Register to reply

Proper definition of world lines in Galilean and Minkowskian spacetime

Share this thread:
Mar1-12, 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)
\bar{w}\colon \mathbb{R}\to \mathbb{R}^{4}\colon \tau \mapsto (t(\tau),\bar{x}(\tau))
for which a curve in Euclidean space
\bar{x}\colon \mathbb{R}\to \mathbb{R}^{3}\colon \tau \mapsto \bar{x}
and an injective map (because a world line shouldn't contain simultaneous events)
t\colon \mathbb{R}\to \mathbb{R}\colon \tau \mapsto t

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
\bar{w}\colon \mathbb{R}\to \mathbb{R}^{4}\colon \tau \mapsto (x^{0}(\tau),x^{1}(\tau),x^{2}(\tau),x^{3}(\tau))
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
(\frac{dx^{1}}{d\tau})^{2}+(\frac{dx^{2}}{d\tau})^{2}+(\frac{dx^{3}}{d\ tau})^{2} <(\frac{dx^{0}}{d\tau})^{2}

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.
Phys.Org News Partner Science news on
Scientists develop 'electronic nose' for rapid detection of C. diff infection
Why plants in the office make us more productive
Tesla Motors dealing as states play factory poker
Mar1-12, 11:15 AM
Sci Advisor
Matterwave's Avatar
P: 2,851
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.

Register to reply

Related Discussions
Classification of events and curves in Minkowskian spacetime Special & General Relativity 13
Proper Definition of a System Classical Physics 11
Definition of *straight* lines... Differential Geometry 7
Spacetime diagrams and proper time Cosmology 3