Metrics

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

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

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

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.

Scalar Product  The metric tensor provides the definition of the scalar product on the vector space. In Euclidean space:

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 |)

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