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Title:

Metrics

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Metrics Euclidean Geometry Distance map x, y, z En d: En En [0, ) Satisfies three properties d(x, y) = 0 if and only if x = y d(x, z) = d(z, x) d(x, y ... – PowerPoint PPT presentation

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Title: Metrics


1
Metrics
2
Euclidean Geometry
  • Distance map
  • x, y, z ? En
  • d En ? En ? 0, ?)
  • Satisfies three properties
  • d(x, y) 0 if and only if x y
  • d(x, z) d(z, x)
  • d(x, y) d(y, z) ? d(x, z)
  • The Pythagorean relationship defines Euclidean
    geometry

x2
d
x1
3
Lorentz Geometry
  • A distance measure exists in Lorentz space.
  • x0 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

x0
s
x1
4
Vector Map
  • The displacement vector Dx 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 vDx
  • Eg. V a ? En, F R
  • Map g V ? V ? R

V
a
g
s
F
5
Metric Tensor
  • A metric is a map from two vectors in a vector
    space to its field.
  • Bilinear tensor
  • May be symmetric or antisymmetric
  • The Lorentz metric can be written as a matrix.

6
Scalar Product
  • The metric tensor provides the definition of the
    scalar product on the vector space.

In Euclidean space
7
Metric Space
  • A pair (X, d)
  • A set X
  • A function d X ? X ? 0, ?)
  • d meets the definition of a metric.
  • Euclidean spaces are metric spaces
  • A metric for a circle
  • S1 q 0 ? q lt 2p
  • d inf (q2 q1, 2p-q2 q1)

8
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 symmetric
  • Unitary symmetric with complex conjugation
  • Symplectic antisymmetric

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