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Gravitational Fields with 2-Dimensional Killing Leaves and the Gravitational Interaction of Light

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General Relativity and John Archibald Wheeler

Part of the book series: Astrophysics and Space Science Library ((ASSL,volume 367))

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Abstract

Gravitational fields invariant for a non Abelian Lie algebra generating a 2-dimensional distribution, are explicitly described. When the orthogonal distribution is integrable and the metric is not degenerate along the orbits, these solutions are parameterized either by solutions of a transcendental equation (the tortoise equation), or by solutions of Darboux equation. Metrics, corresponding to solutions of the tortoise equation, are characterized as those that admit a 3-dimensional Lie algebra of Killing fields with 2-dimensional leaves. It is shown that the remaining metrics represent nonlinear gravitational waves obeying to two nonlinearsuperposition laws. The energy and the polarization of this family of waves are explicitly evaluated; it is shown that they have spin$-1$ and their possible sources are also described. Old results by Tolman, Ehrenfest, Podolsky and Wheeler on the gravitational interaction of photons are naturally reinterpreted.

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Notes

  1. 1.

    Ricci flat manifolds of Kleinian signature appear in the ‘no boundary’ proposal of Hartle and Hawking[27] in which the idea is suggested that the signature of the space–time metric may have changed in the early universe. Some other examples of Kleinian geometry in physics occur in the theory of heterotic N = 2 string (see[5, 37]) for which the target space is 4-dimensional.

  2. 2.

    Be square integrable and, consequently, it would be unbounded somewhere. In this case, l μ could not represent an infinitesimal diffeomorphism.

  3. 3.

    Harmonic functions are not globally square integrable, usually there are two possibilities: either they do not decrease at infinity or they have singularities. In the second case however, they can be square integrable if one excludes the singularities from the spacetime.

  4. 4.

    As it will be now shown, the singularities are needed in order to have spatially asymptotically flat solutions.

  5. 5.

    Of course the usual alternative procedure described in almost all textbooks could be followed.

  6. 6.

    Aichelburg-Sexl solution[3] belongs to this class of solutions.

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Acknowledgements

The results here exposed have been obtained in collaboration with F. Canfora, G. Sparano, A. Vinogradov and P. Vitale.

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Authors and Affiliations

  1. Dipartimento di Fisica, Università di Salerno, Istituto Nazionale di Fisica Nucleare, Sezione di Napoli, Via S. Allende, I-84081, Baronissi, Salerno, Italy

    Gaetano Vilasi

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  1. Gaetano Vilasi
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Correspondence to Gaetano Vilasi .

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Editors and Affiliations

  1. Dipto. Ingegneria dell’Innovazione, Università del Salento, Via per Monteroni, Lecce, 73100, Italy

    Ignazio Ciufolini

  2. Dept. Physics, Center for Relativity, University of Texas, Austin, Austin, 78712-1081, USA

    Richard A. Matzner

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Vilasi, G. (2010). Gravitational Fields with 2-Dimensional Killing Leaves and the Gravitational Interaction of Light. In: Ciufolini, I., Matzner, R. (eds) General Relativity and John Archibald Wheeler. Astrophysics and Space Science Library, vol 367. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-3735-0_13

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