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.(adsbygoogle = window.adsbygoogle || []).push({});

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 aninjectivemap (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 asdifferentiablecurves

[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 aretimelike(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.

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# Proper definition of world lines in Galilean and Minkowskian spacetime

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