Walker, A. G.

2019-02-15T14:25:40Z

2019-02-15T14:25:40Z

1933

The theory of relative co-ordinates w s first introduced by the author (1), but only a few of the results of Part 1 of this work were given. The theory was afterwards applied ton study of distance in relativity (2), the method outlined being fundamentally that used in Part III. In the following work, the notation of the above two papers has been considerably altered so that the same notation can be used consistently throughout this work. With very few exceptions, symbols Till retain their meanings in order to avoid continual references to previous definitions. In the study of twisted curves in ordinary Euclidean space, a very useful theory is that of moving axes. At a point of a given curve, a set of axes is formed by the tangent, principal normal and binormal of the curve, so, as the point moves along, the curve, we have a set of axes moving in space. Any curves or surfaces associated with the given curve can now be referred to these moving axes: e.g., a curve is described by a point with given co-ordinates relative to these axes: thus, the locus of centres of curvature of the given curve is described by the point (o.p.o.) where p is the radius of curvature. The object of the following work is to develop systems of reference similar to the above at points of a curve in a general Riemannian space Vₙ. The first difficulty is to find the most convenient system of reference at a given point, and to overcome this, we shall use the theory of normal co-ordinates (§3). The second difficulty is to decide how the system of reference at one point of the given curve is to be related to the system of reference at any other point. This leads us to examine the theory of transport along, the curve and this will be discussed at length in §2. Part I All be devoted entirely to the geometrical development and application of the theory of relative co-ordinates. We believe that these co-ordinates have a purely geometrical Interest, and several methods and results will be discussed at length on this assumption. With their aid, we shell give short proofs of several known results and we shall also state and prove some now theorems. Part II is concerned with the study of motion in relativity. From the theory of observation in relativity, it will be seen that relative co-ordinates play a very important part, and observable quantities are naturally expressed in terms of these co-ordinates. In Parts, relative co-ordinates play a secondary part. The problem of defining distance in relativity has been greatly discussed recently, end we shall not attempt to criticise the work already published on the subject. We shall be mainly concerned with two of the definitions of distance and with the aid of some of the results of PART 1, we shall formulate these definitions more naturally and give a practical method of calculating the results in a given space-time. To demonstrate the advantage of this method we shall find the formulae for distance in a very general form of space-time, but the subject of this work is not concerned with a detailed discussion of these results. INCLUDES SELECTED PUBLICATIONS: The second curvature of a sub-space. By A. G. Walker. Reprinted from THE QUARTERLY JOURNAL OF MATHEMATICS OXFORD SERIES. Volume 3 No. 12 December 1932. Spatial distance in general relativity. By A. G. WALKER. Reprinted from THE QUARTERLY JOURNAL OF MATHEMATICS OXFORD SERIES. Volume 4 No. z 3 March 1933. Relative co-ordinates. By A. G. Walker Reprint from THE PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH. SESSION 1931 -1932. VOL. LII PART III (No. 21). On small deformation of sub-spaces of a flat space. By A. G. WALKER Extracted from the PROCEEDINGS OF THE EDINBURGH MATHEMATICAL SOCIETY Series 2 Vol. III Part II.

http://hdl.handle.net/1842/34308

The University of Edinburgh

Annexe Thesis Digitisation Project 2019 Block 22

Theory of relative co-ordinates in Riemannian Geometry

The theory of relative co-ordinates in Riemannian Geometry

Thesis or Dissertation

Doctoral

PhD Doctor of Philosophy