Atlas: The effects of gravitation on clocks, proved in axiomatic relativity by Judit X. Madarász

Logic in Hungary, 2005
August 5-10, 2005
Janos Bolyai Mathematical Society
Budapest, Hungary

Organizers
A. Hajnal, J. Suranyi (honorary chair) H. Andreka, I. Juhasz, P. Komjath, I. Nemeti (co-chair) G. Sagi (secretary) L. Csirmaz, M. Ferenczi, M. Redei, I. Sain, L. Soukup (member)

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The effects of gravitation on clocks, proved in axiomatic relativity
by
Judit X. Madarász
Alfréd Rényi Institute of Mathematics, Budapest
Coauthors: Gergely Székely

Abstract

One of the main themes of, say, black hole physics is that gravity causes time to run slow: clocks closer to the black hole run slower than ones far away. This kind of effect is called in popular language "time warp", while the official name is gravitational time dilation (or gravitational red shift). Using Einstein's equivalence principle ("gravity"="acceleration"), we seek to develop a rather simple axiomatization of the theory Th of accelerated observers such that, in Th , the gravitational time dilation effect becomes provable. Our aim is to keep the theory Th as simple, streamlined, and transparent as possible, e.g. we make efforts to remain within the framework of first order logic. One of the purposes with this is to provide a logic based conceptual analysis of the gravitational time dilation effect of relativity, i.e. we seek a logical explanation for why this effect is there. In other words, we seek to find a minimal number of axioms which are "responsible" for this effect.

Indeed, we will present a streamlined, purely kinematical axiom system which proves the gravitational time dilation effect in such a way that none of the axioms can be omitted. In some sense, our research project is analogous with what is known as "reverse mathematics" in the foundations of mathematics (seeking which theorems depend on what axioms of set theory). In our work we build on the first order logic based analysis of special relativity theories in [1] and [2].

References

[1] Madarász, J.X., Németi, I. and Székely, G.: Twin paradox and the logical foundation of special relativity-theory, Foundations of Physics, to appear.

[2] Andréka, H., Madarász, J.X., Németi, I.: Logical axiomatizations of space-time. Samples from the literature. In: János Bolyai Memorial Volume, Kluwer, to appear.

Date received: June 1, 2005