Semi-Riemann Geometry and General Relativity by Shlomo Sternberg

Author:Shlomo Sternberg
Language: eng
Format: epub
Tags: Riemannian Geometry,

j'(t)=c(t)u(t) + r(t).

Then

liy(*)l| 2 = |c(i)| 2 +||r(i)|| 2

so

W(t)\ > \c(t)\

and hence

as was to be proved.

CHAPTER 6. GAUSS'S LEMMA.

Chapter 7

Special relativity

7.1 Two dimensional Lorentz transformations.

We study a two dimensional vector space with scalar product ( , ) of signature

H .A Lorentz transformation is a linear transformation which preserves the

scalar product. In particular it preserves

(where with the usual abuse of notation this expression can be positive negative or zero). In particular,every such transformation must preserve the "light cone" consisting of all u with ||u|| 2 = 0.

All such two dimensional spaces are isomorphic. In particular, we can choose our vector space to be R 2 with metric given by

The light cone consists of the coordinate axes, so every Lorentz transformation must carry the axes into themselves or interchange the axes. A transformation which preserves the axes is just a diagonal matrix. Hence the (connected component of) the Lorentz group consists of all matrices of the form

So the group is isomorphic to the multiplicative group of the positive real num-

l|u|| 2 :- (u,u)

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