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Published bySandra Gordon Modified over 9 years ago
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Metrics
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Euclidean Geometry Distance map x, y, z E nx, y, z E n d: E n E n → [0, )d: E n E n → [0, ) Satisfies three properties d ( x, y ) = 0 if and only if x = yd ( x, y ) = 0 if and only if x = y d ( x, z ) = d ( z, x )d ( x, z ) = d ( z, x ) d ( x, y ) + d ( y, z ) d ( x, z )d ( x, y ) + d ( y, z ) d ( x, z ) The Pythagorean relationship defines Euclidean geometry x2x2 x1x1 d
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Lorentz Geometry A distance measure exists in Lorentz space. x 0 is timelike coordinate s is the distance function This distance function can be true for all points in a coordinate system. The coordinate system is Lorentzian Geometry is Lorentzian x0x0 x1x1 s
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Vector Map The displacement vector x is a an element in the vector space. The distance function maps the displacement vector into the field of the vector space. Treat as two copies of v = xTreat as two copies of v = x Eg. V = { a E n }, F = REg. V = { a E n }, F = R Map g: V V RMap g: V V R g V F a s
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Metric Tensor A metric is a map from two vectors in a vector space to its field. Bilinear tensorBilinear tensor May be symmetric or antisymmetricMay be symmetric or antisymmetric The Lorentz metric can be written as a matrix.
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Scalar Product The metric tensor provides the definition of the scalar product on the vector space. In Euclidean space:
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Metric Space A pair (X, d ) A set XA set X A function d: X X → [0, )A function d: X X → [0, ) d meets the definition of a metric.d meets the definition of a metric. Euclidean spaces are metric spaces A metric for a circle S 1 = { : 0 < 2 } d = inf (| 2 – 1 |, 2 | 2 – 1 |)
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Transformation Groups The group of Jacobian transformations of real vectors Gl(N,r) does not generally preserve a metric. Some subsets of transformations do preserve metrics. Orthogonal – symmetricOrthogonal – symmetric Unitary – symmetric with complex conjugationUnitary – symmetric with complex conjugation Symplectic – antisymmetricSymplectic – antisymmetric next
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