# Metrics - PowerPoint PPT Presentation

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## 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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