Abstract
Based on the quasi-local energy definition of Brown and York, we compute the integral of the trace of the extrinsic curvature over a codimension-2 hypersurface. In particular, we study the difference between the uncompactified Minkowski spacetime and the toroidal Kaluza-Klein compactification. For the latter, we find that this quantity interpolates between zero and the value for the uncompactified spacetime, as the size of the compact dimension increases. Thus, the quasi-local energy is able to discriminate between the two spacetimes.
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References
E. Alvarez, Quantum gravity: a pedagogical introduction to some recent results, Rev. Mod. Phys. 61 (1989) 561 [INSPIRE].
R.L. Arnowitt, S. Deser and C.W. Misner, The dynamics of general relativity, Gen. Rel. Grav. 40 (2008) 1997 [gr-qc/0405109] [INSPIRE].
H. Bondi, M.G.J. van der Burg and A.W.K. Metzner, Gravitational waves in general relativity. 7. Waves from axisymmetric isolated systems, Proc. Roy. Soc. Lond. A 269 (1962) 21.
T. Kaluza, Zum Unitätsproblem der Physik, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys. ) 1921 (1921) 966 [arXiv:1803.08616] [INSPIRE].
O. Klein, Quantum Theory and Five-Dimensional Theory of Relativity. (In German and English), Z. Phys. 37 (1926) 895 [Surveys High Energ. Phys. 5 (1986) 241] [INSPIRE].
E. Witten, Instability of the Kaluza-Klein Vacuum, Nucl. Phys. B 195 (1982) 481 [INSPIRE].
L.B. Szabados, Quasi-Local Energy-Momentum and Angular Momentum in General Relativity, Living Rev. Rel. 12 (2009) 4 [INSPIRE].
S. Hawking, Gravitational radiation in an expanding universe, J. Math. Phys. 9 (1968) 598 [INSPIRE].
R. Penrose, Quasi-Local Mass and Angular Momentum in General Relativity, Proc. Roy. Soc. Lond. A 381 (1982) 53.
R. Bartnik, New definition of quasilocal mass, Phys. Rev. Lett. 62 (1989) 2346 [INSPIRE].
J.D. Brown and J.W. York, Jr., Quasilocal energy and conserved charges derived from the gravitational action, Phys. Rev. D 47 (1993) 1407 [gr-qc/9209012] [INSPIRE].
S.A. Hayward, Quasilocal gravitational energy, Phys. Rev. D 49 (1994) 831 [gr-qc/9303030] [INSPIRE].
C.-C.M. Liu and S.-T. Yau, Positivity of Quasilocal Mass, Phys. Rev. Lett. 90 (2003) 231102 [gr-qc/0303019] [INSPIRE].
M.-T. Wang and S.-T. Yau, Quasilocal mass in general relativity, Phys. Rev. Lett. 102 (2009) 021101 [arXiv:0804.1174] [INSPIRE].
L.P. Eisenhart, Riemannian Geometry, Princeton University Press, Princeton U.S.A. (1964).
R.H. Brandenberger and C. Vafa, Superstrings in the Early Universe, Nucl. Phys. B 316 (1989) 391 [INSPIRE].
S. Watson and R. Brandenberger, Stabilization of extra dimensions at tree level, JCAP 11 (2003) 008 [hep-th/0307044] [INSPIRE].
G.T. Horowitz and B.G. Schmidt, Note on gravitational energy, Proc. Roy. Soc. Lond. A 381 (1982) 215.
L.E. Ibanez and A.M. Uranga, String theory and particle physics: An introduction to string phenomenology, Cambridge University Press, Cambridge U.K. (2012).
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Alvarez, E., Anero, J., del Bosch, G.M. et al. Quasi-local energy and compactification. J. High Energ. Phys. 2018, 69 (2018). https://doi.org/10.1007/JHEP06(2018)069
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DOI: https://doi.org/10.1007/JHEP06(2018)069