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65 Years in and Around Relativity

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Book cover At the Frontier of Spacetime

Part of the book series: Fundamental Theories of Physics ((FTPH,volume 183))

Abstract

At the very beginning I must thank all of the contributors to this book for taking their valuable time to add to it.

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Notes

  1. 1.

    And I believe so did Einstein. See the later discussion of this issue to which a good bit of my thesis was addressed.

  2. 2.

    This ultimately led to the oxymoronic phrase in my thesis title: “...variable...constant.”.

  3. 3.

    He was also concerned that since electromagnetic radiation has \(T=0\), an electromagnetic radiation field would not contribute as a source for \(\phi .\) This question was shoved aside and not further investigated, I believe.

  4. 4.

    Since then I have done work which turned out to be on both sides of this phenomenon.

  5. 5.

    In fact some 20 years later inflationary cosmology was to lead to renewed interest in the addition of a classical scalar field to the metric.

  6. 6.

    There can be no global non-zero vector field on \(S^4\) for topological reasons, and thus no Minkowski signature metric.

  7. 7.

    Of course in the spherical case the “radial” coordinate is not indefinitely continuable because it is essentially an angular one. However, this is not the sort of coordinate anomaly we are addressing here and can certainly be accommodated in standard models.

  8. 8.

    However there are explicit metrics on Milnor’s exotic \(S^7\) [72], and it is known that a Riemannian metric exists on any smooth \(\mathbb {R}^4.\) The Lorentz signature case is different however, since the existence of a nowhere zero timelike vector would result in a smooth foliation of the manifold which would then reduce it to standard, so \(\mathbb {R}\times \mathbb {R}^3=\mathbb {R}^4\), and thus any such metric must have a singularity.

  9. 9.

    Here we do not distinguish between a fermionic quantum field and a fermion.

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Authors and Affiliations

  1. Loyola University, 6363 St. Charles Avenue, Box 92, New Orleans, 70118, USA

    Carl H. Brans

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Correspondence to Carl H. Brans .

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  1. PT-SW, DLR Projektträger, Berlin, Germany

    Torsten Asselmeyer-Maluga

Appendix

Appendix

Here I will summarize a selected list of topics from my thesis, finished in 1960, but formally presented and accepted by Princeton in May 1961.

  • Mach’s Principle I reviewed what I knew of it, and especially what I thought Dicke assumed. This of course required a careful look at the question of how a backgrund metric would affect the motion of particles, both point and extended. So this led to the next point.

  • Equations of Motion During this time the questions associated with the equations of motion of both point and extended (fluid type) particles had been extensively studied by Einstein, Infeld, Papapetrou and others. So, in trying to confirm that I was using an operationally significant procedure for measuring and comparing inertial and gravitational mass for a given background metric. As I recall, Einstein and infeld were primarily concerned with point masses, while Papapetrou looked at fluid type stress-energy tensors.

  • How to measure “mass” I tried to settle this for inertial mass by looking at motion in an external electric field.

  • Equivalence Principles I explored both strong and weak as presented to me by Dicke.

  • Variational principles and field equations I had already presented and explored the formalism Bob and I would use to arrive at equations satisfying the oxymoronic phrase “varying ...constant.” Then, hearing of the work of Jordan and his group, I first reviewed their formalisms.

  • Static, spherically symmetric vacuum I looked at the solution in Jordan’s formulation known as the Heckmann solution, and then did the same for our formalism but in isotropic coordinates. This led to careful analysis of four qualitatively different metric forms.

  • Does \({{\varvec{G}}}\) vary? I then used various idealized operational procedure to look at what could be said about the dependence of \(G\sim 1/\phi \) on the matter distribution in the universe.

  • Boundary Conditions Of course these had to be carefully defined and were eventually defined in the usual way in terms of some “going to zero or other constant at infinity” procedure.

  • Cosmology I could find no exact analog to the FRW metric of the Einstein equations, so I simple did some very extensive power series expansions for each of various ranges of arbitrary constants. A very few of you may recall that at that time we were amazed to have a human driven mechanical computer that could both multiply and divide long numbers within 10 s or so.

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Brans, C.H. (2016). 65 Years in and Around Relativity. In: Asselmeyer-Maluga, T. (eds) At the Frontier of Spacetime. Fundamental Theories of Physics, vol 183. Springer, Cham. https://doi.org/10.1007/978-3-319-31299-6_1

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