The Motion and Structure of Singularities in General Relativity (1970), Roger Posadas's dissertation

The question

In Newton's physics you need two separate laws: one says how masses create gravity, the other says how bodies move under it. Einstein's is different. Its field equations, which describe how matter curves spacetime, also dictate how that matter must move, so a separate law of motion is not needed. Einstein and Jakob Grommer showed this for small test bodies in 1927, and worked out the case of heavy bodies in 1938.

Turning that principle into actual equations of motion is hard. The standard methods of the 1960s pictured a flat, empty background spacetime with gravity as small ripples on top of it. The best earlier attempt to include electric charge, by Leopold Infeld and Philip Wallace in 1940, had to drop part of the theory and make arbitrary choices about which field belonged to the particle. Roger's thesis calls that method “unsatisfactory.”

The approach

Roger and Newman tried something different. They modeled a particle as a , a point where the gravitational field becomes infinite, travelling along a line through spacetime. From every moment on that line, spread outward. Instead of measuring the particle against a flat background, they read its motion from the shape of that family of light cones, using the , the toolkit Newman had built with in 1962. In the thesis's words, the method gives

an intrinsic description of the motion of a singularity in its own space-time, with no reference to a regular background space.

Chapter 6, Summary and conclusion, page 105

He tested it in three settings: the and their electrically charged counterparts; a general space empty of matter; and the full , in which gravity and electromagnetism act on each other.

Light cones. Draw time going up and space going sideways. The particle's path is the curving line; the light leaving each moment spreads out in a cone. Roger described the particle entirely through this family of cones.

Ripples, not a grid. The usual methods measured a particle against a fixed background grid. Roger's method let the grid go and read the motion from the light the particle sends out: where the rings crowd together is the way it was heading.

What he found

Two results stood out. First, the singularity turned out to have an internal structure, with its own equation for how that structure changes over time: in this picture, a point particle is not quite featureless. Second, for a charged particle, the equations reduced, to lowest order, to the , the classic law for a charge that loses energy by radiating as it accelerates. The radiation reaction force, which usually has to be put in by hand or rescued with , appeared on its own.

This derivation of the Lorentz-Dirac equation from the Einstein-Maxwell theory plus the discovery of an internal structure I for an elementary singularity constitute the major acheivements [sic] of our approach to equations of motion.

Chapter 6, Summary and conclusion, page 106

He closed by naming what was left to do: the full non-linear effects, particles that spin, and systems of two or more singularities interacting with each other.

The work also appeared as papers with Newman: “Equations of motion and the structure of singularities” (opens in a new tab) in Physical Review Letters and “Motion and structure of singularities in general relativity” (opens in a new tab) in Physical Review, both in 1969, with a sequel (opens in a new tab) in the Journal of Mathematical Physics in 1971.

Shake a charge, and it loses energy. A charged particle that is shaken sends out waves of light, and the waves carry energy away, so the shaking dies down as if the charge were pushed back by its own light. That push, the radiation reaction, came out of Roger's equations on its own.

Where, and with whom

Roger came to Pittsburgh in 1965 on a Rockefeller Foundation fellowship and went straight for the doctorate, without a master's degree. A major part of the thesis was done in London, at King's College, where he was a guest of . The foreword, one paragraph long, reads in full:

I wish to thank my advisor, Dr. Ezra T. Newman, for suggesting this research problem and for his valuable guidance, assistance, and encouragement throughout the course of this work. I would also like to acknowledge the hospitality extended to me by Prof. Felix Pirani and King's College, University of London, where a major part of this work was done. Finally, I would like to express my gratitude to the Rockefeller Foundation for the financial support of my graduate study at the University of Pittsburgh.

Foreword, page ii

He defended on April 2, 1970. His committee was Ezra T. Newman, as chair, with , P. Stehle, E. Krefetz and J. R. Porter. The title page gives his name as Roger R. Posadas, with one earlier degree: “B.S., , 1964.”

What is inside, with the dissertation's page numbers
Chapter	Page
1. Introduction. The problem of motion, from Einstein to the late 1960s.	1
2. General formalism. The spin-coefficient formalism, spin-weighted spherical harmonics, a special null coordinate system, and the basic assumptions.	9
3. Motion of free singularities. The Robinson–Trautman solutions and their charged, Robinson–Trautman–Maxwell counterparts.	37
4. Motion in a general empty space.	59
5. Motion in the Einstein–Maxwell theory.	83
6. Summary and conclusion.	105
Appendices A to C, and references.	107

Pages from the dissertation

This is the University Microfilms scan of the dissertation (order no. 70-20,354), posted by his family. Its text was recognized by machine, so equations and some words will read poorly with a screen reader.

Read the dissertation (PDF, 129 pages) (opens in a new tab) Back to the Dad page