Tim Maudlin: "The Relativistic account of space-time geometry makes the light-cone structure of space-time a fundamental part of its geometry. This, rather than the "constancy of the speed of light" lies at the heart of the theory."

This is wrong, Tim Maudlin - the constancy of the speed of light does lie at the heart of the theory, even if you want to hide it behind the light-cone structure of space-time. And since the speed of light is not constant (you know that don't you?), you will have to join Steve Giddings, Nima Arkani-Hamed and Lee Smolin in their rejection of Einstein's space-time.

Pentcho Valev

Dear Tim Maudlin,

I really enjoyed the thread of your argument: simple arithmetic and geometry, based on our acquaintance with the physical world, leading to complex mathematical concepts, and these in-turn having some role in physics, and thus the Wigner's puzzle being solved. The whole thing would have been a smooth curve, but for your sudden jump: in explaining geometry, you jumped from the 'geometry of the bodies' to the 'geometry of the space'.

"And as soon as items are countable, other mathematical concepts can be brought to bear: ratios and proportions for example." 'Countable' is a factor; but, imagine the universe has a finite number of atoms, but does not change with time, what structure will it have? For any change, these countable entities should move, or 'motion' is the most fundamental factor. Motion is a space- time relation that obeys mathematical laws, and so the changing world obeys the laws of mathematics. That provides the simplest solution for Wigner's puzzle. Please go through my essay: A physicalist interpretation of the relation between Physics and Mathematics.

"We can even seek to construct new mathematical languages that are better suited to represent the physical structure of the world". As you have pointed out, any new mathematics depends on the basic mathematics, and cannot be 'genuinely new'. The theory of linear structures, in fact, is not new mathematics; the 'newness' is in the 'directional aspect' which is a property, a property which you assume. You have assumed the conceptual primitive, regarding space, to be lines instead of points. The mathematical rules remain the same, but the resultant structures are different; so there is justification in calling it a new mathematical language used to describe space-time.

The physicalist point that I propose is that the properties we assume, whether about space or about atoms, should be in conformity with what we observe directly. As pointed out by you, everything started from the simple observations in the nature. The concept can become more and more complex, but should not go against that simple facts. The essence of your arguments in the first part of the essay is also the same.

    Dear Jose Koshy,

    I agree that the motion of matter is described by some mathematical law, but the formulation of that law is beyond the scope of my present project. If we start with the general idea that physics is the theory of matter in motion, this presents two targets for an exact theory: matter and motion. The "motion" of a localized object seems to be best understood (and this is not how Newton would have conceptualized it) as a trajectory through space-time. So that leaves us with the problem of describing the geometry of the space-time, and the problem of understanding how the trajectories of objects are produced. You have to solve the first in order to even approach the second. That is, you need to understand the geometry of the space-time before you can begin to write down a dynamics. This project is then even more basic: not yet "what is the geometry of space-time?" but rather "what is the best mathematical language to use to describe the geometry of space-time?". If the fundamentals of Relativity are correct, this looks like an especially promising sort of language. The language is new, in the sense that (as far as I know) no one has written it down before. But at a basic level, I am a Platonist about mathematics, and in that sense nothing is really new.

    Regards,

    Tim

    Dear Tim Maudlin,

    Your stand is clear: if the fundamentals of (General) Relativity are correct, then there is four-dimensional space-time, and you are proposing a new language to describe the geometry of the space-time. I have just downloaded "New foundations for physical geometry" to know exactly what you referred as the project.

    My argument is that physicists have not explored all the possibilities classical three-dimensional space offers. For example, the model proposed by me (refer finitenesstheory.com) views that motion at speed 'c' is the fundamental property of matter, and the reaction to this motion creates gravity. Consequently, the path of a body, in a classical 3-dimensional space, is bent by its own gravity. This can give similar results like that of GR.

    From a cursory glance at the material downloaded, I get the impression that you consider going back to classical three-dimensional space is a retrograde step. However, I think that you will ultimately arrive at that conclusion.

    Dear Jose Koshy,

    I would not at all consider quite different space-tmes from GR as "retrograde": I am quite open to all sorts of possibilities. Here are two considerations. One is that if one rejects the idea that space-time is a continuum and uses a discrete space-time instead, the most natural way to implement that idea using this formalism results in the basic geometrical structure of space-time being all light like (null) paths. It would then follow automatically that any matter in continuous motion (i.e. following a continuous path) is always "traveling at the speed of light", and massive particles, such as electrons, must really engage in Zitterbewegung: rapid vibration. The other consequence of going to a discrete space-time is that the geometrical structure has a natural unique foliation, which can be considered a "backward" step to a more classical structure. (Not full Newtonian picture, with a single space persisting through time, but a structure with an intrinsic "simultaneity" built in, together with a light-cone structure.) Is that bad? It is actually the most straightforward way to be able to implement the non-locality implied by violations of Bell's inequality. So I am very open to these possibilities. As I said, in the first place this is just a new mathematical language. There are many things that can be described using it.

    Reagrds,

    Tim Maudlin

    Dear Tim Maudlin,

    One last question: Devoid of physics, can the Theory of Linear Structures be regarded as a new branch of mathematics related to topology? Or is there already any related branch in mathematics, and you are only trying to use that in physics?

    The aim of the Theory of Linear Structures is to provide a precise a formalized mathematical language (axioms for the basic structure and strict definitions) that make precise the same informal notions addressed by standard topology (continuity, connectedness, closure, limit points, etc.). These new definitions are different from those of standard topology...for example some functions that are continuous according to the standard definition are not continuous according to my definitions. Conceptually, I think my definitions are better: they match better what our informal judgments are. But more importantly, in the case of physical space-time, one can see from a physical perspective why the geometry should be well described in the this language, and it is not clear why it should be well described in the standard language. So: it is a new branch related to topology and, in some sense, competing with standard topology, which I am trying to use to do physics. It also provides a common mathematical language for describing both continua and discrete spaces. Standard topology does not provide this. So insofar as we are unsure whether physical space-time is continuous or discrete, we can can still use the basic conceptual tools provided here knowing they will work in either case. This also provides clues about how to formulate discrete approximations to continuous structures.

    Dear Tim Mauldin

    An enjoyable essay to read. I agree that "one could easily write a companion paper to Wigner's called "The Unreasonable Relevance of Some Branches of Mathematics to Other Branches"". For me there are 2 places where this particular unreasonableness of maths transfers over to physics in a sort of pincer movement that constrains physics and potentially your proposed descriptive language.

    The first is causation for any continuous physics over a space with a metrical structure: these conditions specify the maths description must be in terms of the norm-division algebras. Both General Relativity and Standard Model are in terms of NDA valued fields, which constrains every attempt to unify them to be able to reproduce this NDA description. But the physics conditions also constrains all alternative descriptions to be capable of reproducing the NDA based description.

    The second side of the pincer comes from requiring any theory to reproduce quantum theory results. My hidden propagator dynamics analysis came from having a particular theory in mind: one with discrete topological defects in a space with compactified dimensions. This is a discrete theory with a potentially discrete space. However, I was surprised to discover that my HPD analysis revealed that in order to connect with experimental results the details of the discrete nature of the theory must necessarily be erased, leaving only the same NDA-based description of experimental results as quantum theory. This is a general result for any theory with discrete particles: to connect with experimental results all discrete elements of the theory are erased, leaving only an NDA based description. This means that the entire class of HPD theories are experimentally indistinguishable from each other, as they must all reproduce the same descriptive form for experimental results.

    This pincer movement would seem to include your new descriptive language of the Theory of Linear Structures. Any new descriptive language must reproduce the descriptions of existing theory. In this bigger picture context of connecting with experimental results that are already successfully described by NDA valued fields - won't the new description be lost in the process of experimental prediction?

    Michael Goodband

      Dear Professor Maudlin,

      this is a very interesting essay and a very interesting program of research. It reminds me very much of the causal set research program of Rafael Sorkin. In particular, like causal sets it is a beautiful and austere way of looking at spacetime structure, but perhaps a bit too austere for my taste. Below I append the sort of questions that I usually ask about the causal set program which I think also apply to your program. If you have time to reply to any of these questions, that would be very much appreciated.

      --David Garfinkle

      (1) Your method gives a conformal structure, which in the case of a spacetime is equivalent to the conformal class of the metric. There is then a standard result that also giving the volume element will determine the metric. Is there a simple and natural way of specifying a volume element within your formalism?

      (2) Presumably a model for a spacetime within your formalism would be a set with your structure which also (exactly or approximately) admits a differential structure (thus making it a manifold) and a metric. But most of the sets with your structure will not be manifolds, even in an approximate sense. Is their a way within your formalism of figuring out which sets are manifolds? (or approximately manifolds).

      (3) I admire the austere beauty of your approach. But I'm pretty happy with the standard notion of spacetime as a manifold with a metric and with the usual definition of a manifold as a topological space with an atlas. What do I lose by not adopting your approach?

        Dear Michael,

        I do not feel confident that I am in command of the technical details of your analysis, but let me say at least this. The sense in which any new theory must "reproduce the descriptions of existing theory" is obviously a matter of approximation, not exact derivation. That is, all experimental data come with error bars, and so recovering the predictive success of present theories (such as quantum theory or the General Theory of Relativity) requires matching their predictions where they have actually been tested to within the tolerance of the errors. If one were to demand a higher degree of match than this, then the new description would of course be "lost". But the expectation is that the new description will deviate in its predictions from the present theory, but only to a small degree and only in certain circumstances. What that degree is, and which circumstances are relevant depends on the theory itself.

        Regards,

        Tim Maudlin

        Dear Professor Garfinkle,

        I completely understand seeing a similarity between this approach and causal sets, but at the level of implementation and exact detail they are quite different. So, for example, in the causal set approach there is essentially zero chance for any pair of events to be null related, and in this approach to a discrete structure all the fundamental space-time geometry is null. Also, as you note, in the causal set approach the idea is to get the conformal structure from the causal graph and then fill out the rest of the geometry by a volume measure. But in this approach to a discrete Relativistic space-time (which is not described here) one does not add a volume measure but rather a measure corresponding to the Interval measure on continuous paths. In a word, the basic metrical notion is length rather than volume.

        Let me answer your questions.

        Well, actually what I have said above answers 1) for the discrete case. The discrete case is nice because (as Riemann observed) the discrete case comes equipped with a measure: counting measure. But you have to be careful about what you count! In this case, one does not count nodes (which is what the causal set people do to recover a volume measure), and what you are quantifying is not volume but length. This may be a bit cryptic, but in short in the discrete case one can define a "corner" in a continuous path through the space-time as opposed to an "unbent segment", and when quantifying the length of a path one counts corners rather than nodes in the path. The result is that the number of corners on a path that lies on the light cone is zero (even though the number of nodes may be unboundedly large). The connection to the Interval should be obvious.

        In a continuum, the measure must be imported from outside (just as Riemann said). I have not tried yet to somehow analyze that measure in terms of anything else in the continuum case: it is just an additional piece of space-time geometry supplementing the conformal structure, just as in GTR.

        To the second question:

        Investing a space with a Linear Structure automatically invests it with a standard topology, via the definitions I give in the appendix. So it is easy to tell if you have a manifold. But in fact, even familiar continuous space-times (e.g. Minkowski) turn out not to be manifolds once one takes account of the directionality of time. One of my points is that the whole idea of a manifold arose in the context of purely spatial (Riemannian) geometry, and the use of those mathematical tools to deal with space-time (Lorentzian) structures is a mistake. So I am not even aiming at recovering manifolds: I am aiming to recover geometries with an intrinsic light-cone structure, which a manifold does not have.

        To the third:

        One of the main advantages of my approach is that one can deal with contiinua and discrete spaces using the very same analytical tools and definitions. No one knows whether at (say) Planck scale space-time is discrete or not. So it would be nice to articulate theories in a mathematical form that can be adapted to each possibility. As things are, one uses manifolds and differential geometry for continua and graph theory for discrete spaces. Graph theory (including infinite graphs) is a special case of the Theory of Linear Structures, but the Theory of Linear Structures can be used to analyze continua as well. It also allows you to take ideas developed in the context of continua and see how they play out in a discrete setting. So I think that is one thing that is an advantage.

        One other advantage, which I do touch on, is that the Theory of Linear Structures allows for a distinction between intrinsically directed and intrinsically undirected geometries, which is relevant to the description of time. Indeed (as I mention) it even allows one to make a distinction between intrinsically directed and intrinsically undirected topologies, which (to my knowledge) no one working in standard topology has made, and maybe cannot even be drawn without using the resources of the Theory of Linear Structures. So if space-time is discrete (which it may be) and if time is intrinsically directional (which I think is obvious!) then the analytical tools available are better than standard topology and hence manifolds.

        Cheers,

        Tim Maudlin

        the central point here is :(Q) what properties mus physical reality have for math to be applicable? As the author knows the brings in some traditional problems of scientific realism. In the simple cases treated first we know reality and then apply integers or geometric forms. In the difficult cases we know physical reality through the way successful physical theories represent reality. Here Q i still applicable but more difficult to answer. You eventually sharpen Q to: What physical features space or space-time must have to be represented by the topology of an open set? This leads into your development of a theory of linear structures. I don't feel qualified to comment on that, though it looks good. I tend t think of open sets as a mathematical trick with no direct physical significance. Maybe the development of your theory will chang that.

        Ed MacKinnon

        Dear Tim Maudlin,

        You have mentioned that for mathematics to be used as the language of physics, physical world has that sort of structure to be represented mathematically? That depends on the mathematical language being used Physical characteristics are required for mathematical structures to describe a physical situation.

        Yes I agree with you and thats why I have propounded Mathematical Structure Hypothesis to explain their origin in the same line.

        Question is - Who decides the symphonic structure of that language? For any mathematical structure to be compatible to explain the physical structure, we need to match their intrinsic "laws of invariance" otherwise their applications would be wrong.

        This is why in context of Skolem's paradox: "A particular model fails to accurately capture every feature of the reality of which it is a model. A mathematical model of a physical theory, for instance, may contain only real numbers and sets of real numbers, even though the theory itself concerns, say, subatomic particles and regions of space-time. Similarly, a tabletop model of the solar system will get some things right about the solar system while getting other things quite wrong."

        You have classified the mathematical structures into two categories based on Wigner's essay

        1) One which are naturally suited to physical world e.g. Integers and what does their suitability imply about the physical world?

        2) Others which are not e.g. advanced concepts e.g. complex numbers should have use in physics.

        I have explained on the basis of Mathematical Structure Hypothesis that whether it falls in any category, its basically physical characteristics behind the development mathematical language which describes the physical characteristics of the physical world i.e. whether Integer or Complex numbers.

        Wigner talks about Complex numbers as advanced concept but what decides the structure of complex numbers and why they are so effective in Quantum Mechanics.

        Eugene Merzbacher in his book on QM has explained by deriving that for certain physical characteristic to be satisfied( for quantum waves,any displacement in the space & time dimension should not alter the physical characteristics of waves) and to satisfy these criteria, the mathematical parameters turns out to be "i"(complex number).

        Here is the reason the structure of mathematical language has been matched/molded to suit the physical characteristic of quantum waves(physical world).Infact, its not the mathematics describing physics here rather their corresponding law of invariance. So, what is the law of invariance behind complex number. Its answer lies in the definition of why negative multiplied by negative turns out to be positive? Why not positive multiplied by positive also become negative? Here is hidden laws of physics behind the definition of mathematical operators structure and vice versa.

        This is because mathematical structures abstractness and physical reality both are creations of the same thing Vibration, which my Mathematical Structure Hypothesis has propounded.

        Anyway, your essay is indeed great.

        Thanks & Regards,

        Pankaj Mani

          Dear Pankaj Mani,

          Thank you for the comments. The use of complex numbers in quantum theory is a very interesting case, which needs a lot of discussion. My own work here is just on space-time structure, so does not touch on quantum theory directly. But I think it may help to recall that time-revesal is implemented in quantum formalism by taking the complex conjugate of the wave function. This immediately suggests a connection between the use of complex numbers and the temporal structure, indeed a connection with the direction of time.

          It is harder to deal with quantum theory because there is no agreement at all about just what physical entities the theory is committed to, particularly what Bell called the "local beables" of the theory. The observable behavior of laboratory apparatus should be determined by the behavior of these local beables at microscopic scale. If you don't even know what these are, then interpreting the significance of the mathematical apparatus becomes essentially impossible. Pure space-time theory is a bit more straightforward.

          Regards,

          Tim Maudlin

          Dear Tim,

          Concerning you criticisms in my Essay page, some clarifications could be needed. For rotating frame in my Essay I mean the frame in which the observer sees the detector at rest (the absorber orbits around the source). Clearly, in that frame photons propagate in the radial direction. You are of course correct in highlighting that Equivalence Principle has local behavior. On the other hand, rotating frames generate the centrifuge acceleration in the radial direction cited above, which, in turn, defines a locally accelerated frame. Thus, it seems to me that the application of Equivalence Principle is completely legitimate. I also stress that the use of the Equivalence Principle in rotating frames in general and in the Mössbauer rotor experiment in particular has a long, more than fifty-year-old, history. In the paper of Kündig, i.e. ref. [3] in my Essay, which is dated 1963, one reads verbatim: "when the experiment is analyzed in a reference frame K attached to the accelerate observer, the problem could be treated [7] by the principle of equivalence of the general theory of relativity". Reference [7] in the paper of Kündig is the historical book of Pauli on the theory of relativity dated 1958. Thus, it seems that you were wrong in those criticisms. Here the key point is not the viability of the Equivalence Principle in treating this problem, but the issue that previous literature did not take into due account clock synchronization.

          I will read, comment and score your Essay soon. I wish you best luck in the Contest.

          Cheers, Ch.

          Dear Tim,

          I was reading your excellent essay with growing interest as you were getting closer and closer to the geometry of spacetime. You argue that standard geometry has been built on the wrong conceptual foundation to apply optimally to spacetime and you propose an alternative geometrical language. That is really promising as I am just looking for such languages. You finally claim that the Theory of Linear Structures (TLS) is capable of describing the geometry of continua [...]. Is TLS designed exclusively for a specific spacetime description or is it possible to describe e.g. Thurston geometries (the geometrization conjecture, proved by Perelman)? This is double-dealing question because the Thurston geometries, in my opinion, we can treat as a space-like, totally geodesic submanifolds of a 3+1 dimensional spacetime. In three dimensions, it is not always possible to assign a single geometry to a whole space. Instead, the conjecture states that every closed 3-manifold can be decomposed into pieces that each have one of eight types of geometric structure, resulting in an emergence of some attributes that we can observe. 3+1, in turn, means that the constant curvature geometries (S3, H3, E3) can arise as steady states of the Ricci flow, the other five homogeneous geometries arise naturally where the dynamics of the Ricci flow is more complicated and where topological changes (neck pinching or surgery) happen. That way the space flows with time and becomes a dynamical medium - spacetime. Then if TLS can really suit, what is the profit from that approach in comparison to the standard?

          I agree that "Physicists seeking such a mesh between mathematics and physics can only alter one side of the equation." However if we use a proper correspondence rule with the empirical domain than it is really sufficient to discover the set of geometric structures isomorphic to physical reality. Details in my essay.

          I would appreciate your comments.

          Best regards,

          Jacek

            Dear Jacek,

            As I mentioned in the appendix, there are topologies that cannot be recovered from any underlying Directed Linear Structure, but I am morally certain that these have no bearing at all on Thurston's conjecture, which deals with manifolds. I see no reason at all to doubt that every one of the manifolds that the conjecture deals with is generated by at least one (and in fact infinitely many) distinct Directed Linear Structures. (Indeed, by infinitely many Linear Structures, with no direction). And the way to implement the analog of surgery theory in the Theory of Linear Structures is completely straightforward: one specifies how two Linear Structures are to be combined by specifying which lines in one are continuations of lines in the other.

            As for the profit: the Theory of Linear Structures describes geometrical structure in finer detail than standard topology: that is why many distinct Linear Structures typically can generate the same standard topology.

            I'm not sure what you mean when you say that one cannot always assign a geometry to the whole space. I think you must be using that term in a non-obvious way. As you say, the theorem is about decomposing closed 3-manifolds into a set of pieces, each with a specified geometry. But there is obviously a geometry (in the sense of a topological and differentiable structure) assigned to all of these objects: that's why they count as 3-manifolds.

            Regards,

            Tim

            Thank you Tim for inspiration and clarifying as the TLS is new for me. I do not feel it yet and I have to catch up.

            When I say that one cannot always assign a (single) geometry to the whole space I mean this is not possible in three dimensions. This is specific feature of three dimensions pointed out in the geometrization conjecture. Differently, for two-dimensional surfaces you can freely assign a single geometry to a whole space. It was really not clearly stated.

            Best regards,

            Jacek