- #1
Wox
- 68
- 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)
<br /> \bar{w}\colon \mathbb{R}\to \mathbb{R}^{4}\colon \tau \mapsto (t(\tau),\bar{x}(\tau))<br />
for which a curve in Euclidean space
<br /> \bar{x}\colon \mathbb{R}\to \mathbb{R}^{3}\colon \tau \mapsto \bar{x}<br />
and an injective map (because a world line shouldn't contain simultaneous events)
<br /> t\colon \mathbb{R}\to \mathbb{R}\colon \tau \mapsto t<br />
We used the fact that \mathbb{R}^{3} has the Euclidean structure and that a basis was choosen in \mathbb{R}^{4} so that all vectors (0,\bar{x}) form a subspace of Galinean space \mathbb{R}^{4} which is isomorphic with \mathbb{R}^{3} (i.e. Euclidean inner product defined on this subspace).
2. In Minkowskian spacetime with signature (-+++), world lines are defined as differentiable curves
<br /> \bar{w}\colon \mathbb{R}\to \mathbb{R}^{4}\colon \tau \mapsto (x^{0}(\tau),x^{1}(\tau),x^{2}(\tau),x^{3}(\tau))<br />
which are timelike (because a world line shouldn't contain simultaneous events) meaning that the velocity of the world line is a timelike vectors (\eta(\bar{w}',\bar{w}')<0) or in other words
<br /> (\frac{dx^{1}}{d\tau})^{2}+(\frac{dx^{2}}{d\tau})^{2}+(\frac{dx^{3}}{d\tau})^{2} <(\frac{dx^{0}}{d\tau})^{2}<br />
We used the fact that \mathbb{R}^{4} has an inner product \eta which is non-degenerate instead of the usual positive-definite.
3. It seems that we always choose t(\tau)=\tau (Galilean) and x^{0}(\tau)=c\tau (Minkowskian) but I'm not sure how these choices are justified.
1. In Galilean space, world lines are defined as curves (continuous maps)
<br /> \bar{w}\colon \mathbb{R}\to \mathbb{R}^{4}\colon \tau \mapsto (t(\tau),\bar{x}(\tau))<br />
for which a curve in Euclidean space
<br /> \bar{x}\colon \mathbb{R}\to \mathbb{R}^{3}\colon \tau \mapsto \bar{x}<br />
and an injective map (because a world line shouldn't contain simultaneous events)
<br /> t\colon \mathbb{R}\to \mathbb{R}\colon \tau \mapsto t<br />
We used the fact that \mathbb{R}^{3} has the Euclidean structure and that a basis was choosen in \mathbb{R}^{4} so that all vectors (0,\bar{x}) form a subspace of Galinean space \mathbb{R}^{4} which is isomorphic with \mathbb{R}^{3} (i.e. Euclidean inner product defined on this subspace).
2. In Minkowskian spacetime with signature (-+++), world lines are defined as differentiable curves
<br /> \bar{w}\colon \mathbb{R}\to \mathbb{R}^{4}\colon \tau \mapsto (x^{0}(\tau),x^{1}(\tau),x^{2}(\tau),x^{3}(\tau))<br />
which are timelike (because a world line shouldn't contain simultaneous events) meaning that the velocity of the world line is a timelike vectors (\eta(\bar{w}',\bar{w}')<0) or in other words
<br /> (\frac{dx^{1}}{d\tau})^{2}+(\frac{dx^{2}}{d\tau})^{2}+(\frac{dx^{3}}{d\tau})^{2} <(\frac{dx^{0}}{d\tau})^{2}<br />
We used the fact that \mathbb{R}^{4} has an inner product \eta which is non-degenerate instead of the usual positive-definite.
3. It seems that we always choose t(\tau)=\tau (Galilean) and x^{0}(\tau)=c\tau (Minkowskian) but I'm not sure how these choices are justified.