ar X iv : 0 70 6 . 43 24 v 1 [ gr - q c ] 2 8 Ju n 20 07 Conformal coordinates for a constant density star

The constant density, spherically symmetric, perfect fluid solution to Einstein’s equations was given in 1916 by Schwarzschild[1]. Despite being a stalwart of introductory General Relativity, the source of much of the knowledge we have about the Schwarzschild star interior solution is shrouded in obscurity. When he was writing about it in 1971, Buchdahl[2] decried the fact that no standard texts available to him even mentioned that the metric for this geometry is conformally flat. To our knowledge, the first documented record concerning this fact was published only a few years earlier by Shepley and Taub[3], in a work which Buchdahl actually did not cite. In fact, both of these papers seem to have been remarkably little read, or at least referred to, in the, almost forty, intervening years. Like many others from around that time, Shepley and Taub apparently came across the conformal flatness of the Schwarzsdchild interior solution in the course of establishing a different, somewhat more general, geometric result – including its unique properties among perfect fluid matter sources[2, 4]. For his part, Buchdahl wrote down coordinates in which the conformal flatness was manifest, but he neither established their existence specifically nor discussed their properties in any detail. Instead he concentrated on describing ensuing optical properties, for the systematic study of which he is justly held in high regard. In this paper, we fill in specific gaps in Buchdahl’s discussion. We find constructively the flat coordinates in which the conformal flatness is manifest and we discuss their domain of applicability and related properties in detail. Forming a backdrop to our presentation is the need to show a modern application (which will be published elsewhere) of calculating the self force of a static electric charge placed inside/outside a Schwarzschild star. Just as importantly, we also wish to prevent the earlier awareness [2] of this conformal flatness and its underlying implications from fading into oblivion. The layout of the paper is as follows. We first demonstrate the conformal flatness of the spatial part of the metric by relating it to the metric on the 3-sphere. Then we find the family of conformal factors and related coordinates in which the conformal flatness of the 4-geometry is manifest, and discuss their domains of applicability. We then briefly address the implications of this work for the problem of static electromagnetic sources in the interior Schwarzschild star. Finally, we mimic our procedure to indicate the conformal coordinates for another conformally flat metric (Robertson-Walker).