I think I’m more confused than ever so please pardon my apparent lug-headedness.

    All lug-headedness is very much pardoned here, if coming with the willingness to learn.

    Tim gave some good answers already. If you feel like understanding the answers to these questions more formally, you might start looking into what is called “ADM formalism”. See the index of your copy of Misner-Thorne-Wheeler for that. There is an “ADM mass” and also an “ADM angular momentum” defined for asymptotically flat spacetimes. I think MTW supply lots of heuristic discussion for this in their chapter.

    (And do not look at the Wikipedia entries on this. Like almost all Wikipedia entries on physics – and in stark contrast to those on math – they are no good.)

    • CommentRowNumber11.
    • CommentAuthorIan_Durham
    • CommentTimeMay 20th 2010

    Kerr spacetime is a family of vacuum solutions with two parameters M and a, if you set M = a = 0 you get the Minkowski spacetime, so question is: can we use these small, strange effects to measure M and a? (I guess you already know the answer, what I intend to achieve is to provide context that serves as a common ground where we can meet :-)



    Thanks Tim. I think it is becoming much, much clearer how you all view this. I still think there's some kind of missing layer in my understanding. Again, pardon my density here. I'm really trying to understand this and I think it will be incredibly important for me in then understanding how to apply categories to this whole thing (because, as an empiricist, I fundamentally think in terms of physical processes and "visualizations.") So thanks for your patience.

    Let's see if I can describe this. So from my work in GR years ago I always envisioned the Kerr metric exactly how you describe it (in fact, I can "visualize" the difference between the Kerr and Schwarzschild metrics by visualizing the geodesic dropping straight into the latter while sort of "curling" around into the former, but the geodesics themselves don't move). In other words if the geodesics were like little tracks for marbles, the marbles would fall straight into the Schwarzschild metric, but they'd sort of "orbit" the Kerr metric. So, in that sense, I see how the pure geometry - which is stationary - imparts an angular momentum to the "marbles." But there are a couple of details here that I don't quite get.

    The first is that, since photons travel on geodesics, and photons orbiting in the opposite direction of the "rotation" (or whatever) behave differently, geodesics going one way around a Kerr solution must be fundamentally different in some way from those going around in the other direction. How is this explained geometrically if the spacetime is static (which it has to be unless there is a fifth macroscopic dimension)?

    The second is that this idea that spacetime itself can impart an angular momentum seems to suggest it possesses some itself which seems odd since it's static. Now, if you think in terms of the marbles, the naive way to explain it is to say that the spacetime is this manifold on which the marbles (matter) move (so a real marble traveling around a curved wooden track gets its angular momentum from constant microscopic interactions with the track which is also made of matter and so momentum and energy are conserved). But spacetime is not made of matter (obviously) so the analogy breaks down. I guess it's a bit of a chicken-and-egg type situation - you could say that, physically, the motion of a photon defines a geodesic but then you could say that the geodesic defines the motion of the photon.

    That probably made no sense. Sometimes I hate the internet.

    Anyway, regarding mass and vacuum, I'm still working under what apparently is the old adage, "space tells matter how to move, matter tells space how to curve," or whatever the quote was (was that Wheeler?).
    • CommentRowNumber12.
    • CommentAuthorIan_Durham
    • CommentTimeMay 20th 2010
    • (edited May 20th 2010)
    Thanks for the pointer to the ADM stuff Urs. I will have to take a look at that. I have no trouble asking questions and I'll admit my previous approach was a bit heavy-handed (side-note: the best complement I ever got from a student was that my physics class had taught her to question everything, so go figure).

    I think what threw me for a loop was that I had worked on GR for ten years and thus was under the impression that I knew it (which I obviously don't in the same way you do). Mine should be a cautionary tale for the future: a) experimentalists/empiricists and pure theorists need to talk to each other more and b) we need to do a better job with undergraduate and even early graduate-level education!

    Edit: I still harbor some philosophical misgivings, but maybe my response to Tim will lead to getting some of that cleared up. I also think those of you immersed in QFT for so long may sometimes forget some of the "prejudices" non-field theorists have ingrained into them since it is becoming more clear to me that a field-theoretic viewpoint eliminates many of the problems.
    • CommentRowNumber13.
    • CommentAuthorIan_Durham
    • CommentTimeMay 20th 2010

    you certainly can influence and change the gravitational field in your spatial environment...



    Right, and I think that nails my misunderstanding right on the head. So, would you say spacetime is defined by the associated metric and all the associated curvature stuff (Reimann tensor, Ricci scalar, etc.) and the field is defined by the stress-energy tensor and then Einstein's equations link them? I think, then, the misunderstanding can be directly traced to the interpretation of that equal sign in Einstein's field equations or, perhaps, to the left-hand side (the curvature stuff). My head's spinning from watching my worldview crumble so I won't say I have a stance on it yet (but I used to), but let me ask then, do you view that as saying that the field and the spacetime are one and the same thing or that they are separate but related by Einstein's equations?
    • CommentRowNumber14.
    • CommentAuthorTim_van_Beek
    • CommentTimeMay 20th 2010

    …I’m still working under what apparently is the old adage, “space tells matter how to move, matter tells space how to curve,” or whatever the quote was (was that Wheeler?).

    That’s still true! (I, too, think that it was Wheeler). In GR, everything that is not the gravitational field, is matter. And you can get a non-trivial gravitational field without matter, like the Kerr spacetime. This is not a contradiction to Wheeler’s statement.

    …do you view that as saying that the field and the spacetime are one and the same thing or that they are separate but related by Einstein’s equations?

    If you state the field equations of GR as

    G ab=8πT ab G _{ab}= 8 \pi T_{ab}

    (using the notation of the book by Wald), then the (gravitational) field and the spacetime are both defined by the left side, while all matter is on the right side (your question seems to imply that you see the field or spacetime on the right side). The field equations define the influence of gravity on matter and of matter on gravity, just like Wheeler said :-)

    The second is that this idea that spacetime itself can impart an angular momentum seems to suggest it possesses some itself which seems odd since it’s static.

    Static means that the whole future and past is described by spacetime. But that travelling on a timelike curve changes your state by interaction with gravitation is not a surprise at all, you experience it when you fall down :-) A more dramatic example is a spacecraft that is torn apart by tidal forces, but I’ll set that aside for the next Bruckheimer blockbuster…

    The first is that, since photons travel on geodesics, and photons orbiting in the opposite direction of the “rotation” (or whatever) behave differently, geodesics going one way around a Kerr solution must be fundamentally different in some way from those going around in the other direction. How is this explained geometrically…?

    That’s tricky, finding some exact but easy to understand explanation of this situation. If I come up with something useful I’ll mention it, but please don’t be disappointed if I don’t :-)

    • CommentRowNumber15.
    • CommentAuthorIan_Durham
    • CommentTimeMay 21st 2010
    • (edited May 21st 2010)

    And you can get a non-trivial gravitational field without matter, like the Kerr spacetime.



    OK, so the empiricist in me then asks, what causes the curvature in this situation? And what happens to the mass of a rotating star when it collapses to a Kerr black hole if the Kerr solution is for a vacuum? (Sorry. I'm like a little kid who keeps asking "but, why?" after every answer. :-))

    your question seems to imply that you see the field or spacetime on the right side



    ...and that must be why that whole original argument started. (Is there an emoticon for a lightbulb going on over one's head?) Spacetime and the gravitational field are one and the same then. (As a note, there's a whole sub-culture of people working on GR that don't think this, but it does make more sense this way.)

    I'll set that aside for the next Bruckheimer blockbuster...



    Funny aside: not long after my wife and I were married the film Event Horizon came out. Before we went to see it, all I had seen was a poster for it and I naively thought it was about black holes...

    That's tricky, finding some exact but easy to understand explanation of this situation. If I come up with something useful I'll mention it, but please don't be disappointed if I don't :-)



    Yes, please let me know if you do. I think this is a sticking point with many people. On the one hand, it's hard to imagine how a static spacetime manifold does this (and thus many people give up and just assume it rotates), but on the other hand, if the spacetime did rotate, one would have to ask "with respect to what?" It's a little bit of a Catch-22 in a way.

    Nevertheless, I think a lot of these issues are very subtle and not easily grasped even after reading (and re-reading and teaching and writing dissertations on... :-)) GR and GR texts.
    • CommentRowNumber16.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 21st 2010

    Ian wrote:

    The second is that this idea that spacetime itself can impart an angular momentum seems to suggest it possesses some itself which seems odd since it’s static. Now, if you think in terms of the marbles, the naive way to explain it is to say that the spacetime is this manifold on which the marbles (matter) move

    if you are visualising marbles, then you are looking at point-in-time snapshots, and so it is not appropriate to talk about spacetime. Spacetime contains geodesics which represent the entire history of the marbles, and the interpretation that marble ’move along’ the geodesics requires a second ’time’ dimension to allow this process to occur. If you are in the reference frame of the marble, then the spacelike slice (just a local bit of it) is not static: it changes dynamically. This should then account for the fact the marble changes it angular momentum.

    Time vB wrote:

    you certainly can influence and change the gravitational field in your spatial environment…

    And Ian replied

    Right, and I think that nails my misunderstanding right on the head. So, would you say spacetime is defined by the associated metric and all the associated curvature stuff (Reimann tensor, Ricci scalar, etc.) and the field is defined by the stress-energy tensor and then Einstein’s equations link them?

    This is where I repeat my warning: this whole discussion should take place with no reference to the gravitational field. I think Tim was being slightly colloquial: really he was saying (if I may take the liberty of putting words in his mouth) that you can curve spacetime in your vicinity, i.e. you create a disturbance in the Force (whoops, wrong movie ;-)

    OK, so the empiricist in me then asks, what causes the curvature in this situation? And what happens to the mass of a rotating star when it collapses to a Kerr black hole if the Kerr solution is for a vacuum?

    Note that the Kerr spacetime is ’just’ a manifold with a metric that satisfies some equations, with a ’hole’ where the singularity should be. From a more physical pov, technically the only place in the Kerr spacetime which is not vacuum is the worldline of the singularity, and then it is very much a philosophical question whether the singularity is part of spacetime or not. The metric is not defined at the singularity, and so (at least mathematically) that bit is removed. Really we know that a quantum theory of gravity is needed to deal with what happens at the singularity, and so this is not a question we are allowed to ask under strictly classical GR.

    • CommentRowNumber17.
    • CommentAuthorTim_van_Beek
    • CommentTimeMay 21st 2010
    • (edited May 21st 2010)

    Ian asked:

    OK, so the empiricist in me then asks, what causes the curvature in this situation? And what happens to the mass of a rotating star when it collapses to a Kerr black hole if the Kerr solution is for a vacuum?

    David answered:

    …Really we know that a quantum theory of gravity is needed to deal with what happens at the singularity, and so this is not a question we are allowed to ask under strictly classical GR.

    Indeed, as an empiricist you should view the Kerr spacetime as a mathematical idealization, much like in classical mechanics where you would model the sun and the planets in the solar system as points to calculate their trajectories. And although it is common sense that GR predicts the existence of black holes, and astronomors are sure that such objects exist, we should keep in mind that we do not really know what an observer would observe after falling through the event horizon, because we do not really know what “physically existing black holes” look like (BTW: the movie was about black holes, wasn’t it? I mean the “Event Horizon” had a wormhole drive).

    David wrote some time ago:

    the stuff with horizons is all geometry: the Schwarzschild spacetime is, at its most complicated, topologically $\mathcal{R}^4$−{0} (as far as I understand it - I’ve not studied topology in GR properly). The interesting stuff depends on the manifold/metric structure.

    Interesting question, but I think you mean something like ℛ 4\mathcal{R}^4 with a one dimensional submanifold removed, or else the singularity would exist for an infinitesimal time only :-) Note that both the Schwartzschild and the Kerr metric can be used to describe the (vacuum) spacetime outside of a spherically symmetric mass distribution of radius >0\gt 0, so there does not need to be a singularity.

    In differential geometry we usually think of a topological manifold that gets additional geometric structure like a metric, in GR the situation is somewhat reversed: You assume that spacetime is a four dimensional Lorentzian manifold (Lorentzian by the equivalence principle), then the field equations spit out the metric. You can then go ahead and try to deduce information about the topology. For the Kerr spacetime it is possible to classify the spacetime up to homotopy, see Barrett O’Neill: “The geometry of Kerr black holes” chapter 3.9 “Topology of Kerr Spacetime”. (It has pictures!)

    • CommentRowNumber18.
    • CommentAuthorIan_Durham
    • CommentTimeMay 21st 2010

    Indeed, as an empiricist you should view the Kerr spacetime as a mathematical idealization...



    :-) Indeed, I view all of mathematical physics that way, to some extent, but I wasn't sure what folks here thought so I kept my mouth shut about it. Though I will say that I still question what aspects of math, if any, are "real." At some point the mathematical description becomes so "abstract" it's hard to know where physical reality ends and math begins (for lack of a better way of saying it). I tried getting a sense for what mathematicians thought about this over at MO, but they "ran me out of town" as it were.

    the movie was about black holes, wasn't it? I mean the "Event Horizon" had a wormhole drive



    All I remember of that movie was an eye-less Sam Neill saying "where we're going we won't need eyes!" I think my wife has forgiven me for making her sit through it.

    then the spacelike slice (just a local bit of it) is not static: it changes dynamically. This should then account for the fact the marble changes it angular momentum.



    I still find that vaguely unsettling, but maybe this another example of why we need a theory of quantum gravity. It's the "interaction" (if you will) between the object and spacetime (or the space-like slice of it) that bugs me.

    This is where I repeat my warning: this whole discussion should take place with no reference to the gravitational field.



    Right, though what happens if gravitons are discovered? I guess it comes down to the truth of the paradigm of four fundamental forces - is gravity really a force (and in GR it's not or, rather, it's what some people call a "fictitious" force).

    then it is very much a philosophical question whether the singularity is part of spacetime or not.



    Right, ok. That makes sense. Most of the folks I know think of it as part of the spacetime (I would assume) since they deal in very practical (?) matters of the masses of such things (e.g. the mass of the "black hole" at the center of the galaxy).

    For the Kerr spacetime it is possible to classify the spacetime up to homotopy, see Barrett O’Neill: "The geometry of Kerr black holes" chapter 3.9 "Topology of Kerr Spacetime". (It has pictures!)



    Oh, cool. I'll have to get my hands on this.

    And, yes, things like event horizons can, of course, be "eliminated" by changing one's coordinate system, though it presumably doesn't eliminate the physical effect (which is why we need to create a black hole in the laboratory so we can see what's going on here).
    • CommentRowNumber19.
    • CommentAuthorTim_van_Beek
    • CommentTimeMay 21st 2010

    I said:

    Indeed, as an empiricist you should view the Kerr spacetime as a mathematical idealization…

    Ian answered:

    :-) Indeed, I view all of mathematical physics that way, to some extent…

    Okay :-) But I was referring to the more concrete question “what rotates in the Kerr spacetime?” and meant “nothing, because that was a abstracted away when we let the radius of our rotating star go to zero”.

    Right, though what happens if gravitons are discovered?

    That discovery would leave the theorists with the same open question that they face now: “How do we reconcile quantum theories with gravity?” The existence of “black holes” (if you accept this as a fact) already shows that there is a large gap in our understanding, the discovery of gravitons would only support this.

    Most of the folks I know think of it as part of the spacetime (I would assume) since they deal in very practical (?) matters of the masses of such things (e.g. the mass of the “black hole” at the center of the galaxy).

    There is no contradiction here, a mathematician would say “spacetime is a Lorentzian manifold, ergo points with divergent metric cannot be part of it”, but that is just a mathematical definition. Wether or not you exclude the singularity itself from spacetime, the mass and angular momentum are still well defined and measurable. I think that is a point that Urs had in mind when he mentioned Misner-Thorne-Wheeler: Chapter 19 is about how to measure both e.g. in the weak field approximation far away.

    …things like event horizons can, of course, be “eliminated” by changing one’s coordinate system, though it presumably doesn’t eliminate the physical effect…

    There are “phenomena” that depend on the chosen coordinate system and phenomena that don’t. A “singularity” of the metric may depend on the chosen coordinate system, then it is not a true singularity that has any physical effect, or it does not depend on the chosen coordinate system. The difference becomes manifest if one takes a look at measurable quantities like the Riemann curvature tensor (divergent for a true singularity, not divergent for a coordinate singularity). The existence of an event horizon does not depend on the chosen coordinate system (“the event horizon is the boundary of the region where the escape to infinity is possible”).

    Understood in this sense, the statement “the event horizon can be eliminated by changing one’s coordinate system” is wrong, strictly speaking. (But I guess you had something in mind that is similar to what I say above?).

    • CommentRowNumber20.
    • CommentAuthorIan_Durham
    • CommentTimeMay 22nd 2010
    • (edited May 22nd 2010)

    Indeed, as an empiricist you should view the Kerr spacetime as a mathematical idealization...I was referring to the more concrete question "what rotates in the Kerr spacetime?" and meant "nothing, because that was a abstracted away when we let the radius of our rotating star go to zero".



    Well, let me put it this way, I see the Kerr spacetime as a mathematical model. It appears to match physical data, it's consistent with other "models" (theories), and it has predictive power. Empirically, I'm still interested in what's physically there and what it's doing, i.e. physically measurable quantities, stuff that's got some concrete ontological status. So your argument is akin to the standard argument that is given for electron spin - it's not really rotating (forget the bizarre discreteness issue for a moment) since it's a point particle, i.e. its radius is zero (which it has to be in order to not violate relativity). As standard an argument as this is, I have long had misgivings about this as well for similar reasons. However, by not involving spacetime, it does avoid some of the stickier issues associated with spacetime (though let me be clear in saying I still think it's a point particle, I just have misgivings about the idea that point-particles necessarily give up certain ontological qualities that extended objects possess).

    Understood in this sense, the statement "the event horizon can be eliminated by changing one's coordinate system" is wrong, strictly speaking. (But I guess you had something in mind that is similar to what I say above?).



    Yeah, I meant you can get rid of the apparent mathematical problem, but the physical event horizon is still there in the same place it always was. All it really buys you is a better set of coordinates.

    So, I had to attend our annual honors convocation tonight and afterward I got to talking with one of my colleagues from the math department. He knows I've been on-and-off-and-on-again interested in category theory and he said that, even though he's a pure mathematician, he finds category theory too abstract for him. He then thought I was nuts (at first) for trying to develop this theory of "relational empiricism" that I'm working on combining categories with empiricism since he sees them as so completely opposite - categories are highly abstract while physical measurements are about as real as you can get. Anyway, if anyone finds this general idea intriguing, Eric gave me a scratch pad on his personal nLab site and you're more than welcome to contribute.
    • CommentRowNumber21.
    • CommentAuthorTim_van_Beek
    • CommentTimeMay 22nd 2010
    • (edited May 22nd 2010)

    Ian said:

    Empirically, I’m still interested in what’s physically there and what it’s doing…

    Yes of course! This is a very fascinating question in theoretical physics, I share that feeling. But Kerr spacetime and more generally general relativity won’t tell us. If you ask “what’s really behind the event horizon?” the answer won’t come from general relativity.

    So your argument is akin to the standard argument that is given for electron spin - it’s not really rotating (forget the bizarre discreteness issue for a moment) since it’s a point particle, i.e. its radius is zero (which it has to be in order to not violate relativity).

    Yes, true, I say that the theory won’t answer this kind of question and that we have to take this for granted when working with it. But I am still fascinated by the question if there is some deeper truth to be discovered here, and I don’t consider this to be a contradiction (believe or don’t believe?).

    BTW: If I have some good ideas and some time I will expand the Kerr spacetime entry and announce that here, right now it only has a table with the metric in canonical coordinates for the Minkowski, Schwartzschild and Kerr spacetimes (any help with formatting this table is welcome :-).

    …he finds category theory too abstract for him.

    That’s unfortunate, did you notice that John Baez is giving a talk in Oxford today that may be of interest to you? And that Bob Coecke explained some of his thoughts and results on the nCafé here?

    • CommentRowNumber22.
    • CommentAuthorEric
    • CommentTimeMay 22nd 2010

    Mike made a comment recently that I meant to respond to, but work’s been hectic lately and it slipped by. If I can find it, I’ll point to it. The subject was “passing to cohomology”. It seems like the way a cohomologist might look at the world depends on his choice for what cohomology means. There are various flavors of cohomology and although they all have common roots, they can give different impressions of the same observations. This reminded me of quantum mechanics and experimentation. A choice of a cohomology theory seems like a choice of an experimental set up (loosely and vaguely speaking).

    As we know from wave-particle duality, the outcome of an experiment depends to a large degree on how the experiment is set up and what its looking for. If you’re looking for waves, you’ll see waves. If you’re looking for particles, you’ll see particles. It would be neat to relate wave-particle duality to choices of cohomology. The link is tenuous at best, but my gut says there might be something there.

    Anyway, I think what matters to an empirical physicist is what can be measured. Category theory seems perfectly suitable as a theory of measurement. You have probes testing spaces, etc. Have a look at:

    motivation for sheaves, cohomology and higher stacks

    Have you been following John’s recent discussions about copying classical states?

    • CommentRowNumber23.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 23rd 2010

    @Ian, Eric

    Have either of you read Chris Isham’s Topos Methods in the Foundations of Physics? I find it fascinating, and it’s certainly backed up with a bunch of ’real maths’ (the article linked to is described as a conceptual description). But this is off topic.

    • CommentRowNumber24.
    • CommentAuthorIan_Durham
    • CommentTimeMay 23rd 2010

    If you ask "what's really behind the event horizon?" the answer won't come from general relativity.



    Well, true enough, though you can take GR pretty far (e.g. you can study wormholes in a purely classical manner). I think in my mind it goes back to the nature of spacetime again. So we can say things like "nothing is rotating in the Kerr spacetime" or "the Kerr spacetime is a vacuum solution" because of the way the equations seem to behave. But then we can also say that it's just a mathematical model so, empirically, there's nothing wrong with. But the trouble is spacetime since it is part of the mathematical model and yet it is something we seem to experience. In other words, given a model like GR, what rules are employed when assigning an ontological status to things in the model?

    That's unfortunate, did you notice that John Baez is giving a talk in Oxford today that may be of interest to you? And that Bob Coecke explained some of his thoughts and results on the nCafé here?



    Thanks for the heads up. I didn't know Bob had posted stuff to the nCafé (when I chat with Bob the conversation usually ends up being about beer :-)). As for the Baez talk, I wonder if there will be a recording of it posted somewhere.

    The subject was "passing to cohomology".



    Was it Wheeler who had gotten into cohomology as part of geometrodynamics? I have this recollection that someone had gotten into this link between cohomology and GR back in the '50s or something. In any case, I have never heard your take on it. That's really interesting. I don't know enough about cohomology yet to really say that much about it (but summer is here and I have nothing to do!).

    It would be neat to relate wave-particle duality to choices of cohomology. The link is tenuous at best, but my gut says there might be something there.



    Could be, though see my previous comment about my lack of knowledge in cohomology. In addition, you might be interested in a lengthy discussion on my blog recently about the ontological status of the wavefunction. I took a poll and the vast majority of people don't view it as possessing any ontological status. By extension, I would bet many of those same folks extrapolate that to mean that the term "wave-particle duality" is misleading. To them it's all particles and the difference is simply the shape of the probability distribution.

    Have you been following John's recent discussions about copying classical states?



    Unfortunately no. I have been very busy lately doing some work with the oil spill in the Gulf and the semester just wrapped up so I haven't had much time. But things start to calm down tomorrow.

    Have either of you read Chris Isham's Topos Methods in the Foundations of Physics?



    Not yet, but it is on my "to-read" list. I first heard about his work on topos theory through some FQXi announcement or article or something. Incidentally, I've heard he's a really nice guy and quite responsive to queries.
    • CommentRowNumber25.
    • CommentAuthorTim_van_Beek
    • CommentTimeMay 25th 2010

    Kerr spacetime now has a definintion of the Boyer-Lindquist blocks, and I think I can already make some of my statements a little more precise with the little material that is there:

    1. Spacetime itself cannot move, of course, that is true for all spacetimes - to have it move you would need an extra time dimension, and maybe some ambient space, that is something that I called “stationary” above, but

    2. in GR stationary is usually defined to be something else. Using the Boyer-Lindquist coordinates we can see that ∂ t\partial_t is a Killing vector, so Kerr spacetime is invariant under “time translations”, but

    3. the problem with the last statement is of course that t is close to a classical global time coordinate on the Boyer-Lindquist block I only. When I said that GR won’t tell us what is behind the event horizon I meant that, of course, we can calculate what is going on in the blocks II and III, but do the results tell us that GR breaks down there or that the universe is very crazy? :-) I don’t know, I think we should expect that a “physically existing Kerr black hole” behaves differently from what pure GR predicts about blocks II and III.

    4. There is nothing rotating in Kerr spacetime means that you can take the coordinate r from infinity to zero without finding any matter at any point :-)

    Next, if time permits, I will add some additional statements about the blocks and then maybe a little bit about different geodesics…

    • CommentRowNumber26.
    • CommentAuthorIan_Durham
    • CommentTimeMay 25th 2010

    There is nothing rotating in Kerr spacetime means that you can take the coordinate r from infinity to zero without finding any matter at any point :-)

    Well, the problem I guess is whether the value of r=0r=0 is an actual point on the geodesic since an empiricist would argue that matter does exist at that point.

    • CommentRowNumber27.
    • CommentAuthorTim_van_Beek
    • CommentTimeMay 26th 2010

    To take the discussion one step further we will need a better definition of singularity and some facts about geodesics in Kerr spacetime, it will take me a while to prepare those. So, if the thread falls asleep, that does not imply that I lost interest :-)

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