Inversion of a ‘discontinuous coordinate transformation’ in general relativity

Abstract

Penrose [R. Penrose, The geometry of impulsive gravitational waves, in General Relativity, Papers in Honour of J. L. Synge, L. O'Raifeartaigh, ed., Clarendon Press, Oxford, 1972, pp. 101–115], – in a purely formal way – introduced a ‘discontinuous coordinate transformation’, which relates a continuous representation of the metric of impulsive pp-waves to a discontinuous one. On the basis of the invertibility concept for generalized functions developed recently by Erlacher [E. Erlacher, Inversion of Colombeau generalized functions, preprint (2010)], we show that this discontinuous coordinate transformation indeed represents an invertible generalized function in the appropriate sense.

Keywords:

• discontinuous coordinate transformation
• impulsive pp-wave
• distributional metric
• Colombeau algebra
• inverse generalized function

AMS Subject Classifications::

• 83C35
• 46F30
• 26B10

Acknowledgements

The authors were supported by START-project Y-237 and by project P20525 of the Austrian Science Fund.