![]() So we will restrict the range of a coordinate system ψ to make it single valued, just as we restrict the range of the arcsine “function.” The second problem is more interesting. One may describe Pasadena as having colatitude 0.98 (radians) and longitude −2.06 but it also has longitude −2.06 + 2π = 4.22. First is that the mapping ψ : S 2 → R 2 is multiple-valued. The example of S 2 immediately exhibits two problems with this idea. The number of coordinates needed is the dimension n of the surface. So one might begin by trying to characterize a curved surface M by its coordinate system: a point in the manifold corresponds via a coordinate system (or chart) ψ to a point (x 1. A hint is provided by the familiar way in which we refer to points on S 2 – by their “coordinates,” the colatitude θ and longitude φ. MANIFOLDS We have an intuitive notion of what a curved surface is – but our first step will be to sharpen this definition, and in particular to make sense of the idea without regard to an externally defined flat space. Note that we will cover the “Track 1” material in the book first selected examples of “Track 2” material will be introduced later in the course. The lecture will go into slightly more detail than the book, but does neither aspires to mathematical rigor. This presents an overview of the subject of mathematics in curved spacetime. The recommended reading for this lecture is: Therefore we will need a new set of tools to speak meaningfully of vectors and their derivatives. In general relativity, 4-dimensional curved spacetime simply is – it is not embedded in any flat higher-dimensional space. Most importantly, the unit sphere S 2 is embedded in R 3, and in spherical trigonometry the geometry of R 3 is inherited by S 2. the Earth’s surface), and we will use this example frequently. It is time now for a mathematical digression: how do we do geometry and vector calculus in curved spacetime? In some ways, this is analogous to geometry on the surface of a sphere (e.g. OVERVIEW Thus far we have studied mathematics and physics in flat spacetime extensively. Lecture V: Vectors and tensor calculus in curved spacetime Christopher M.
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