By Richard J. Szabo.
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Extra resources for Busstepp Lectures on String Theory
Paralleltransport X along dxλ . At its extremity, X will have the value X µ − Γµ νλ X ν dxλ • Parallel-transport that value along dx ν . It will lead to a value which we denote X µ . • Back at the starting point, parallel-transport X now ﬁrst along dxν and then along dx λ . This will lead to a ﬁeld value which we denote X µ. 53 • In a ﬂat case, X µ = X µ . On a curved case, the diﬀerence between them is non-vanishing, and given by δX µ = X µ − X µ = − Rµ ρλν X ρ dxλ dxν . 47) • This is the inﬁnitesimal case, in the limit of vanishing parallelogram and with the value of Rµ ρλν at the left-lower corner.
Geometries” have been synonymous of “kinds of metric manifolds”. This comes from the impression that we measure something (say, distance from the origin) when attributing coordinates to a point. We do not. Only homeomorphisms are needed in the attribution, and they have nothing to do with metrics. We hope to have made it clear that a metric on a diﬀerentiable manifold is chosen at convenience. 48 Minkowski spacetime is a 4-dimensional connected manifold on which a certain indeﬁnite metric (the “Lorentz metric”) is deﬁned.
We ∂y µ shall later examine general tetrads in detail. 55 A connection deﬁnes covariant derivatives of general tensorial objects. It goes actually a little beyond tensors. A connection Γ deﬁnes a covariant derivative of itself. This gives, rather surprisingly, a tensor, the Riemann curvature tensor of the connection: Rκ λρσ = ∂ρ Γκ λσ − ∂σ Γκ λρ + Γκ νρ Γν λσ − Γκ νσ Γν λρ . 46) It is important to notice the position of the indices in this deﬁnition. Authors diﬀer in that point, and these diﬀerences can lead to diﬀerences in the signs (for example, in the scalar curvature deﬁned below).
Busstepp Lectures on String Theory by Richard J. Szabo.