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Lubos Motl's question about absolute spacetime

  1. Nov 29, 2003 #1


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    In the "resource" sticky-thread in string/loop forum Lubos Motl described Brian Greene's new book "Fabric..." and restates a basic question that (as Lubos alludes) has been around since Leibniz and Newton debated it and even before that

    Is space (and spacetime) a "something" or is it just a bookkeeping device to remember the relations between different events? Our answer to this question has changed many times as the centuries went...

    Do space and spacetime have an absolute stand-alone reality as somethings that could exist on their own, without reliance on the gravitational field and matter?

    Newton is sometimes mentioned as the main person responsible for establishing the idea of an absolute space and time. An opposing view, that space and time are relational, is ascribed to Leibniz, Descartes, and others. Einstein's apparently identified space with the gravitational field itself---a "relationalist" point of view---and so I suppose his thought would, if it had to be put on one side or another, be more closely allied with Leibniz and Descartes. Mach's name has sometimes served as a rubric for relationalism and was invoked by Einstein. But it seems to me that the relational idea of space only begins to make sense with General Relativity. As I see it the split is between Newton's vision (absolute space and time, the frame in which fields are defined, modernly now Minkowski spacetime) on the one hand and Einstein's vision on the other.

    People sometimes quoted as relationalist antecedents to GR, such as Ernst Mach, Leibniz, Aristotle, dont come across to me coherently.

    I will include an exerpt from Lubos Motl's post below to add context.

    One thing I find really interesting as a BTW remark is that in 1905 Einstein crystalized a new model Absolute Space----Minkowski spacetime---giving the Newtonian viewpoint a seemingly perfect venue.
    Special relativity is a kind of perfection of Newton's reality.

    And then in 1915 he presented us with what is effectively the antithesis of absolute space and time. In a sense the question refreshed for us by Lubos is: do we favor the 1905 picture or the 1915 picture. There's a basic split between the two that is still alive and kicking today.

    Here is the relevant part of Lubos' post
    Hi physics boys,

    I think that you deserve to be insiders. Brian Greene has finished his second popular book, The Fabric of the Cosmos,


    I've read it and it looks great. There is some older material about string/M-theory in it, a newer material on cosmology (and stringy cosmology), inflation, the arrow of time, the speculations and facts on time travels etc., but also a great story of the space. Is space (and spacetime) a "something" or is it just a bookkeeping device to remember the relations between different events? Our answer to this question has changed many times as the centuries went...
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  3. Nov 30, 2003 #2


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    search for a balance empirical perspective

    the quest for a quantum model of gravity does raise issues of the nature of space and time----may even raise philosophical questions (as the development of science has at certain points in its history)

    the question Lubos relayed in his comment on BG's new book is one of several basic ones that the development of quantum gravity will almost certainly have to address----another is the nature of local Lorentz symmetry.

    Amelino-Camelia talks about this in a recent survey of the different approaches to quantum gravity and their possibilities for empirical verification.

    "The three perspectives on the quantum-gravity problem and their implications for the fate of Lorentz symmetry"

    http://arxiv.org/gr-qc/0309054 [Broken]

    From the abstract:

    "Each approach to the quantum-gravity problem originates from expertise in one or another area of theoretical physics. The particle-physics perspective encourages one to attempt to reproduce in quantum gravity as much as possible of the successes of the Standard Model of particle physics, and therefore, as done in String Theory, the core features of quantum gravity are described in terms of graviton-like exchange in a background classical spacetime.

    From the general-relativity perspective it is natural to renounce to any reference to a background spacetime, and to describe spacetime in a way that takes into account the in-principle limitations of measurements. The Loop Quantum Gravity approach and the approaches based on noncommutative geometry originate from this general-relativity perspective.

    The condensed-matter perspective, which has been adopted in a few recent quantum-gravity proposals, naturally leads to scenarios in which some familiar properties of spacetime are only emergent, just like, for example, some emergent collective degrees of freedom are relevant to the description of certain physical systems only near a critical point.

    Both from the general-relativity perspective and from the condensed-matter perspective it is natural to explore the possibility that quantum gravity might have significant implications for the fate of Lorentz symmetry in the Planckian regime. From the particle-physics perspective there is instead no obvious reason to renounce to exact Lorentz symmetry, although (“spontaneous”) Lorentz symmetry breaking is of course possible.

    A fast-growing phenomenological programme looking for Planck-scale departures from Lorentz symmetry can contribute to this ongoing debate."

    The paper is in part derived from an invited talk given at the Tenth Marcel Grossmann Meeting on General Relativity (Rio de Janeiro, July 2003) It strikes me as an attempt to see the whole picture in a balanced way particularly as relates to the possibility of testing.
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  4. Nov 30, 2003 #3


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    from Amelino-Camelia's conclusion

    "...approach and the perspective that generated it. This exercise appears to suggest that what one finds in a given quantum-gravity approach might be more directly connected with the perspective which has been adopted, rather than with something intrinsic in the quantum-gravity problem.

    Many features of quantum-gravity approaches that originate from the particle-physics perspective, such a String Theory, simply reflect the intuition that one develops working with the Standard Model of particle physics.

    Analogously quantum-gravity approaches that originate from the general-relativity perspective or from the condensed-matter perspective carry a strong trace of intuition developed in working in those fields.

    The expectations concerning the fate of Lorentz symmetry in quantum gravity appear to be a natural way to discriminate between the different perspectives. As I stressed here, it is rather obvious that a particle-physics perspective should lead to quantum-gravity approaches in which there is no a priori reason for departures from ordinary Lorentz symmetry. And it is equally obvious that instead from the condensed matter perspective Lorentz symmetry should only emerge as an approximate symmetry.

    Somewhat more subtle are the indications of the general-relativity perspective for the fate of Lorentz symmetry, and therefore I devoted a significant portion of these notes to the general-relativity perspective. Results obtained in recent years, some of which I
    reviewed here, suggest that also from the general-relativity perspective some departures from Lorentz symmetry might naturally emerge. If one adopts a description based fundamentally on noncommutative geometry (as some aspects of the general-relativity perspective could invite us to do) departures from Lorentz symmetry are really very natural, perhaps inevitable.

    And there is now growing evidence that also when the general-relativity perspective leads to discretized-spacetime approaches, as in Loop Quantum Gravity, departures from Lorentz symmetry are naturally encountered. I here also presented additional observations, of more general validity, that favour the presence of departures from Lorentz symmetry in approaches based on the general relativity perspective.

    Tests of possible Planck-scale departures from Lorentz symmetry, besides giving us a chance of finding the first experimental facts about the quantum-gravity realm, are therefore also a way to check which one of these three perspectives should be favoured. Remarkably, as discussed in Section 6, there will be several opportunities in
    these coming years for experimental searches of Planck-scale departures from Lorentz symmetry..."

    as I read more of this paper I continue to be impressed by the philosophical clarity. What will be (or already are being) tested are actually different philosophical perspectives on space and time! These different expectations and intuitions about space and time derive from experience in General Relativity on the one side and Standard Model Particle Physics on the other. And there is even a third approach he identifies which is now emerging from condensed matter physics. Almost independently of the details of particular mathematical models---upcoming tests of Lorentz symmetry are likely (A-C suggests) to distinguish between different sets of philosophical predispositions which (consciously or not) prevail in different specialty-groups of physicists.
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  5. Nov 30, 2003 #4


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    probably it is philosophical questions like the issue mentioned by Lubos Motl (which we may hope is clarified in the new "Fabric of the Cosmos" book) that will come to the forefront in attempts to merge QM and GR, because these fields differ at the most basic foundations level

    Here is an image that has some bearing on these "what is space/what is time?" issues. This is from page 7 of Rovelli's book "Quantum Gravity" (online draft in PDF at his website) section 1.1.3 "The physical meaning of general relativity".

    At this point he isnt talking about quantum gravity but merely about classical (1915 vintage) GR:

    "GR is the discovery that spacetime and the gravitational field are the same entity. What we call 'spacetime' is itself a physical object, in many respects similar to the electromagnetic field. We can say that GR is the discovery that there is no spacetime at all. What Newton called 'space' and Minkowski called 'spacetime' is unmasked: it is nothing but a dynamical object...
    ...This implies that physical entities--particles and fields--are not all immersed in space and moving in time. They do not live on spacetime [my comment: because in GR an absolute space time for them to live in doesnt exist]. They live, so to say, on one another.

    It is like if we had observed in the ocean many animals living on an island...Then we discover that the island itself is in fact a great whale. Not anymore animals on the island, just animals on animals. Similarly the universe is not made by fields on spacetime; it is made by fields on fields."
    Last edited: Nov 30, 2003
  6. Nov 30, 2003 #5


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    distinguishing Rovelli's view from Amelino-C's

    we have to carfully distinguish Amelino-Camelia's viewpoint from the way Rovelli characterizes General Relativity.
    A-C's main interest now is Quantum Gravity Phenomenology, which is apparently heating up as an empirical research area. This means taking an impartial overview and discerning differences among the various theories in the predictions they lead to.

    Rovelli, on the other hand, is focussed on Loop Gravity and wants his readers to have a firm grasp of the Relationalist roots in classical GR that the Loop Gravity approach stems from.

    In some branches of physics one habitually starts with an absolute Minkowski space time. Fields are functions defined on that space. In specific cases one may depart from custom slightly and use some specified curved spacetime instead, but the idea of defining a field "in midair" so to speak, without any spacetime for it to live on would be difficult to accept.

    It is also hard to accept from a lay perspective. Maybe the obstacles to understanding are akin to those around the idea of how can a knot be defined without a specific space for it to live in---the abstract knot (which is what the knot-theorist wants to study) is an equivalence class of particular knots in ordinary 3D space. One may use a conventional space to start from, with some definite coordinates and some specified geometry, but in the end one can throw away the space and just keep the knot.

    For Rovelli you answer the question of whether some particular star is rotating not by judging its rotation relative to some particular point or point in space---you say it is rotating or not rotative with respect to the gravitational field. Points in whatever spacetime continuum have no physical identity--no absolute meaning--and the field is what the theory orients to and gives it reference.

    this is tough to assimilate and make intuitive but people have been living with it since 1915---part of the classical theory. So Rovelli spends a fair amount of the early part of the book just on philosophy. Getting over the mental hurdles associated with background independence and the GR concept of the gravitational field.

    The image of the whale, on the back of which the other animals live, is not a complete explanation but it is a start.

    When we compare the Absolute Spacetime and the Relationist perspectives part of the job is to listen to both stories receptively.
    Probably in both cases one can say that spacetime is a "real something". But in the Relationalist case the something is a whale rather than an island. So if one has been taught in school to expect an island, then one may have the impression that the whale is not real (simply because he is not an island).
  7. Dec 10, 2003 #6
    The definition of time!

    Hi Marcus!

    You seem to have an interest in the definition of time as it relates to physics.

    I would very much like to hear your comments on my feeling about the subject.

    http://home.jam.rr.com/dicksfiles/flaw/Fatalfla.htm [Broken]

    Looking forward to your comments -- Dick
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  8. Dec 11, 2003 #7


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    Interesting post Marcus,

    A couple of things came up that I think are worth going into detail.

    First, the statement: GR is the discovery that spacetime and the gravitational field are the same entity.

    This often causes confusion, because gravitational field seems to have 2 different meanings. Since the time of Newton, there exists a gravitational field between things with mass. This mysterious field was really only a mathematical description of gravity, as no mechanism for how this field work (or what it actually was) is provided under that Newtonian framework. Then GR comes along and provides a geometric explanation for the field. Instead of being some mysterious force, gravity is a result of spacetime curvature.

    Now correct me if I'm wrong, but from what I've read spacetime curvature (the explanation for newton's fields) is not the same thing as the more fundemental gravitational field. The claim is that even in the complete absence of any mass, the field can exist as flat spacetime, along the lines of an empty DeSitter universe. But this seems like an odd choice of words. If the field is not necessarily the manifold plus metric, why the name? Can you elaborate on why this is?

    The second point has to do with how LQG would address the philosophical question of space. Correct me if I've misread something here. Points in space have no meaning except where the field lines intersect. But if we claim that spacetime is nothing but the relations of these events, then it seems to be a bad case of circular logic.

    IOW, spacetime has no meaning except as relations among events. And events have no meaning except as intersections of the field lines. Am I missing something here? From that one would conclude that field lines, and not points are fundemental to the existence of spacetime.
  9. Dec 11, 2003 #8


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    There is no gravity field over and above the curvature (the possible cosmological constant is an entirely different subject, and I'm going to ignore it here).

    Einstein's field equations for empty space say that the sorta kinda second order divergence of the curvature is equal to the energy-momentum-stress at any spacetime point (except singularities). The field IS the curvature. The reason you can have a non-zero result with no matter is that other things than matter can create energy-momentum-stress. Light for one example, and gravity waves for another.
  10. Dec 11, 2003 #9
    It's a matter of terminology, but I'd probably say that the gravitational field is the metric (or metric modulo diffeomorphisms) ... it's the thing you're solving for in Einstein's gravitational field equation. But we tend to say that we observe the "force of gravity" when the metric isn't flat.
  11. Dec 11, 2003 #10


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    I find the philosophical discussion of space and time in Rovelli's book fascinating. Go back over parts of it occasionally and always get more out of it.

    the draft available for download at his website is currently the
    December 2 version. It is due to go to the publisher (Cambridge U. P.) fairly soon and then might not be available subsequently---dont know for sure about this.

    the relational idea of space does not seem to be logically circular!
    Einstein took considerable trouble to be clear about this
    and so have others who have made General Relativity their specialty---"Relativists"---over the years

    the underlying manifold or continuum has no meaning or existence on its own, but the field does
    the field can be thought of as a metric or as a cartan frame field
    there is no absolute space----only the gravitational field itself and it is with respect to IT that other fields and motions are defined

    if something is rotating it is not rotating with respect to absolute space (as Newton imagined) but is instead rotating with respect to the gravitational field

    these are GR truths that have been around since 1915 and have been reiterated by Einstein and various other respected Relativists. Within the field of GR they are not controversial, but they are not always understood OUTSIDE GR and indeed seem paradoxical.

    Since Newton we are all used to thinking of absolute space. Cant imagine how anything can be defined rigorously without it. But GR is an example of something logically and mathematically rigorous that does not need absolute space to define its things.

    I believe it would be better if you just look at Rovelli's book for this (instead of discussing my sketchy repetition of key ideas out of context) No use bothering with my paraphrase, where I can easily say something carelessly and give the wrong idea---when Rovelli says it all carefully.

    Look at his section 1.1.3 "The physical meaning of general relativity". This is a short exposition about a page long, around page 7 or so.

    Also the historical and philosophical parts of Chater 2 "General Relativity"
    For example 2.2 "The conceptual path to the theory" somewhere around page 34 onwards
    Especially 2.2.2 "Einstein's 2nd problem: Relativity of motion" and
    2.2.3 "The key idea"
    around page 40 and following
    also 2.3.2 "The disappearance of spacetime"
    this is a one-page summing up section, around page 52

    My page references are approx because I have the November draft instead of the December 2 draft.

    Anyway there is a part that is less than 20 pages long from around page 34 to around page 52
    in whatever draft version you have it would be sections 2.2.1 thru
    section 2.3.2

    The reasoning is verbal with several good concrete examples (a bucket of water, a pair of stars in an otherwise empty universe, a cluster of galaxies

    You may already have downloaded Rovelli's book, if not just google with the name "Rovelli" and get his homepage and look down the page to where it says "the book: PDF file"
    it says November there but if you click you actually get the December version. With my slow modem and computer it takes 5 to 10 minutes.
    but then you can save it on your desktop and not have to do it again
  12. Dec 11, 2003 #11


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    Whoah! I see that selfAdjoint and ambitwistor were just here while I was writing that.

    Earlier there was a poster named Doctor who was asking about time.
    In Rovelli's book there is a section 2.4.4 called "Meanings of time"

    It has about 10 different ideas of time, some of which occur in various branches of physics, and some 9 different features that a timescale may or may not have.

    there is also a section 2.4.6 about the GPS (global positioning system). Rovelli likes to use very basic concrete examples to illustrate stuff.

    Since there are several different meaning or kinds of time I think section 2.4.4 which lists them is a good place to start any discussion of time---so I recommend it to the other poster (Doctor).

    I suppose it is useless, at least at present, to aim for the one unique 'correct' idea of time.

    So far none of this refers explicitly to Quantum Gravity----only to the classical or un-quantized version of GR. These issues do not requir Quantum GR to address. But when one gets into Quantum GR it might turn out that time becomes discrete---goes in tiny jerks. This is an intuitive disaster for me. It is not how I grew up thinking of it. Maybe for you other people too. But fortunately we can have this discussion and reply to Eh without departing from the classical version.

    Here is Rovelli's book in case anyone wants to look at those sections on time, or GPS, or how space equals the gravitational field,


    this will save having to go to google for it.
    Last edited: Dec 11, 2003
  13. Dec 11, 2003 #12


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    Maybe you can shed some light on this. A while back, I sent an e-mail to the astronomer that runs the website http://itss.raytheon.com/cafe/qadir/qanda.html, asking about vacuum universes. Here is the reply:

    Curvature is a different thing than the gravitational field. The field can be constant (euclidean) with no curvature, and still exist as a field. Curvature just means that gravity changes from place to place which is true near sources. Einstein's interpretation of his own mathematics is that space and time have no independent existence other than as 'structural qualities of the gravitational field', which is a deeper issue than merely the statement of curvature and geometry. The fact that the elements to g-mu-nu can be interpreted as the gravitational potential is not really the point in the deeper issue of space/time and gravity.

    Any thoughts?
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  14. Dec 11, 2003 #13


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    That's ok, but I'm wondering specifically about statements made by proponents of LQG. One example is (this not an exact quote): points have no meaning outside of where field lines intersect, but you can only talk of "length" as a relation between points. That certainly seems circular.
  15. Dec 11, 2003 #14


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    I would have to see the actual quote. It is very carelessly stated because what are "field lines" and how could field lines, if they are like magnetic field lines, EVER intersect

    the world lines of two material objects can intersect, if the two collide---or they can almost intersect if there is some kind of close encounter

    that could define an event. that is how events are defined in GR
    since points of the manifold have no physical meaning

    But Eh, this is all CLASSICAL talk. It would not necessarily be a "LQG proponent" who talks about worldlines of two particles intersecting----it could be ANYBODY who wants to give an idea of what ordinary GR involves. In the case of LQG people, they spend a lot of time being clear about classical GR because LQG is based on it---is in fact the attempt to quantize it. But other people expound the classical theory too.

    To define a physical length in GR you need something to work with. Like a material object. Perhaps in absence of matter you could define a length using the field itself, e.g. the wavelength of some gravity waves! But that seems pretty far fetched and difficult. Best to think of having some matter around, if one wants to define meaningful lengths, surface areas, volumes. Hopefully someone else will correct me if I am giving misconception.

    I dont think this is logically circular---seems most UNcircular. Use matter to define events, matter to define physical distances. Because the manifold itself is "gauge": just a convenience to be factored out when no longer needed.
  16. Dec 11, 2003 #15


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    I will have to dig for it, because I'm not sure where I read that bit.

    It may have been something written by Smolin though. In Three Roads to Quantum Gravity, he explains that GR is ultimately about the gravitational field. And this field, he explains, is nothing but the relationships amoung three sets of field lines. For every physical event, there is a corresponding way in which the field lines link and knot. But I'll have to check that out to be sure.
  17. Dec 12, 2003 #16


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    Eh it is fine either way. If you find it, great. But I can get the general drift whether or not. Forgetting completely about QUANTIZING for the moment, are you familiar with a formalism for classical GR where the field is represented by a "tetrad" or a "cartan moving frame".

    Or sometimes the 4-manifold is split into a 3-manifold and a time coordinate and the gravitational field on space is represented by a "triad" or a moving frame with just 3 vectors

    I am trying to approach the picture you evoked of 3 vector fields.
    If you connected things up to make 3 sets of field lines then these
    field lines would cross (a sort of flexible XYZ frame) at every point.

    I guess it is one kind of pictorial concept of how the gravitational field looks, an image of it that someone (like Henri Cartan?) might even have used

    another image of the gravitational field is a METRIC defined at each point of the manifold---but if you picture it as a coordinate frame defined at every point then you can GET the metric (there is a little formula they always give---a chance for me to use LaTex, which I will decline at the moment)

    Anyway I dont know for sure what the quote was that your question was about but it doesnt seem too far fetched that it was about a way of visualizing the classical 1915 gravitational field (but not using the metric formalism)
  18. Jan 13, 2004 #17
    I can't follow the details of the physics here but it seems to me inevitable that the question of the absoluteness or 'relativeness' of spacetime is a philosphical question, as a few people suggest above, not a scientific one.

    This is true for any problem concerning 'essences' as opposed to attributes. It is true simply by the logic of the problem, which is that it is impossible in principle to explain things in terms of other things and not have something left over. (Aristotle, Popper etc. and maybe Goedel as well)

    This is related closely to the doctrine of 'dependent existence' which states that nothing physical exists (nor anything conceptual)except by virtue of its relation to other things that exist. It is often asserted for spiritual, experiential or mystical reasons. But it can also be argued that it follows from strict logical reasoning.

    In effect it is the 'problem of attributes' in disguise, which asks what is left of a thing when its attributes (its relationships) are removed. (If you don't know this one it's worth picking an object and trying to answer it. It's baffles most analytic philsophers).

    The doctrine of dependent existence states that physical existence is possible because there is one thing that lies beyond relational existence. This is NOT God, (although it is a little like Spinoza's idea of God).

    I don't want to drag the thread off topic, and I know some people are allergic to philosophy, but it seems to me that we have learnt so much about the science of the world that science is now being forced to face up to metaphysical problems (consciousness, spacetime, Big Bang etc.). After all science is not independent of metaphysics, since some of its axioms are answers to metaphysical questions.

    These questions represent the problems that science put to one side a couple of hundred years ago, with spectacular results. However as science advances it increasingly bumps up against its self-defined boundaries, and these problems are becoming less and less easy to ignore, as this thread illustrates.

    On the absoluteness of spacetime I think science has no choice but to leave the question unanswered or get its hands dirty by doing some metaphysics.

  19. Jan 14, 2004 #18


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    I wish just for once you philosophical people would actually download and read the philosophical parts (the stuff with no mathematical formulas) in Rovelli's new book "Quantum Gravity"

    Store it on your computer when you download because though it is available free online now it may not always be so and the book is a treasure.

    Vast amounts of it are in ordinary English.

    Newton and Einstein also considered philosophical questions (like about what is spacetime) and wrote a lot in ordinary English or sometimes German.

    Rovelli's point is that there are moments of revolution in science when in order to make progress the scientist has to think about the foundations
    and not just work at a mathematical level
    has to get down and think deeply about what the symbols actually mean
    instead of just cranking the old mathematical crank
    and performing the usual parlormagic

    And this is probably what you Canute are saying in your own philosophical lingo, so why dont you get Rovelli's book from his website and read the early nonmathematical sections like in chapter 2

    or the overview conclusions at the end in chapter 10.

    you might like to encounter a scientist who agrees with you.

    indeed they are not going to make gravity compatible with quantum phyics without a major overhaul of what you call the "metaphysics" and I would simply say the philosophical underpinnings.
  20. Jan 15, 2004 #19
    Ok. 'Just for once' I had a look. I immediately came across this.

    "“GR is the discovery that spacetime and the gravitational field are the same entity. What we call ‘spacetime’ is itself a physical object, in many respects similar to the electromagnetic field. We can say that GR is the discovery that there is no spacetime at all. What Newton called ‘space’, and Minkowski called ‘spacetime’, is unmasked: it is nothing but a dynamical object -the gravitational field- in a regime in which we neglect its dynamics.

    In Newtonian and special relativistic physics, if we take away the dynamical entities -particles and fields- what remains is space and time. In general relativistic physics, if we take away the dynamical entities, nothing remains. The space and time of Newton and Minkowski are reinterpreted as a configuration of one of the fields, the gravitational field. This implies that physical entities -particles and fields- are not all immersed in space and moving in time. They do not live on spacetime. They live, so to say, on one another.

    It is as if we had observed in the ocean many animals living on a island: animals on the island. Then we discover that the island itself is in fact a great whale. Not anymore animals on the island. Just animals on animals. Similarly the universe is not made by fields on spacetime; it is made by fields on fields.”

    Carlo Rovelli, 'Quantum Gravity' – pre-pub draft ( Cambridge University Press)

    Imho here we see a physicists take on the 'dependent existence' of Buddhism, the 'turtles on turtles' of Terry Pratchet's Discworld, and the solution to Zeno's paradoxes of motion.

    Thanks for the reference. I'll plod on with it.

  21. Jan 15, 2004 #20


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    You immediately came across one of my favorite passages in the whole book! The parable of the animals.
    I am glad to know that the draft is still available online at Rovelli's site. Now that it is in the works at Cambridge, it wouldnt have to be still available in draft form, but apparently it is.

    Canute, thanks for giving the book a try-out. It is a pleasure having someone to share enjoyment of the non-mathematical chunks of Rovelli with.
    Last edited: Jan 15, 2004
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