In mathematical physics, vanishing scalar invariant (VSI) spacetimes are Lorentzian manifolds with all polynomial curvature invariants of all orders vanishing. Although the only Riemannian manifold with VSI property is flat space, the Lorentzian case admits nontrivial spacetimes with this property. Distinguishing these VSI spacetimes from Minkowski spacetime requires comparing non-polynomial invariants[1] or carrying out the full Cartan–Karlhede algorithm on non-scalar quantities.[2][3]

All VSI spacetimes are Kundt spacetimes.[4] An example with this property in four dimensions is a pp-wave. VSI spacetimes however also contain some other four-dimensional Kundt spacetimes of Petrov type N and III. VSI spacetimes in higher dimensions have similar properties as in the four-dimensional case.[5][6]

References

  1. ↑ Page, Don N. (2009), "Nonvanishing Local Scalar Invariants even in VSI Spacetimes with all Polynomial Curvature Scalar Invariants Vanishing", Classical and Quantum Gravity, 26 (5): 055016, arXiv:0806.2144, Bibcode:2009CQGra..26e5016P, doi:10.1088/0264-9381/26/5/055016, S2CID 118331266
  2. ↑ Koutras, A. (1992), "A spacetime for which the Karlhede invariant classification requires the fourth covariant derivative of the Riemann tensor", Classical and Quantum Gravity, 9 (10): L143, Bibcode:1992CQGra...9L.143K, doi:10.1088/0264-9381/9/10/003, S2CID 250904726
  3. ↑ Koutras, A.; McIntosh, C. (1996), "A metric with no symmetries or invariants", Classical and Quantum Gravity, 13 (5): L47, Bibcode:1996CQGra..13L..47K, doi:10.1088/0264-9381/13/5/002, S2CID 250905968
  4. ↑ Pravda, V.; Pravdova, A.; Coley, A.; Milson, R. (2002), "All spacetimes with vanishing curvature invariants", Classical and Quantum Gravity, 19 (23): 6213–6236, arXiv:gr-qc/0209024, Bibcode:2002CQGra..19.6213P, doi:10.1088/0264-9381/19/23/318, S2CID 11958495
  5. ↑ Coley, A.; Milson, R.; Pravda, V.; Pravdova, A. (2004), "Vanishing Scalar Invariant Spacetimes in Higher Dimensions", Classical and Quantum Gravity, 21 (23): 5519–5542, arXiv:gr-qc/0410070, Bibcode:2004CQGra..21.5519C, doi:10.1088/0264-9381/21/23/014, S2CID 17036677.
  6. ↑ Coley, A.; Fuster, A.; Hervik, S.; Pelavas, N. (2006), "Higher dimensional VSI spacetimes", Classical and Quantum Gravity, 23 (24): 7431–7444, arXiv:gr-qc/0611019, Bibcode:2006CQGra..23.7431C, doi:10.1088/0264-9381/23/24/014, S2CID 85442360


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