On Kruskal-Novikov co-ordinate systems

Document Type

Article

Publication Date

3-1-1980

Abstract

We rederive the transformation from Schwarzschild coordinates to Novikov co-ordinates, which appear to be equivalent to Kruskal co-ordinates. Our derivation takes place in two steps. We first replace the Schwarzschild time co-ordinate T with a new time coordinate τ measured by radially moving geodesic clocks, but keep the Schwarzschild radial co-ordinate R. The transformation from T to τ results in a nondiagonal metric which is regular at the Schwarzschild radius, R=2 M, and geodesics can be followed across the Schwarzschild radius in terms of the new (R, τ) co-ordinates. However, the metric does contain an expected co-ordinate singularity, arising because of the limitation of the reference system that one cannot have a geodesic clock with a turning point smaller than the Schwarzschild radius. Because of this, the co-ordinate system is incomplete in the sense one can find geodesic trajectories that cannot be followed to the intrinsic singularity at R=0. In our second step, the Schwarzschild radial co-ordinate R is replaced with a new spatial co-ordinate R i equal to the maximum Schwarzschild radius of each geodesic clock forming the reference system, a constant uniquely associated with the world-line of each co-ordinate clock. In terms of (R i, τ) co-ordinates the metric assumes a diagonal form, but still maintains the previous co-ordinate singularity and is still geodesically incomplete. It is shown that the co-ordinate singularity in the metric can be removed by the mathematical procedure of replacing the Schwarzschild value R i with a derived co-ordinate R * monotonically related to R i by R *=(R i /2 M-1)1/2. Because the metric written in terms of (R *, τ) now contains no appearance of co-ordinate singularities, it is possible to consider letting R * take on negative values. However, arguments are given for excluding both negative values of R * and the corresponding left-hand side of a Novikov diagram that would be generated by these negative values. Similar arguments for excluding the left-hand side of Kruskal diagrams are also given. © 1980 Società Italiana di Fisica.

Identifier

51249180741 (Scopus)

Publication Title

Il Nuovo Cimento B Series 11

External Full Text Location

https://doi.org/10.1007/BF02738358

e-ISSN

18269877

ISSN

03693554

First Page

49

Last Page

71

Issue

1

Volume

56

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