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Affine spaces and time-varying vector fields

  1. Feb 3, 2012 #1

    Wox

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    Consider the following affine space [itex]\mathbb{G}[/itex]
    1. a four-dimensional vector space [itex]G_{v}^{4}[/itex] over field [itex]\mathbb{R}[/itex] which acts (sharply transitive) on a set [itex]G_{p}^{4}[/itex]
    2. a surjective linear functional from [itex]G_{v}^{4}[/itex] to its field, which kernel is isomorphic with three-dimensional Euclidean vector space
    [tex]t_{L}\colon G_{v}^{4}\to \mathbb{R}\colon\text{Ker}(t_{L})\cong E_{v}^{3}[/tex]
    (I will treat Euclidean space [itex]\mathbb{E}^{n}[/itex] in the strict sense: an affine space where inner product space [itex]E_{v}^{n}[/itex] acts on point space [itex]E_{p}^{n}[/itex])


    This affine space is known as Galilean space-time as defined for example here and here.

    This definition induces two metrics in Galilean point space [itex]G_{p}^{4}[/itex]. The first is called the time difference:
    [tex]d_{t}\colon G_{p}^{4}\times G_{p}^{4}\to \mathbb{R}\colon (p,q)\mapsto d_{t}(p,q)=t_{L}(q-p)[/tex]
    The following equivalence relation is derived from this metric: the points in [itex]G_{p}^{4}[/itex] (called events) are "simultaneous" when their time difference is zero. This partitions [itex]G_{p}^{4}[/itex] in equivalence classes
    [tex]\text{Cl}_{\text{sim}}(p)=\{q\in G_{p}^{4}\colon d_{t}(p,q)=0\}[/tex]
    The difference vectors in a class are given by [itex]\text{Ker}(t_{L})[/itex] so a second metric is induces in each equivalence class, which is called the distance between simultaneous events
    [tex]d_{e}\colon \text{Cl}_{\text{sim}}(p)\times \text{Cl}_{\text{sim}}(p)\to \mathbb{R}\colon (p_{1},p_{2})\mapsto d_{e}(p_{1},p_{2})=d(p_{1},p_{2})=||p_{2}-p_{1}||[/tex]
    where "d" the Euclidean distance in Euclidean point space, which is itself induced by the norm in Euclidean vector space [itex]E_{v}^{3}[/itex]. So Galilean point space [itex]G_{p}^{4}[/itex] is actually a union of classes which are all isomorphic with Euclidean point space [itex]E_{p}^{3}[/itex].

    In this rather basic space-time I want to start working with scalar and vector fields, which are defined as the functions
    [tex]f\colon\mathbb{R}^{3}\to\mathbb{R} \colon\vec{x}\mapsto f(\vec{x})[/tex]
    [tex]\vec{F}\colon\mathbb{R}^{3}\to\mathbb{R}^{3}\colon\vec{x}\mapsto\vec{F}(\vec{x})[/tex]
    where [itex]\mathbb{R}^{3}[/itex] Euclidean coordinate space isomorphic with [itex]E_{v}^{3}[/itex].

    Now my question: how do I connect in a strictly mathematical sense the concept of Galilean space-time with scalar and vector fields (which will be time-varying)?

    I tried concidering Galilean coordinate space [itex]\mathbb{R}^{4}[/itex] which is isomorphic with Galilean vector space [itex]G_{v}^{4}[/itex] so that
    [tex]t_{L}\colon \mathbb{R}^{4}\to \mathbb{R}\colon\text{Ker}(t_{L})\cong \mathbb{R}^{3}[/tex]
    so I can define a scalar or vector field on the kernel of [itex]t_{L}[/itex] but I don't know how to formalize the time-varying aspect of the scalar or vector field. Any ideas?
     
  2. jcsd
  3. Feb 3, 2012 #2

    mathwonk

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    i dont really follow your drift too well, but a time dependent vector field on R^n is just a map RxR^n--->R^n.

    i.e. you just put in an extra variable to be time. will that work for you?
     
  4. Feb 6, 2012 #3

    Wox

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    That works on a practical level, but my question is more theoretical: how do I bring this practical/intuitive idea of time varying vector fields in the mathematical description of space-time. The purpose of my drift was to indicate on which level I wanted to "understand the connection".

    Of course Galilean space-time is obsolete in a way, but it is a valid construction to describe classical laws.
     
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