Cambridge University

Ph. D. Thesis B. Carter,  1967

Pembroke College

It has been difficult to offer any way of looking up my Cambridge doctoral thesis, except via the library of the Department of applied Mathematics and Theoretical Physics , where - under the direction of Dennis Sciama - the work was completed in 1967. The problem arose from a system that was antiquated even by the primitive standards of that time, when laser printers had not yet been invented, and photocopying was considered too expensive for large scale use. A Cambridge thesis was normally produced only in a very limited edition consisting of as few as 6 copies: one for the University library, one for the relevant Departmental Library, one for the External Examiner, one for the Internal Examiner, one for the Thesis Director, and finally one to be kept by the Candidate. Since the main new results would in any case be properly published in regular journals, there was little incentive for extra thesis copies, whose production was inhibited not just by the trouble of printing and binding, but also by the faint and faulty quality of the output. (This was because the analogue of what is known today as the source file consisted of wax stencil sheets on which the main text was inscribed using a mechanical typewriter - a major source of misprints - leaving gaps in which Greek and other mathematical symbols had to be scratched in laboriously by hand.)

In the complete thesis, entitled STATIONARY AXISYMMETRIC SYSTEMS IN GENERAL RELATIVITY, there were four chapters:   I A systematic notation for space-time manifolds and the elements of etiology;   II Relationships connecting horizons, causality and isometry groups;   III Hamilton-Jacobi and Schrodinger separable solutions of Einstein's equations; IV Global properties of the Hamilton-Jacobi separable spaces.  

This site provides access to the photographed pages of just the last two chapters, whose subheadings are as follows.
Chapter III:   1. Introduction,   [1 - 2],   [3 - 4],   [5 - 6];   2. Hamilton-Jacobi and Schrodinger equations, [7 - 8],   [9 -10];   3. Separability conditions, [11-12],   [13-14],   [15-16],   [17-18],   [19-20],   [21-22],   [23-24];   4. The Hamilton-Jacobi separable case, [25-26],   [27-28],   [29-30],   [31-32],   [33-34],   [35-36];   5. The Schrodinger separable case, [37-38],   [39-40],   [41-42];   6. Source-free Maxwell-Einstein Solutions. [43-44],   [45-46],   [47-48],   [49-50],   [51],  
Chapter IV:   1. Formal integration of the geodesic equations,   [0 - 1],   [2 - 3],   [4 - 5],   [6 - 7];   2. Geodesic incompleteness,   [8 - 9],   [10-11];   3. General analytic extensions,   [12-13],   [14-15],   [16-17],   [18-19],   [20-21],   [22-23],   [24-25],   [25A-25B],   [25C-25D],   [25E-25F],   [25G-26],   [27-28],   [29-30],   [31-32];   4. Global canonical forms, [33-34],   [35-36],   [37-38],   [39-40];   [41-42];   5. The family of solutions,   [43-44],   [45-46],   [47-48];   6. The spheroidal subfamily,   [49-50],   [51-52],   [53-54];   Appendix: Causality, geodesics and the ring singularity in the Kerr solution, [55-56],   [57-58],   [59-60],   [61-62],   [63-64],   [65-66],   [67-68].

The subheadings of the other two chapters were as follows.
Chapter I : 1. Etiology; 2. Fundamental definitions; 3. Time ordering relationships; 4. Time symmetric relationships; 5. Internal connectivity properties; 6. Cauchy developments; 7. Unions and intersections; 8. Topology and higher degree causality. 9. Impisonment; 10. Internam connectivity theorems; 11. Cauchy sets, partial Cauchy sets, and slices; 12. Axioms in etiological spaces; 13. Covering spaces; 14. Horizons; 15. Causality and chronology horizons; 16. Normal space-time manifolds; 17. Consequences of normalizability.
Chapter II : 1. Surfaces of transitivity; 2. Causality and the Lie Algebra; 3. Global and local isometry horizons; 4. An existence theorem for local isometry horizons where an orthogonally transitive group has null surfaces of transitivity; 5. Orthogonal transitivity inan (n-2)-parameter Abelianisometry group with invertible Ricci tensor; 6. Convective circulation.

The main results are contained in the publications listed as follows.
Chapter I: " Causal Structure in Space-Time", Gen. Rel. and Grav. 1 (1971) 349-391.
Chapter II: "Killing Horizons and Orthogonal Transitivity in Space-Time", J. Math. Phys. 10 (1969) 70-81.
Chapter III: "Hamilton-Jacobi and Schrodinger Separable Solutions of Einstein's Equations", Commun. Math. Phys. 10 (1968) 280-310.
Chapter IV: "Global Structure of the Kerr Family of Gravitational Fields", Phys. Rev. 174, (1968) 1559-1571.

Further technical details are included in "Black Hole Equilibrium States", in Black Holes (proc. 1972 Les Houches Summer School) ed. B. and C. DeWitt, (Gordon and Breach, New York, 1973) 57-210, and in "Domains of Stationary Communications in Space-Time", Gen. Rel. and Grav. 9 (1978) 437-450.