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Proper definition of world lines in Galilean and Minkowskian spacetime 
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#1
Mar112, 08:26 AM

P: 71

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 nondegenerate instead of the usual positivedefinite. 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
Mar112, 11:15 AM

Sci Advisor
P: 2,953

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 reparametrizations 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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