Abstract
There does not exist a notion of time which could be transferred straightforwardly from classical to quantum gravity. For this reason, a method of time quantification which would be appropriate for gravity quantization is being sought. One of the existing proposals is using the evolving matter as an intrinsic ‘clock’ while investigating the dynamics of gravitational systems. The objective of our research was to check whether scalar fields can serve as time variables during a dynamical evolution of a coupled multicomponent matter-geometry system. We concentrated on a neutral case, which means that the elaborated system was not charged electrically nor magnetically. For this purpose, we investigated a gravitational collapse of a self-interacting complex and real scalar fields in the Brans-Dicke theory using the 2+2 spacetime foliation. We focused mainly on the region of high curvature appearing nearby the emerging singularity, which is essential from the perspective of quantum gravity. We investigated several formulations of the theory for various values of the Brans-Dicke coupling constant and the coupling between the Brans-Dicke field and the matter sector of the theory. The obtained results indicated that the evolving scalar fields can be treated as time variables in close proximity of the singularity due to the following reasons. The constancy hypersurfaces of the Brans-Dicke field are spacelike in the vicinity of the singularity apart from the case, in which the equation of motion of the field reduces to the wave equation due to a specific choice of free evolution parameters. The hypersurfaces of constant complex and real scalar fields are spacelike in the regions nearby the singularities formed during the examined process. The values of the field functions change monotonically in the areas, in which the constancy hypersurfaces are spacelike.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
B.S. DeWitt, Quantum theory of gravity. 1. The canonical theory, Phys. Rev. 160 (1967) 1113 [INSPIRE].
C. Rovelli, Time in quantum gravity: physics beyond the Schrödinger regime, Phys. Rev. D 43 (1991) 442 [INSPIRE].
C. Rovelli and L. Smolin, The physical Hamiltonian in nonperturbative quantum gravity, Phys. Rev. Lett. 72 (1994) 446 [gr-qc/9308002] [INSPIRE].
M. Domagala, K. Giesel, W. Kaminski and J. Lewandowski, Gravity quantized: loop quantum gravity with a scalar field, Phys. Rev. D 82 (2010) 104038 [arXiv:1009.2445] [INSPIRE].
J. Lewandowski, M. Domagala and M. Dziendzikowski, The dynamics of the massless scalar field coupled to LQG in the polymer quantization, PoS(QGQGS 2011) 025 [INSPIRE].
S. Alexander, J. Malecki and L. Smolin, Quantum gravity and inflation, Phys. Rev. D 70 (2004) 044025 [hep-th/0309045] [INSPIRE].
A. Ashtekar, T. Pawlowski and P. Singh, Quantum nature of the big bang, Phys. Rev. Lett. 96 (2006) 141301 [gr-qc/0602086] [INSPIRE].
A. Ashtekar, T. Pawlowski and P. Singh, Quantum nature of the big bang: improved dynamics, Phys. Rev. D 74 (2006) 084003 [gr-qc/0607039] [INSPIRE].
M.P. Dabrowski and A.L. Larsen, Quantum tunneling effect in oscillating Friedmann cosmology, Phys. Rev. D 52 (1995) 3424 [gr-qc/9504025] [INSPIRE].
P.W. Graham, B. Horn, S. Kachru, S. Rajendran and G. Torroba, A simple harmonic universe, JHEP 02 (2014) 029 [arXiv:1109.0282] [INSPIRE].
A.T. Mithani and A. Vilenkin, Collapse of simple harmonic universe, JCAP 01 (2012) 028 [arXiv:1110.4096] [INSPIRE].
A.T. Mithani and A. Vilenkin, Tunneling decay rate in quantum cosmology, Phys. Rev. D 91 (2015) 123511 [arXiv:1503.00400] [INSPIRE].
L. Perlov, Wheeler-DeWitt equation for 4D supermetric and ADM with massless scalar field as internal time, Phys. Lett. B 743 (2015) 143 [arXiv:1412.4740] [INSPIRE].
J.D. Brown and K.V. Kuchar, Dust as a standard of space and time in canonical quantum gravity, Phys. Rev. D 51 (1995) 5600 [gr-qc/9409001] [INSPIRE].
X. Zhang, Y. Ma and M. Artymowski, Loop quantum Brans-Dicke cosmology, Phys. Rev. D 87 (2013) 084024 [arXiv:1211.4183] [INSPIRE].
C.R. Almeida, A.B. Batista, J.C. Fabris and P.R. L.V. Moniz, Quantum cosmology with scalar fields: self-adjointness and cosmological scenarios, Gravit. Cosmol. 21 (2015) 191 [arXiv:1501.04170] [INSPIRE].
C. Brans and R.H. Dicke, Mach’s principle and a relativistic theory of gravitation, Phys. Rev. 124 (1961) 925 [INSPIRE].
V. Faraoni, Illusions of general relativity in Brans-Dicke gravity, Phys. Rev. D 59 (1999) 084021 [gr-qc/9902083] [INSPIRE].
C.M. Will, The confrontation between general relativity and experiment, Living Rev. Rel. 17 (2014) 4 [arXiv:1403.7377] [INSPIRE].
C. Romero and A. Barros, Does Brans-Dicke theory of gravity go over to the general relativity when ω → ∞?, Phys. Lett. A 173 (1993) 243 [INSPIRE].
B. Chauvineau, On the limit of Brans-Dicke theory when ω → ∞, Class. Quant. Grav. 20 (2003) 2617 [INSPIRE].
B. Chauvineau, A.D. A.M. Spallicci and J.-D. Fournier, Brans-Dicke gravity and the capture of stars by black holes: some asymptotic results, Class. Quant. Grav. 22 (2005) S457 [gr-qc/0412053] [INSPIRE].
B. Chauvineau, Stationarity and large ω Brans-Dicke solutions versus general relativity, Gen. Rel. Grav. 39 (2007) 297 [INSPIRE].
B. Bertotti, L. Iess and P. Tortora, A test of general relativity using radio links with the Cassini spacecraft, Nature 425 (2003) 374 [INSPIRE].
A. De Felice, G. Mangano, P.D. Serpico and M. Trodden, Relaxing nucleosynthesis constraints on Brans-Dicke theories, Phys. Rev. D 74 (2006) 103005 [astro-ph/0510359] [INSPIRE].
J.C. Fabris, S.V.B. Goncalves and R. de Sa Ribeiro, Late time accelerated Brans-Dicke pressureless solutions and the supernovae type-IA data, Grav. Cosmol. 12 (2006) 49 [astro-ph/0510779] [INSPIRE].
L.-E. Qiang, Y. Gong, Y. Ma and X. Chen, Cosmological implications of 5-dimensional Brans-Dicke theory, Phys. Lett. B 681 (2009) 210 [arXiv:0910.1885] [INSPIRE].
O. Hrycyna, M. Szydlowski and M. Kamionka, Dynamics and cosmological constraints on Brans-Dicke cosmology, Phys. Rev. D 90 (2014) 124040 [arXiv:1404.7112] [INSPIRE].
A. Avilez and C. Skordis, Cosmological constraints on Brans-Dicke theory, Phys. Rev. Lett. 113 (2014) 011101 [arXiv:1303.4330] [INSPIRE].
Y.-C. Li, F.-Q. Wu and X. Chen, Constraints on the Brans-Dicke gravity theory with the Planck data, Phys. Rev. D 88 (2013) 084053 [arXiv:1305.0055] [INSPIRE].
V. Acquaviva, C. Baccigalupi, S.M. Leach, A.R. Liddle and F. Perrotta, Structure formation constraints on the Jordan-Brans-Dicke theory, Phys. Rev. D 71 (2005) 104025 [astro-ph/0412052] [INSPIRE].
T. Damour and K. Nordtvedt, General relativity as a cosmological attractor of tensor scalar theories, Phys. Rev. Lett. 70 (1993) 2217 [INSPIRE].
T. Damour and K. Nordtvedt, Tensor-scalar cosmological models and their relaxation toward general relativity, Phys. Rev. D 48 (1993) 3436 [INSPIRE].
T. Damour, F. Piazza and G. Veneziano, Violations of the equivalence principle in a dilaton runaway scenario, Phys. Rev. D 66 (2002) 046007 [hep-th/0205111] [INSPIRE].
D. La and P.J. Steinhardt, Extended inflationary cosmology, Phys. Rev. Lett. 62 (1989) 376 [Erratum ibid. 62 (1989) 1066] [INSPIRE].
C. Mathiazhagan and V.B. Johri, An inflationary universe in Brans-Dicke theory: a hopeful sign of theoretical estimation of the gravitational constant, Class. Quant. Grav. 1 (1984) L29 [INSPIRE].
M. Arik and M.C. Calik, Primordial and asymptotic inflation in Brans-Dicke cosmology, JCAP 01 (2005) 013 [gr-qc/0403108] [INSPIRE].
M. Arik, M.C. Calik and M.B. Sheftel, Friedmann equation for Brans-Dicke cosmology, Int. J. Mod. Phys. D 17 (2008) 225 [gr-qc/0604082] [INSPIRE].
S. Sen and A.A. Sen, Late time acceleration in Brans-Dicke cosmology, Phys. Rev. D 63 (2001) 124006 [gr-qc/0010092] [INSPIRE].
L.-E. Qiang, Y.-G. Ma, M.-X. Han and D. Yu, 5-dimensional Brans-Dicke theory and cosmic acceleration, Phys. Rev. D 71 (2005) 061501 [gr-qc/0411066] [INSPIRE].
J.P. de Leon, Late time cosmic acceleration from vacuum Brans-Dicke theory in 5D, Class. Quant. Grav. 27 (2010) 095002 [arXiv:0912.1026] [INSPIRE].
J. Cortez, G.A. Mena Marugan, J. Olmedo and J.M. Velhinho, A unique Fock quantization for fields in non-stationary spacetimes, JCAP 10 (2010) 030 [arXiv:1004.5320] [INSPIRE].
Y. Bisabr, Cosmic acceleration in Brans-Dicke cosmology, Gen. Rel. Grav. 44 (2012) 427 [arXiv:1110.3421] [INSPIRE].
Y. Bisabr, On the chameleon Brans-Dicke cosmology, Phys. Rev. D 86 (2012) 127503 [arXiv:1212.2709] [INSPIRE].
L.L. Samojeden, F.P. Devecchi and G.M. Kremer, Fermions in Brans-Dicke cosmology, Phys. Rev. D 81 (2010) 027301 [arXiv:1001.2285] [INSPIRE].
D.-J. Liu, Dynamics of Brans-Dicke cosmology with varying mass fermions, Phys. Rev. D 82 (2010) 063523 [arXiv:1005.5508] [INSPIRE].
S. Nojiri and S.D. Odintsov, Unifying phantom inflation with late-time acceleration: scalar phantom-non-phantom transition model and generalized holographic dark energy, Gen. Rel. Grav. 38 (2006) 1285 [hep-th/0506212] [INSPIRE].
S. Capozziello, S. Nojiri and S.D. Odintsov, Unified phantom cosmology: inflation, dark energy and dark matter under the same standard, Phys. Lett. B 632 (2006) 597 [hep-th/0507182] [INSPIRE].
M.R. Setare, The holographic dark energy in non-flat Brans-Dicke cosmology, Phys. Lett. B 644 (2007) 99 [hep-th/0610190] [INSPIRE].
M.R. Setare and M. Jamil, Holographic dark energy in Brans-Dicke cosmology with chameleon scalar field, Phys. Lett. B 690 (2010) 1 [arXiv:1006.0658] [INSPIRE].
H. Farajollahi, J. Sadeghi, M. Pourali and A. Salehi, Stability analysis of agegraphic dark energy in Brans-Dicke cosmology, Astrophys. Space Sci. 339 (2012) 79 [arXiv:1201.0007] [INSPIRE].
S. Chattopadhyay, A. Pasqua and M. Khurshudyan, New holographic reconstruction of scalar field dark energy models in the framework of chameleon Brans-Dicke cosmology, Eur. Phys. J. C 74 (2014) 3080 [arXiv:1401.8208] [INSPIRE].
O. Hrycyna and M. Szydlowski, Dynamical complexity of the Brans-Dicke cosmology, JCAP 12 (2013) 016 [arXiv:1310.1961] [INSPIRE].
A. Paliathanasis, M. Tsamparlis, S. Basilakos and J.D. Barrow, Classical and quantum solutions in Brans-Dicke cosmology with a perfect fluid, arXiv:1511.00439 [INSPIRE].
A. Einstein and E.G. Straus, The influence of the expansion of space on the gravitation fields surrounding the individual stars, Rev. Mod. Phys. 17 (1945) 120 [INSPIRE].
N. Sakai and J.D. Barrow, Cosmological evolution of black holes in Brans-Dicke gravity, Class. Quant. Grav. 18 (2001) 4717 [gr-qc/0102024] [INSPIRE].
M. Novello and S.E.P. Bergliaffa, Bouncing cosmologies, Phys. Rept. 463 (2008) 127 [arXiv:0802.1634] [INSPIRE].
J.C. Fabris, R.G. Furtado, N. Pinto-Neto and P. Peter, Regular cosmological solutions in low-energy effective action from string theories, Phys. Rev. D 67 (2003) 124003 [hep-th/0212312] [INSPIRE].
D.A. Tretyakova, A.A. Shatskiy, I.D. Novikov and S. Alexeyev, Non-singular Brans-Dicke cosmology with cosmological constant, Phys. Rev. D 85 (2012) 124059 [arXiv:1112.3770] [INSPIRE].
D.A. Tretyakova, B.N. Latosh and S.O. Alexeyev, Wormholes and naked singularities in Brans-Dicke cosmology, Class. Quant. Grav. 32 (2015) 185002 [arXiv:1504.06723] [INSPIRE].
M. Artymowski, Y. Ma and X. Zhang, Comparison between Jordan and Einstein frames of Brans-Dicke gravity a la loop quantum cosmology, Phys. Rev. D 88 (2013) 104010 [arXiv:1309.3045] [INSPIRE].
A. Nakonieczna and J. Lewandowski, Scalar field as a time variable during gravitational evolution, Phys. Rev. D 92 (2015) 064031 [arXiv:1508.05578] [INSPIRE].
R. Torres and F. Fayos, Singularity free gravitational collapse in an effective dynamical quantum spacetime, Phys. Lett. B 733 (2014) 169 [arXiv:1405.7922] [INSPIRE].
R. Torres and F. Fayos, On the quantum corrected gravitational collapse, Phys. Lett. B 747 (2015) 245 [arXiv:1503.07407] [INSPIRE].
C. Vaz, Quantum gravitational dust collapse does not result in a black hole, Nucl. Phys. B 891 (2015) 558 [arXiv:1407.3823] [INSPIRE].
R. Gambini and J. Pullin, An introduction to spherically symmetric loop quantum gravity black holes, AIP Conf. Proc. 1647 (2015) 19 [arXiv:1312.5512] [INSPIRE].
M.A. Scheel, S.L. Shapiro and S.A. Teukolsky, Collapse to black holes in Brans-Dicke theory. 1. Horizon boundary conditions for dynamical space-times, Phys. Rev. D 51 (1995) 4208 [gr-qc/9411025] [INSPIRE].
M.A. Scheel, S.L. Shapiro and S.A. Teukolsky, Collapse to black holes in Brans-Dicke theory. 2. Comparison with general relativity, Phys. Rev. D 51 (1995) 4236 [gr-qc/9411026] [INSPIRE].
D.-I. Hwang and D.-H. Yeom, Responses of the Brans-Dicke field due to gravitational collapses, Class. Quant. Grav. 27 (2010) 205002 [arXiv:1002.4246] [INSPIRE].
J. Hansen and D.-H. Yeom, Charged black holes in string-inspired gravity: I. Causal structures and responses of the Brans-Dicke field, JHEP 10 (2014) 040 [arXiv:1406.0976] [INSPIRE].
J. Hansen and D.-H. Yeom, Charged black holes in string-inspired gravity: II. Mass inflation and dependence on parameters and potentials, JCAP 09 (2015) 019 [arXiv:1506.05689] [INSPIRE].
T. Koivisto and D.F. Mota, Vector field models of inflation and dark energy, JCAP 08 (2008) 021 [arXiv:0805.4229] [INSPIRE].
K. Becker, M. Becker and J.H. Schwarz, String theory and M-theory. A modern introduction, Cambridge University Press, Cambridge U.K. (2007) [INSPIRE].
T.P. Sotiriou and V. Faraoni, f (R) theories of gravity, Rev. Mod. Phys. 82 (2010) 451 [arXiv:0805.1726] [INSPIRE].
M. Gasperini, Elements of string cosmology, Cambridge University Press, Cambridge U.K. (2007).
L. Randall and R. Sundrum, A large mass hierarchy from a small extra dimension, Phys. Rev. Lett. 83 (1999) 3370 [hep-ph/9905221] [INSPIRE].
J. Garriga and T. Tanaka, Gravity in the brane world, Phys. Rev. Lett. 84 (2000) 2778 [hep-th/9911055] [INSPIRE].
H. Kim, B.-H. Lee, W. Lee, Y.J. Lee and D.-H. Yeom, Nucleation of vacuum bubbles in Brans-Dicke type theory, Phys. Rev. D 84 (2011) 023519 [arXiv:1011.5981] [INSPIRE].
R.S. Hamadé and J.M. Stewart, The spherically symmetric collapse of a massless scalar field, Class. Quant. Grav. 13 (1996) 497 [gr-qc/9506044] [INSPIRE].
A. Borkowska, M. Rogatko and R. Moderski, Collapse of charged scalar field in dilaton gravity, Phys. Rev. D 83 (2011) 084007 [arXiv:1103.4808] [INSPIRE].
A. Nakonieczna and M. Rogatko, Dilatons and the dynamical collapse of charged scalar field, Gen. Rel. Grav. 44 (2012) 3175 [arXiv:1209.3614] [INSPIRE].
A. Nakonieczna, M. Rogatko and R. Moderski, Dynamical collapse of charged scalar field in phantom gravity, Phys. Rev. D 86 (2012) 044043 [arXiv:1209.1203] [INSPIRE].
A. Nakonieczna, M. Rogatko and L. Nakonieczny, Dark sector impact on gravitational collapse of an electrically charged scalar field, JHEP 11 (2015) 012 [arXiv:1508.02657] [INSPIRE].
S. Hod and T. Piran, Mass inflation in dynamical gravitational collapse of a charged scalar field, Phys. Rev. Lett. 81 (1998) 1554 [gr-qc/9803004] [INSPIRE].
S. Hod and T. Piran, The inner structure of black holes, Gen. Rel. Grav. 30 (1998) 1555 [gr-qc/9902008] [INSPIRE].
E. Sorkin and T. Piran, The effects of pair creation on charged gravitational collapse, Phys. Rev. D 63 (2001) 084006 [gr-qc/0009095] [INSPIRE].
E. Sorkin and T. Piran, Formation and evaporation of charged black holes, Phys. Rev. D 63 (2001) 124024 [gr-qc/0103090] [INSPIRE].
Y. Oren and T. Piran, On the collapse of charged scalar fields, Phys. Rev. D 68 (2003) 044013 [gr-qc/0306078] [INSPIRE].
J. Hansen, A. Khokhlov and I. Novikov, Physics of the interior of a spherical, charged black hole with a scalar field, Phys. Rev. D 71 (2005) 064013 [gr-qc/0501015] [INSPIRE].
A. Doroshkevich, J. Hansen, D. Novikov, I. Novikov and A. Shatskiy, Physics of the interior of a black hole with an exotic scalar matter, Phys. Rev. D 81 (2010) 124011 [arXiv:0908.1300] [INSPIRE].
S.E. Hong, D.-I. Hwang, E.D. Stewart and D.-H. Yeom, The causal structure of dynamical charged black holes, Class. Quant. Grav. 27 (2010) 045014 [arXiv:0808.1709] [INSPIRE].
D.-I. Hwang and D.-H. Yeom, Internal structure of charged black holes, Phys. Rev. D 84 (2011) 064020 [arXiv:1010.2585] [INSPIRE].
J. Hansen, B.-H. Lee, C. Park and D.-H. Yeom, Inside and outside stories of black-branes in anti de Sitter space, Class. Quant. Grav. 30 (2013) 235022 [arXiv:1307.0266] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1512.06712
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Nakonieczna, A., Yeom, Dh. Scalar field as an intrinsic time measure in coupled dynamical matter-geometry systems. I. Neutral gravitational collapse. J. High Energ. Phys. 2016, 49 (2016). https://doi.org/10.1007/JHEP02(2016)049
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP02(2016)049