TY - CHAP

T1 - Introduction

AU - Borchers, Hans Jürgen

AU - Sen, Rathindra Nath

PY - 2006/12/1

Y1 - 2006/12/1

N2 - In every textbook on the theory of relativity it is assumed, without any discussion, that space-time is a differentiable manifold, possibly with singularities. Between a point-set and a differentiable manifold there is an enormous gap, and we felt that physics itself could contribute to narrowing this gap. In experimental physics, one can make only a finite number of well-separated measurements, and therefore it is natural to start with a discrete set of points as a candidate for space-time. As one cannot put an upper bound on the number of measurements, this set cannot be finite; it must, at least, be countable. Next, as one cannot place a quantitative limit on experimental accuracy, one has to admit the "density" property that between any two points on a scale lies a third. Finally, one has to ask how one arrives at the continuum. In short, we felt that it should be possible to start from point-sets and find conditions (axioms) -motivated by physics -which would allow us to construct a topological manifold from the point-set. If this turned out to be true, one could become more ambitious and look for conditions which would imply the differentiability of the manifold.

AB - In every textbook on the theory of relativity it is assumed, without any discussion, that space-time is a differentiable manifold, possibly with singularities. Between a point-set and a differentiable manifold there is an enormous gap, and we felt that physics itself could contribute to narrowing this gap. In experimental physics, one can make only a finite number of well-separated measurements, and therefore it is natural to start with a discrete set of points as a candidate for space-time. As one cannot put an upper bound on the number of measurements, this set cannot be finite; it must, at least, be countable. Next, as one cannot place a quantitative limit on experimental accuracy, one has to admit the "density" property that between any two points on a scale lies a third. Finally, one has to ask how one arrives at the continuum. In short, we felt that it should be possible to start from point-sets and find conditions (axioms) -motivated by physics -which would allow us to construct a topological manifold from the point-set. If this turned out to be true, one could become more ambitious and look for conditions which would imply the differentiability of the manifold.

UR - http://www.scopus.com/inward/record.url?scp=33847292783&partnerID=8YFLogxK

U2 - 10.1007/3-540-37681-X_1

DO - 10.1007/3-540-37681-X_1

M3 - Foreword/postscript

AN - SCOPUS:33847292783

SN - 3540376801

SN - 9783540376804

T3 - Lecture Notes in Physics

SP - 1

EP - 6

BT - Mathematical Implications of Einstein-Weyl Causality

ER -