6/26/2023 0 Comments Curved space coordinate radius![]() ![]() Well, not as easy as the way Riemann did it at least: Riemann's Approach The Problem with Gauß' Approach is although it is intuitive when looking from “outside” at the manifold, determining it from inside the manifold involves taking a limit, and it is not so easy to compute and generalize. A nice mental picture for $n=2$ is that if you tried mounting a sheet of paper, and observe that: Since the circle is “stretched”, the circumference and area are larger than expected – this would be an example of negative curvature. More precisely, in a Space with positive curvature – the radius can be shorter than expected, as well! Consider a saddle. If you are on a sphere, the ”perceived radius“ of a circle is slightly larger than the radius corresponding to its circumference / area, so you know you are in a curved space. Gauß' approach is to compare how “circles grow”. Luckily, there have been people such as Gauss and Riemann doing the hard work for you. We need to find tools to measure the failure of our coordinate maps to become constant. “Curvature” is not so easy to define however. If these derivatives are zero everywhere, you already know you are in a flat space. That is extremely useful, since we now can easily take derivatives of it (in the directions of our coordinate basis). Once you introduce a coordinate map, you have a basis for the metric tensor and can represent it by multiple components which are real numbers. Wherever you are, you will find a map giving you a set of real numbers. “Coordinates” Are actually maps from our Manifold to $\mathbb R^n$, in the case of spacetime $n=4$. The latter is called the “Metric tensor field”. To be more precise, we need to use the mathematical terms: Our “space” or “spacetime” becomes a “Riemannian Manifold”, namely an abstract mathematical set with some nice properties and the ability to measure distances locally. ![]() Note however that the vocabulary is extremely vague. When saying “spacetime is curved”, we mean “Spacetime has curvature”, and not only “The coordinates vary”. Curvature is also said to be an “intrinsic property of the space”, meaning exactly that this property does not depend on its representation by coordinates. This means it must be caused by the space itself - If coordinates fail to get straight, we say the “Space has curvature”. In the latter, you can choose any representation you want – you will not get variation-free coordinates! For instance, the closer you get to the poles, your coordinates are forced to get “denser”, if they shall stay continuous. In the first case, obviously a change to cartesian coordinates eliminates all variation in your coordinates. Living on the sphere, using any kind of coordinates.Being in “flat” Euclidean space, but using spherical coordinates.Let me call the latter “variation” instead. How can I know whether the curvature is caused by my choice of coordinates or the space I live in?Īs has been mentioned in other answers, the word “curvature” is referred to as either a property of the space, but also a property of the coordinates. Congratulations! You stumbled upon an important question of differential geometry:
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