Skip to main content
SpringerLink
Log in

Consequences of the Noncompactness of the Lorentz group

  • Published:
International Journal of Theoretical Physics Aims and scope Submit manuscript

Abstract

The following four statements for indefinitemetrics of Lorentz signature do not possess an analogousstatement in the definite Euclidean signature case: (1)All curvature invariants of a gravitational wave vanish, in spite of the fact that itrepresents a nonflat spacetime. (2) The eigennullframecomponents of the curvature tensor (the Cartan“scalars”) do not represent curvaturescalars. (3) The Euclidean topology in the Minkowskispacetime doesnot possessa basis composedofLorentzinvariant neighborhoods. (4) There are pointsin the de Sitter spacetime which cannot be joined toeach other by any geodesic. We show that these fourstatements all follow from the noncompactness of theLorentz group. We conclude that the popular (and oftenuseful) imaginary-coordinate rotation from Euclidean to Lorentzian signature (called Wick rotation) isnot an isomorphism.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  • Dautcourt, G., Jann, K.-P., Riemer, E., and Riemer, M. (1981). Astronomische Nachrichten, 302, 1.

    Google Scholar 

  • Fullwood, D. (1992). Journal of Mathematical Physics, 33, 2232.

    Google Scholar 

  • Geroch, R., Held, A., and Penrose, R. (1973). Journal of Mathematical Physics, 14, 874.

    Google Scholar 

  • Hawking, S., and Ellis, G. F. R. (1973). The Large Scale Structure of Space-Time, Cambridge University Press, Cambridge.

    Google Scholar 

  • Hawking, S., King, A., and McCarthy, P. (1976). Journal of Mathematical Physics, 17, 174.

    Google Scholar 

  • Jordan, P., Ehlers, J., and Kundt, W. (1960). Abhandlungen der Akademie der Wissenschaften, Mainz. Mathematik/Naturwissenschaften, 21.

  • Ludwig, G., and Edgar, S. (1996). General Relativity and Gravitation, 28, 707.

    Google Scholar 

  • Schimming, R. (1974). Mathematische Nachrichten, 59, 129.

    Google Scholar 

  • Schmidt, H.-J. (1984). Abhandlungen der Akademie der Wissenschaften der DDR, Abteilung Mathematik, 2N, 207.

    Google Scholar 

  • Schmidt, H.-J. (1993). Fortschritte der Physik, 41, 179.

    Google Scholar 

  • Schmidt, H.-J. (1995). In Abstracts GR 14 Florence, A61.

  • Schmidt, H.-J. (1996). In New Frontiers in Gravitation, G. Sardanashvily, ed., Hadronic Press, Palm Harbor, Florida, p. 337.

    Google Scholar 

Download references

Authors
  1. Hans-Jurgen Schmidt
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Schmidt, HJ. Consequences of the Noncompactness of the Lorentz group. International Journal of Theoretical Physics 37, 691–696 (1998). https://doi.org/10.1023/A:1026660211608

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1026660211608

Keywords

Search

Navigation