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SR and differential geometry

  1. May 11, 2010 #1

    Fredrik

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    When I say "SR" in this post, I mean the set of classical and quantum theories of particles and fields in Minkowski spacetime.

    I'm trying to come up with a list of topics in SR that can be dealt with in a better way when we have defined Minkowski spacetime as a manifold instead of as a vector space (or an affine space). Maybe there aren't that many? The ones I can think of right away are

    • Born rigidity (I've seen a definition that uses Lie derivatives, and I haven't seen one that would work with a spacetime that doesn't have a manifold structure)
    • A classification of types of fields we might be interested in. (Sections of various vector bundles over Minkowski spacetime).
    • The solid rotating disc, if we need to analyze it much more deeply than anyone wants to (except one person I know :wink:)
    • Definitions of measurable quantities that we'd prefer to be explicitly coordinate independent (i.e. proper time).
    • A coordinate independent definition of geodesics and inertial motion.
    That's pretty much it. Maybe the Lagrangian/Hamiltonian stuff is more natural in this context too? Let me know if you can think of other stuff that you think is easier to explain or can be treated in a more general or more elegant way in the "manifold version" of the theory than in the "vector space version". Also let me know if you think the stuff I've mentioned can be handled just as well in a vector space.
     
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  3. May 13, 2010 #2
    I would add to Your list yet
    geometry in non-inertial frame of reference.

    Very curious. Whom do you mean? Maybe You give a link?
     
  4. May 13, 2010 #3

    dx

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    I'm not sure exactly what you mean by "manifold version" and "vector space version" of SR. Would the following argument about the rotating disc be part of the manifold version?

    If we work in a Minkowski reference frame, then the points of the disc have worldlines parametrized by circular coordinates r and θ:

    [tex] \gamma (r, \theta) : \mathbb{R} \rightarrow \mathbb{R}^3 [/tex]

    with

    [tex] t \rightarrow( r \cos (\theta + \omega t), r \sin (\theta + \omega t),t) [/tex]

    The problem, presumably, is whether there is some kind of curvature here. There is no curvature of spacetime of course, but if we define the distance between two nearby worldlines by their distance in an instantaneous rest frame, then the metric on the (r,θ)-space of worldlines is

    [tex] \mathcal{D} = dr \otimes dr + \frac{r^2}{1 - r^2 \omega^2} d\theta \otimes d\theta [/tex]
     
    Last edited: May 13, 2010
  5. May 13, 2010 #4

    Fredrik

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    The way I see it, SR is defined by axioms like "A clock measures the proper time of the curve in spacetime that represents its motion". We can choose to define "spacetime" as a vector space, as an affine space, or as a smooth manifold...or as some combination, like an affine space that also has manifold structure. Since different choices of what mathematical structure to use give us different axioms, and since I consider a theory to be defined by its axioms, I would say that each choice defines a different theory. They're obviously equivalent in the sense that they make the same predictions about the results of experiments, but they're still different theories, or at least different versions of "the" theory.

    Yes, because in the vector space version of the theory, there's no metric tensor field, only a bilinear form on Minkowski spacetime, and we don't even define the tangent spaces at different points, so there's no way to define the components of the bilinear form in a coordinate system that can't be constructed from a basis of the "Minkowski vector space" in the obvious way.

    It's good that you made me think about this, because now I understand Omega's comment much better. If we define spacetime as a vector space with a bilinear form instead of as a manifold with a metric, it really limits our ability to work with arbitrary coordinate systems (i.e. arbitrary functions from spacetime to [tex]\mathbb R^4[/tex]).

    So I've been told. :smile: What I haven't been able to figure out is why this is interesting. We clearly don't need to define a quotient manifold and study its properties if we're only trying to see e.g. that the material will stretch when we give the disc a spin. So are we doing this just because it's cool, or is the quotient manifold important in some other way? If you or someone else could enlighten me about that (without forcing me to learn everything about the quotient manifold first), I'd appreciate it.
     
    Last edited: May 13, 2010
  6. May 13, 2010 #5

    atyy

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    But even in the "vector space" conception, wouldn't one need to make use of a 4-velocity at some point, which surely brings in the tangent space immediately - so can one really do SR without the idea of a manifold with metric?
     
  7. May 13, 2010 #6

    Hurkyl

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    It's sort of an awkward question; AFAIK pretty much all of the tools of differential geometry were developed for real vector spaces first.

    e.g. the tangent bundle to V is nothing more than VxV; a section is (the graph of) a continuous function V -> V.
     
    Last edited: May 13, 2010
  8. May 13, 2010 #7

    Fredrik

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    A world line is a function [itex]x:[a,b]\rightarrow M[/itex] from an interval of the real numbers into spacetime. If we have given spacetime a vector space structure, we can make sense of

    [tex]x'(t)=\lim_{s\rightarrow 0}\frac{x(t+s)-x(t)}{s}[/tex]

    so we don't need to define tangent vectors the hard way (as derivative operators or as equivalence classes of curves). We don't even have to define a tangent space. The four-velocity is just defined as the derivative above, with the appropriate normalization.
     
  9. May 14, 2010 #8

    dx

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    I don't think we have a choice in definition of SR this way. All the structures are necessary and are assumed at a basic level. No matter how you choose to axiomatize the theory, I think the following must always be a basic assumption: "There exist mappings of events into the real 4-dimensional topological vector/affine space and smooth manifold R4"


    The necessity of such an assumption is shown in the way it is used in defining the basic elements of the theory. For example, the notion of an 'inertial frame' is essentially the following: a mapping from events into Minkowski space which sends 'free particles' to straight lines, and such that 'clock rates' respect the affine structure of Minkowsi space. So the next postulate would have to be that such mappings exist, i.e. that inertial frames exist. Starting from this foundation, we can, in the manner of mathematicians, prove the following theorem:

    Theorem. Let φ and φ' be inertial frames. Then φ'⋅φ-1 is an affine transformation, i.e. the transformations between inertial frames are affine transformations.

    Once we add the conformal structure derived from light cones, the class of symmetries reduces from general affine to Poincare.

    This statement is, of course, not a mathematical one. The domains of the mappings φ and φ' are not defined mathematically. 'Events' are assumed to be something given in experience, and the idea can only be communicated by demonstration, not definition. Basically, our language is always a little part of our theory, and this statement must be interpreted as a linguistic contruct, which describes the situation.


    By 'quotient manifold', I assume you're talking about the manifold of worldlines parametrized by r and θ. The quotient manifold is important because it is the space which is curved. D = dr² + r²dθ²/(1 - r²ω²) is the induced geometry of a class of spatial hyperslices Dt (t in R), i.e. the 3-dimensional intersections of the world-tube of the disc with the set of space-slices of an inertial frame. Surely, this is the central object of discussion in the problem of the rotating disc in special relativity?
     
    Last edited: May 14, 2010
  10. May 14, 2010 #9

    Fredrik

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    I agree that I can't just say that it's "a vector space". Without a topology, we can't make sense of limits, continuity and derivatives like the one in my previous post. But how about this? We define "spacetime" as the set [tex]\mathbb R^4[/tex] with the standard topology, the standard vector space structure, and the bilinear form g defined by [itex]g(x,y)=x^T\eta y[/itex]. We then define coordinate systems as smooth bijections from [tex]\mathbb R^4[/tex] into itself. (The topological vector space structure is sufficient to make sense of "smooth"). Some of the coordinate systems are associated with bases of the vector space in the following way: For each basis [itex]B=\{e_\mu\}[/itex], we define a coordinate system [itex]f_B[/itex] by [itex]f_B(x)=(x_0,x_1,x_2,x_3)[/itex], where the [itex]x_\mu[/itex] are defined by [itex]x=x_\mu e_\mu[/itex]. Since I'm not going to be talking about tensors, I'm putting all indices downstairs. All inertial frames belong to this class. Technically it's the inverses of my [itex]f_B[/itex] functions that should be called "frames", but I'll stick with the standard abuse of terminology and refer to the "nice" coordinate systems as "inertial frames".

    With the definitions I made above, we can define inertial frames mathematically, as coordinate systems that take straight lines to straight lines. (The concept of straight lines is already well-defined by the vector space structure). We can then prove as a theorem that the set of functions of the form [itex]\phi'\circ\phi^{-1}[/itex], where [itex]\phi,\phi'[/itex] are inertial frames, is a group with "multiplication" defined as composition of functions. The identity element is the identity map. We call this group the Poincaré group, and its members Poincaré transformations.

    There are still some issues that must be dealt with when we write down our complete set of axioms for the theory. (The axioms are statements that identify things in the real world with things in the mathematical model. The complete list of axioms is what defines the theory). For example, we seem to need an axiom that identifies zero acceleration in the real world with straight lines in the theory, so now we have to think about how to define "accelerometer" operationally. The language used in the axioms may depend slightly on what mathematical structure we have chosen (topological vector space or manifold), but neither of the choices seems to make any of the issues significantly harder to deal with than the other.

    This confused me at first, but I think I know what you mean now. When you talk about spacetime, events and the domains of those mappings, you're talking about things in the real world, right? That's not how I think about these things. I prefer to have a clear separation between the real world and the mathematics, e.g. no "functions" that take events in the real world to points in [itex]\mathbb R^4[/itex]. I agree that things in the real world can only be "defined" by descriptions in plain English or whatever language you prefer, but when I talk about spacetime, events, coordinate systems, etc, I'm always referring to mathematical concepts.

    Yes, that's what I meant.

    The thing is, I don't see how any of that is relevant to anything. I have to admit that I have never studied these aspects of the rotating disc problem in detail, but the reason is that I have never been able to find a reason why I should. This manifold seems completely irrelevant to me, and I'm wondering if it's only used by people who just think the math is cool and people who incorrectly think that we need it to solve problems that we can solve without it.
     
    Last edited: May 14, 2010
  11. May 14, 2010 #10
    IMHO space is flat. While I do not want to say more. I think Demistifier support.
     
  12. May 14, 2010 #11

    Fredrik

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    If we define "space" as a hypersurface of constant time coordinate in the rotating coordinate system, then you're right, but that's not the manifold we're talking about. We're talking about defining a manifold structure on the set of world lines (of points in the disc), and that manifold is curved. Note that we're not talking about a submanifold of spacetime.
     
  13. May 15, 2010 #12

    atyy

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    So in the manifold version, the metric acts on tangent vectors and the spacetime interval is defined by integrating over a straight worldline. In the vector space version, the spacetime interval is obtained by the scalar product acting on position vectors directly, no need to go to tangent vectors and integration. Is that consistent with how you are defining the two versions?
     
  14. May 15, 2010 #13

    Fredrik

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    Except for the "straight worldline" part, yes. We have to define the proper time of an arbitrary timelike curve and the proper length of an arbitrary spacelike curve in both versions. The definitions of proper time are explained here. See #8 for the topological vector space version and #4 for the manifold version.

    The main limitation of the topological vector space version is that we can't define the components of the metric in a coordinate system that isn't associated with a basis for the vector space as described in my reply to dx above.
     
  15. May 15, 2010 #14
    I am surprised that You agreed. :rolleyes: Usually begin to argue.
    Yes. The standard definition of simultaneity leads to a curvature of space.
    Geometry IMHO is a conditional concept.
     
  16. May 15, 2010 #15

    dx

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    I don't think we can avoid using a phrase like "one can find a map of events into the linear space R4 such that the world-lines of free particles are straight lines", even though 'event' and 'free particle' are not defined in the theory. This is because there is real physical content in this: imagine a picture of two particles coming in; they first come into contact at point A, and then again at point B, and then continue without interacting. Using the above postuate we can conclude that these particles cannot be free particles. This is because there is no continuous map from this set of events into R4 that will straighten out both of the trajectories, as is required by the postulate above. We need similar postulates about clocks and light.


    Here's how I would define "manifold SR": First, we start with the mathematical structure, the real linear space R4. Then we represent 'free particles', 'events', 'clocks', 'light', and postulate the existance of maps such that free particles are straight lines etc. We introduce the idea of inertial frame as a map of events into R4 that respects all the postulates about clocks etc. above. Now the set of maps has been greatly reduced. In fact, it restricts it such that the transformation between any two such reference frames is a linear transformation. If we further restrict the maps by taking into account the 'constancy of the speed of light' in the definition of an inertial frame, i.e. in precise language, the conformal structure of spacetime encoded in the properties of light (from postulates), then we are left with exactly what is required: Lorentz tranformations (of if we had been a little more general and used affine above, Poincare.) So although people usually say Einstein postulates are not well defined and so on, the thing about the constancy of the velocity of light if properly interpreted does contain special relativity. The clock behavior implied by the formula s² = t² - x² - y² - z² is the one which has the correct symmetry (Lorentz symmetry).

    This is all at a basic level. For practical purposes, I don't think we can avoid using tensors in their modern guise. Inertial frameins, in practice are presented as follows: We have four functions t, x, y, z : R4 → R. From these we can contruct the objects ∂t, ∂x, ∂y, ∂z, dt, dx, dy, dz (at each point of R4). Using these objects, we can represent the clock behavior in the Minkowski metric tensor field

    [tex] N = \eta_{\mu \nu} dx^{\mu} \otimes dx^{\nu}[/tex]

    I don't think its necessary to solve anything; in fact, I don't really consider this issue about the geometry of the disc relevant or interesting (unless we discuss it in the context of general relativity.) The metric on this quotient space was defined in a specific way, i.e. by constructing it locally in an intantaneous rest frame moving with the worl-lines at that point. This has nothing to do with any curvature of the rigid disc as it appears in space. The assumption that it is rigidly rotating is essentially equivalent to saying that it appears as a rigidly and uniformly rotating disc in your reference frame. This notion of rigidity is not in any way in harmony with SR, I think. Maybe there would be some interesting features if we try to relate the definition of the metric to born rigidity.
     
    Last edited: May 15, 2010
  17. May 16, 2010 #16
    Doesn't Weinberg have a book on gravitation where he tries to de-emphasize the geometric view?
    I think he says something like "I believe the geometric view has driven a wedge between GR and the theory of elementary particles."

    I don't know how relevant that is with what you're saying, but I remember reading that in the preface.
    (However....when looking through the book, I couldn't find much difference in pedagogy in Weinberg's book over any other GR book. The only thing I remember finding somewhat different is that he had a few chapters on different topics of GR, including discussion of g^uv, affine connections, and gravitational forces in GR before his chapter on tensor analysis)
     
  18. May 16, 2010 #17
    Why do you think it is curved?

    It is true that a rotating disk is not Born rigid and its clocks are not synchronized and neither is there a single rest frame that shows all circumnavigated points on the disk to come back to the same spatial location.

    But all that does not imply the spacetime is curved.
     
    Last edited: May 16, 2010
  19. May 16, 2010 #18

    Fredrik

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    We're probably thinking along pretty similar lines, but I don't like your terminology. A "map" is a mathematical concept, with a very precise definition, and it hurts my eyes to see it used this way. I would define "spacetime" as a specific mathematical structure (either a topological vector space or a smooth manifold). I would define proper time as a mathematical property of a timelike curve in spacetime. I would also define "inertial frame" and "Poincaré group" mathematically. I would define "events", "clocks", and so on, operationally, i.e. by descriptions in plain English. Then I can write down the axioms of the theory. Maybe the term "axiom" hurts someone else's eyes, but I haven't thought of a better one. These "axioms" tell us what pieces of the mathematics that correspond to the operationally defined things in the real world. They would look roughly like this: (I'm quoting myself from one of the rotating disc threads).
    I don't see any reason to think that it matters if I choose to define spacetime as a topological vector space or as a manifold. Different choices require slightly different definitions of proper time, but the axioms would be the same except that I should probably replace "geodesic" with "straight line" in the topological vector space version. And these two theories, or two versions of the same theory if that sounds better, still make the same predictions about results of experiments.

    I probably have to add one more axiom though, one that identifies straight lines with non-accelerating motion. This would require an operational definition of "accelerometer", but then we can just say that the motion of an accelerometer that reads zero is represented by a timelike straight line in spacetime (if we chose to define spacetime as a topological vector space) or by a timelike geodesic in spacetime (if we chose to define spacetime as a manifold).

    We don't need anything more about inertial frames, invariance of the speed of light, and so on, because all of that stuff is already included in the definition of inertial frame, and we just needed this last axiom to relate it to reality.


    We can define tensors in the topological vector space version too. A tensor would be a multilinear map from M*×...×M*×M×...×M into the real numbers, and a tensor field would be a tensor-valued function. So even if we need tensors, it doesn't imply that we need manifolds.

    I think this is an occasion where differential geometry is nice. Without it, we can prove "locally" (i.e. by considering an infinitesimally small region of the disc) that we can't give the disc a spin without stretching the material, but I think "global" proofs are hard. (Is there even a global definition of Born rigidity that doesn't use differential geometry?). If we use techniques from differential geometry, and a differential geometry definition of Born rigidity, a "global" proof is definitely possible. But we don't need the quotient manifold for that (as far as I know), and I'm thinking that if we don't even need it for that, then what do we need it for? I suspect that we don't need it for anything.
     
  20. May 16, 2010 #19

    Fredrik

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    Because people have told me so (and because the circumference of the disc in this manifold isn't 2*pi*r). I haven't even bothered to find out how the metric of spacetime induces a metric on this manifold, because I haven't found a single reason to think this manifold is significant.

    Please read what I said again. I wasn't talking about spacetime or about a submanifold of spacetime.
     
  21. May 16, 2010 #20
    That is related to the difficulty for a single rest frame to interpret what a complete circumnavigation for all outer points of the disk actually means.
     
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