Proper definition of world lines in Galilean and Minkowskian spacetime

In summary, the conversation discusses the definitions of world lines in both Galilean and Minkowskian spacetime. In Galilean space, world lines are defined as curves in Euclidean space with an injective map, while in Minkowskian spacetime they are defined as differentiable curves that are timelike. The conversation also mentions the importance of choosing the correct time function in each case.
  • #1
Wox
70
0
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.
 
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  • #2
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.
 

1. What is the difference between Galilean and Minkowskian spacetime?

Galilean and Minkowskian spacetime are two different mathematical frameworks used to describe the motion of objects in the universe. Galilean spacetime is based on the laws of classical mechanics and describes the motion of objects in a non-relativistic manner. Minkowskian spacetime, on the other hand, is based on Einstein's theory of special relativity and takes into account the effects of time dilation and length contraction at high speeds.

2. How are world lines defined in Galilean spacetime?

In Galilean spacetime, world lines are defined as the path traced by an object in space and time. It is a straight line that represents the motion of an object at a constant velocity, as there is no time dilation or length contraction in this framework.

3. How are world lines defined in Minkowskian spacetime?

In Minkowskian spacetime, world lines are defined as the path traced by an object in space and time, taking into account the effects of special relativity. These world lines are curved, representing the changing velocity and direction of an object as it moves through spacetime.

4. Can world lines cross in Galilean spacetime?

Yes, in Galilean spacetime, world lines can cross as there is no restriction on the maximum speed at which objects can travel. This means that two objects can occupy the same space at the same time, resulting in their world lines crossing.

5. Can world lines cross in Minkowskian spacetime?

No, in Minkowskian spacetime, world lines cannot cross due to the effects of special relativity. As the speed of an object approaches the speed of light, time dilation and length contraction become significant, making it impossible for two objects to occupy the same space at the same time.

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