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Unformatted text preview: LECTURE 3: SPACETIME AND GEOMETRY: AN INTRODUCTION TO SPECIAL RELATIVITY AS204 February 11 2008 I. The Geometry of Space You are used to dealing with vectors in ordinary 3-dimensional space, and to specify loca- tions in space by means of coordinates , like ordinary x, y, z Cartesian coordinates, though there are, of course, many other more complex coordinate systems, such as Cylindrical r, , z and Sphericopolar r, , , and many, many others. The prototype of a vector is the entity which connects neighboring points in space: dr = ( dx, dy, dz ) . Associated with this vector (or any vector) is its length , the square of which is ds 2 = dx 2 + dy 2 + dz 2 , and an associated notion of dot product of two vectors dr = ( dx, dy, dz ) and u = ( u, v, w ), dr u = u dx + v dy + w dz. It is important to understand that vectors have existence quite independent of the coordi- nates used to describe them; their dot products and their lengths are geometric properties of the vectors themselves which must be calculable in any coordinate system and must be independent of the coordinate system used to describe them. In particular, different carte- sian coordinate systems are related to each other by a translation i. e. a choice of origin and a rotation a choice of the directions of the axes. The latter is not quite arbitrary, of course, since ordinary cartesian coordinates are orthogonal i. e. the coordinate axes are mutually perpendicular. The transformation from one cartesian coordinate system to another related by an arbitrary rotation is called an orthogonal transformation: ( u ) = O ( u ) where (u) is the array of the components of the vector u in a particular coordinate system, and (u) the array of the components of the same vector u in the new rotated coordinate system: O is a 3 3 matrix, which, if and only if it is a proper orthogonal transformation, has the property that the matrix product of O and its transpose is the identity: O T O = I where I is the identity matrix, I ab = 1 for a = b ;0 , for a negationslash = b . It is easy to see why O T O = I is the defining property of orthogonal transformations: 1 Let u = ( u 1 , u 2 , u 3 ) and x = ( x 1 , x 2 , x 3 ) be two vectors as described in some coordinate system. Their dot product in the original coordinates is calculated by u x = u 1 x 1 + u 2 x 2 + u 3 x 3 = u a x a , where we introduce the shorthand convention that repeated latin indices are summed over from 1 to 3, i. e. u a x a = i =1 , 3 u a x a . Since u a = O ab u b , and x c = O cd x d we must have u x = u a x a = u b x b = O bc u c O bd x d Since u and x are completely arbitrary, it must be that, since the first and last expressions must be equal, O bc O bd must be 1 for c = d and zero if c negationslash = d , i. e. , the identity. but O bc O bd is just the matrix product O T O ....
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