Proper definition of world lines in Galilean and Minkowskian spacetime

  1. I posted several questions on Galilean and Minkowskian spacetime on this forum lately, but I just don't seem to be able to get a real grip on things. I noticed that the core of my problems mostly arise from the definition of world lines. Therefore I tried formulating a definition of them in both spacetime's and my question is whether these definitions are correct/complete.

    1. In Galilean space, world lines are defined as curves (continuous maps)
    [tex]
    \bar{w}\colon \mathbb{R}\to \mathbb{R}^{4}\colon \tau \mapsto (t(\tau),\bar{x}(\tau))
    [/tex]
    for which a curve in Euclidean space
    [tex]
    \bar{x}\colon \mathbb{R}\to \mathbb{R}^{3}\colon \tau \mapsto \bar{x}
    [/tex]
    and an injective map (because a world line shouldn't contain simultaneous events)
    [tex]
    t\colon \mathbb{R}\to \mathbb{R}\colon \tau \mapsto t
    [/tex]

    We used the fact that [itex]\mathbb{R}^{3}[/itex] has the Euclidean structure and that a basis was choosen in [itex]\mathbb{R}^{4}[/itex] so that all vectors [itex](0,\bar{x})[/itex] form a subspace of Galinean space [itex]\mathbb{R}^{4}[/itex] which is isomorphic with [itex]\mathbb{R}^{3}[/itex] (i.e. Euclidean inner product defined on this subspace).

    2. In Minkowskian spacetime with signature (-+++), world lines are defined as differentiable curves
    [tex]
    \bar{w}\colon \mathbb{R}\to \mathbb{R}^{4}\colon \tau \mapsto (x^{0}(\tau),x^{1}(\tau),x^{2}(\tau),x^{3}(\tau))
    [/tex]
    which are timelike (because a world line shouldn't contain simultaneous events) meaning that the velocity of the world line is a timelike vectors ([itex]\eta(\bar{w}',\bar{w}')<0[/itex]) or in other words
    [tex]
    (\frac{dx^{1}}{d\tau})^{2}+(\frac{dx^{2}}{d\tau})^{2}+(\frac{dx^{3}}{d\tau})^{2} <(\frac{dx^{0}}{d\tau})^{2}
    [/tex]

    We used the fact that [itex]\mathbb{R}^{4}[/itex] has an inner product [itex]\eta[/itex] which is non-degenerate instead of the usual positive-definite.

    3. It seems that we always choose [itex]t(\tau)=\tau[/itex] (Galilean) and [itex]x^{0}(\tau)=c\tau[/itex] (Minkowskian) but I'm not sure how these choices are justified.
     
  2. jcsd
  3. Matterwave

    Matterwave 3,774
    Science Advisor
    Gold Member

    You have to be a little careful with the notation and logic. The tau you wrote in the Galilean case is some universal time function that every observer can measure with a good clock (up to affine re-parametrizations for origin and units). The tau you wrote in the Minkowski case is the proper time as measured by a clock carried by that specific observer. There is a major difference in logic there.

    Otherwise, I don't see any problem with what you've written down.
     
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