From Special Relativity to Feynman Diagrams.pdf

Or if the two points are infinitesimally apart d 2 δ

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or, if the two points are infinitesimally apart, d 2 = δ ij d ξ i d ξ j . (3.29) It is clear that the same considerations and formalism hold if the Euclidean space has a generic number D of dimensions; in this case the indices i , j , . . . would run over D values instead of only three. As we have already remarked in Sect. 1.5 of the first chapter, there is a close analogy between the four-dimensional distance in Minkowski space–time and the Euclidean distance ( 3.28 ) in D = 4 dimensions. Indeed the four-dimensional distance between two events 9 labeled by the four coordinates ξ 0 , ξ 1 , ξ 2 , ξ 3 was written as 10 : s 2 = c 2 τ 2 = ( ξ 0 ) 2 ( ξ 1 ) 2 ( ξ 2 ) 2 ( ξ 3 ) 2 = α,β η αβ ξ α ξ β η αβ ξ α ξ β (α, β = 0 , 1 , 2 , 3 ) (3.30) that is in a way which is strictly analogous to the four-dimensional Euclidean distance, the only difference being the replacement of the metric δ ij with the Minkowski metric η αβ : η αβ = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 , (3.31) where the extra fourth coordinate is related to time, ξ 0 = ct . We stress that these simple expressions of distance in Euclidean space or proper distance in Minkowski 9 Recall that the space–time (four-dimensional) distance was conventionally defined as the negative of the proper distance , see ( 2.44 ). 10 From now on, we use Greek indices to label four dimensional space-time coordinates and Latin ones for the coordinates in Euclidean space.
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74 3 The Equivalence Principle case, are only valid if we use the three-dimensional Cartesian rectangular or the anal- ogous four-dimensional Cartesian rectangular (also referred to as Minkowskian 11 ) coordinates ξ α , the latter being defined rectangular as the coordinates used to describe inertial frames in terms of the spatial Cartesian rectangular coordinates and the usual time coordinate t . In any other coordinate system, not related by a three-dimensional rotation or a Lorentz transformation, respectively, the Euclidean distance ( 3.28 ) or the Minkowski proper distance ( 3.30 ) would take a more complicated form. Suppose indeed that in the Euclidean case we want to use an arbitrary system of curvilinear coordinates x i , i = 1 , 2 , 3 (an example would be the spherical polar coordinates); we would then have: ξ i = ξ i ( x j ). (3.32) In these new coordinates the infinitesimal distance ( 3.30 ) becomes: d 2 = δ ij ∂ξ i x k ∂ξ j x dx k dx . = g k dx k dx , (3.33) where g k = δ ij V i k V j , (3.34) being: V i k . = ∂ξ i x k . (3.35) The dimensionless quantity g k ( x ), replacing δ ij in the formula for the squared dis- tance, is called metric tensor or, more simply, metric 12 in curvilinear coordinates. It is obvious that all the geometric quantities of Euclidean geometry (lengths, angles, areas, etc) do not depend of the particular coordinates used for their descrip- tion. However it is well known that in general it is much simpler to compute geometric quantities using Cartesian coordinates, rather than the curvilinear ones.
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