Wox
Feb3-12, 10:59 AM
Consider the following affine space \mathbb{G}
1. a four-dimensional vector space G_{v}^{4} over field \mathbb{R} which acts (sharply transitive) on a set G_{p}^{4}
2. a surjective linear functional from G_{v}^{4} to its field, which kernel is isomorphic with three-dimensional Euclidean vector space
t_{L}\colon G_{v}^{4}\to \mathbb{R}\colon\text{Ker}(t_{L})\cong E_{v}^{3}
(I will treat Euclidean space \mathbb{E}^{n} in the strict sense: an affine space where inner product space E_{v}^{n} acts on point space E_{p}^{n})
This affine space is known as Galilean space-time as defined for example here (http://www.math.uni-hamburg.de/home/schweigert/ws09/pskript.pdf) and here (http://www.mast.queensu.ca/~andrew/teaching/math439/pdf/chapter1.pdf).
This definition induces two metrics in Galilean point space G_{p}^{4}. The first is called the time difference:
d_{t}\colon G_{p}^{4}\times G_{p}^{4}\to \mathbb{R}\colon (p,q)\mapsto d_{t}(p,q)=t_{L}(q-p)
The following equivalence relation is derived from this metric: the points in G_{p}^{4} (called events) are "simultaneous" when their time difference is zero. This partitions G_{p}^{4} in equivalence classes
\text{Cl}_{\text{sim}}(p)=\{q\in G_{p}^{4}\colon d_{t}(p,q)=0\}
The difference vectors in a class are given by \text{Ker}(t_{L}) so a second metric is induces in each equivalence class, which is called the distance between simultaneous events
d_{e}\colon \text{Cl}_{\text{sim}}(p)\times \text{Cl}_{\text{sim}}(p)\to \mathbb{R}\colon (p_{1},p_{2})\mapsto d_{e}(p_{1},p_{2})=d(p_{1},p_{2})=||p_{2}-p_{1}||
where "d" the Euclidean distance in Euclidean point space, which is itself induced by the norm in Euclidean vector space E_{v}^{3}. So Galilean point space G_{p}^{4} is actually a union of classes which are all isomorphic with Euclidean point space E_{p}^{3}.
In this rather basic space-time I want to start working with scalar and vector fields, which are defined as the functions
f\colon\mathbb{R}^{3}\to\mathbb{R} \colon\vec{x}\mapsto f(\vec{x})
\vec{F}\colon\mathbb{R}^{3}\to\mathbb{R}^{3}\colon \vec{x}\mapsto\vec{F}(\vec{x})
where \mathbb{R}^{3} Euclidean coordinate space isomorphic with E_{v}^{3}.
Now my question: how do I connect in a strictly mathematical sense the concept of Galilean space-time with scalar and vector fields (which will be time-varying)?
I tried concidering Galilean coordinate space \mathbb{R}^{4} which is isomorphic with Galilean vector space G_{v}^{4} so that
t_{L}\colon \mathbb{R}^{4}\to \mathbb{R}\colon\text{Ker}(t_{L})\cong \mathbb{R}^{3}
so I can define a scalar or vector field on the kernel of t_{L} but I don't know how to formalize the time-varying aspect of the scalar or vector field. Any ideas?
1. a four-dimensional vector space G_{v}^{4} over field \mathbb{R} which acts (sharply transitive) on a set G_{p}^{4}
2. a surjective linear functional from G_{v}^{4} to its field, which kernel is isomorphic with three-dimensional Euclidean vector space
t_{L}\colon G_{v}^{4}\to \mathbb{R}\colon\text{Ker}(t_{L})\cong E_{v}^{3}
(I will treat Euclidean space \mathbb{E}^{n} in the strict sense: an affine space where inner product space E_{v}^{n} acts on point space E_{p}^{n})
This affine space is known as Galilean space-time as defined for example here (http://www.math.uni-hamburg.de/home/schweigert/ws09/pskript.pdf) and here (http://www.mast.queensu.ca/~andrew/teaching/math439/pdf/chapter1.pdf).
This definition induces two metrics in Galilean point space G_{p}^{4}. The first is called the time difference:
d_{t}\colon G_{p}^{4}\times G_{p}^{4}\to \mathbb{R}\colon (p,q)\mapsto d_{t}(p,q)=t_{L}(q-p)
The following equivalence relation is derived from this metric: the points in G_{p}^{4} (called events) are "simultaneous" when their time difference is zero. This partitions G_{p}^{4} in equivalence classes
\text{Cl}_{\text{sim}}(p)=\{q\in G_{p}^{4}\colon d_{t}(p,q)=0\}
The difference vectors in a class are given by \text{Ker}(t_{L}) so a second metric is induces in each equivalence class, which is called the distance between simultaneous events
d_{e}\colon \text{Cl}_{\text{sim}}(p)\times \text{Cl}_{\text{sim}}(p)\to \mathbb{R}\colon (p_{1},p_{2})\mapsto d_{e}(p_{1},p_{2})=d(p_{1},p_{2})=||p_{2}-p_{1}||
where "d" the Euclidean distance in Euclidean point space, which is itself induced by the norm in Euclidean vector space E_{v}^{3}. So Galilean point space G_{p}^{4} is actually a union of classes which are all isomorphic with Euclidean point space E_{p}^{3}.
In this rather basic space-time I want to start working with scalar and vector fields, which are defined as the functions
f\colon\mathbb{R}^{3}\to\mathbb{R} \colon\vec{x}\mapsto f(\vec{x})
\vec{F}\colon\mathbb{R}^{3}\to\mathbb{R}^{3}\colon \vec{x}\mapsto\vec{F}(\vec{x})
where \mathbb{R}^{3} Euclidean coordinate space isomorphic with E_{v}^{3}.
Now my question: how do I connect in a strictly mathematical sense the concept of Galilean space-time with scalar and vector fields (which will be time-varying)?
I tried concidering Galilean coordinate space \mathbb{R}^{4} which is isomorphic with Galilean vector space G_{v}^{4} so that
t_{L}\colon \mathbb{R}^{4}\to \mathbb{R}\colon\text{Ker}(t_{L})\cong \mathbb{R}^{3}
so I can define a scalar or vector field on the kernel of t_{L} but I don't know how to formalize the time-varying aspect of the scalar or vector field. Any ideas?