Archive for May, 2014

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The Problem of Space

May 28, 2014

Joshua Eisenthal

Abstract: I define the Problem of Space as the problem of delimiting the range of candidate geometrical descriptions of physical space. I argue that, ever since the development of non-Euclidean geometries, tackling the Problem of Space has become necessary in order to understand thespatial significance of geometrical structures.

I briefly review the nineteenth century approach to this problem, arriving at the so-called “classical solution”. This solution centered around the claim, advanced by Helmholtz and Poincaré, that candidate physical geometries were just those structures which could represent the free mobility of rigid bodies. As noted originally by Riemann, then argued for by Helmholtz and proved rigorously by Lie, congruence relations which could represent such free mobility existed only in geometries of constant curvature. Helmholtz and Poincaré regarded this fact as providing the means of delimiting the range of spatially significant geometrical structures.

However, I then review how this view was fatally undermined by the development of general relativity. I thus turn to explore the twentieth century solution to the Problem of Space advanced by Hermann Weyl. I conclude by reflecting on the significance of this discussion for a relatively recent dispute regarding the status of the metric field in general relativity. I suggest that this dispute has arisen partially due to a failure to properly appreciate the insights made available by the kind of sophisticated analysis of geometrical concepts exemplified by Weyl’s work. More generally, I argue that the nuances of Weyl’s view demonstrate the importance of engaging with the Problem of Space in interpreting general relativity today.

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