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| Detailed information |
| Original study plan |
Bachelor's programme Technical Mathematics 2018W |
| Subject |
It's one of these educationally hard-boiled assumptions that the 'space' of our spacial perceptions is a three dimensional euclidean vector-space and that this space in no way depends on any state of an 'observer'. Eventually we'll keep the first (though dimension isn't important to us) and drop the second. This 'space of spatial perception' in principle differes from the 'space' of classical Newtonian mechanics, i.e. the space objects move in. The former involves 'local data' attached to a local observer or detector, such as the angle formed by two rays of light, which cross exactly at the instantaneous observers position. The latter more or less incorporates all these positions, hence it's the space light lives in, it comprises in particular all these points light has passed from it's source to an observer or detector. The way we perceive an object, its colors and its shape are assumed to be independent from the path of the light, it purely depends on the energy and the tangent vectors of these paths at the observers position. Thus mathematically speaking the 'local space' is modeled as a tangent space of the 'space of all positions' at a particular position. But since we are used to identifying the space of classical Newtonian mechanics with a three dimensional euclidean space, we are inclined to mix up all these 'spaces': a vector at a point (which is a tangent vector at this point) gets identified with a vector at another point (which is a tangent vector at this other point) by what is called 'parallel translation' and by singling out a particular point (the zero vector) of the Newtonian space we may also identify this space with any of its tangent spaces. Well, that's all admissible and it works just fine for our historically shaped common sense knowledge, but it's an immeasurable drawback when it comes to more advanced models. It didn't need relativity for these models to come up, simple experience suggested to replace the classical Newtonian space with Riemannian manifolds (just think of motions under certain constraints such as rolling a ball on a surface). Unlike the Newtonian case tangent spaces couldn't be identified in an obvious way and parallel transport became a central notion in Riemannian geometry, not at all as easily accessible as in the Newtonian case. Also - and that's what we will be concerned with - the tangent spaces to these mainfolds don't carry a euclidean structure, they carry a Lorentz structure, which admits some subspaces to be euclidean. One of the essential features that makes relativity so much different to Newtonian mechanics is the fact that our space of perception is no more the space relevant in theoretical physics, its only part of this. Both the Lorentz and the euclidean structure are particular cases of what is called an inner product. This will be explored in the run of the first chapter. The second chapter is about Lorentz spaces and some crucial notions from relativity.It's one of these educationally hard-boiled assumptions that the 'space' of our spacial perceptions is a three dimensional euclidean vector-space and that this space in no way depends on any state of an 'observer'. Eventually we'll keep the first (though dimension isn't important to us) and drop the second. This 'space of spatial perception' in principle differes from the 'space' of classical Newtonian mechanics, i.e. the space objects move in. The former involves 'local data' attached to a local observer or detector, such as the angle formed by two rays of light, which cross exactly at the instantaneous observers position. The latter more or less incorporates all these positions, hence it's the space light lives in, it comprises in particular all these points light has passed from it's source to an observer or detector. The way we perceive an object, its colors and its shape are assumed to be independent from the path of the light, it purely depends on the energy and the tangent vectors of these paths at the observers position. Thus mathematically speaking the 'local space' is modeled as a tangent space of the 'space of all positions' at a particular position. But since we are used to identifying the space of classical Newtonian mechanics with a three dimensional euclidean space, we are inclined to mix up all these 'spaces': a vector at a point (which is a tangent vector at this point) gets identified with a vector at another point (which is a tangent vector at this other point) by what is called 'parallel translation' and by singling out a particular point (the zero vector) of the Newtonian space we may also identify this space with any of its tangent spaces. Well, that's all admissible and it works just fine for our historically shaped common sense knowledge, but it's an immeasurable drawback when it comes to more advanced models. It didn't need relativity for these models to come up, simple experience suggested to replace the classical Newtonian space with Riemannian manifolds (just think of motions under certain constraints such as rolling a ball on a surface). Unlike the Newtonian case tangent spaces couldn't be identified in an obvious way and parallel transport became a central notion in Riemannian geometry, not at all as easily accessible as in the Newtonian case. Also - and that's what we will be concerned with - the tangent spaces to these mainfolds don't carry a euclidean structure, they carry a Lorentz structure, which admits some subspaces to be euclidean. One of the essential features that makes relativity so much different to Newtonian mechanics is the fact that our space of perception is no more the space relevant in theoretical physics, its only part of this. Both the Lorentz and the euclidean structure are particular cases of what is called an inner product. This will be explored in the run of the first chapter. The second chapter is about Lorentz spaces and some crucial notions from relativity.
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| Criteria for evaluation |
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| Changing subject? |
No |
| Corresponding lecture |
(*)TM1PFVONATU: VO Mathematische Modelle in den Naturwissenschaften (3 ECTS)
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