Modified Fronsdal coordinates for maximally extended Schwarzschild spacetime

Abstract

We introduce a coordinate system that complements the Kruskal--Szekeres extension. Like the standard construction, it covers the maximally extended Schwarzschild manifold in its entirety, while offering an additional advantage of expressing the areal radius as an explicit function of the new coordinates. Its main limitation, however, is that radial null geodesics are no longer represented as 45-degree lines in the Kruskal plane, making the causal structure more difficult to interpret. Nevertheless, the new system offers a compelling aesthetic trade-off: among all known maximally extended systems - including those of Kruskal-Szekeres, Israel, Fronsdal, Novikov, and Synge - it exhibits the highest degree of symmetry with respect to Schwarzschild's original r- and t-coordinate lines. It trades the regular pattern of Kruskal's light cones for a symmetric nesting arrangement of the two-dimensional spheres. The proposed extension sheds new light on the closely related Fronsdal's six-dimensional embedding construction, and clarifies the deep connection that exists between the most important implicit (Kruskal-Szekeres) and explicit (Israel's) procedures for maximal extension of the Schwarzschild geometry that is well known to those working in the field but rarely presented in textbooks on general relativity.