Geometry of a generalized uncertainty-inspired spacetime

Author(s)

Gingrich, Douglas M., Rastgoo, Saeed

Abstract

We examine the geometry of a generalized uncertainty-inspired quantum black hole. The diagonal line element is not $t$-$r$ symmetric, i.e. $g_{00} \ne -1/g_{11}$, which leads to an interesting approach to resolving the classical curvature singularity. In this paper, we show, in Schwarzschild coordinates, the $r = 0$ coordinate location is a null surface which is not a transition surface or leads to a black bounce. We find the expansion of null geodesic congruences in the interior turn around then vanishes at $r = 0$, and the energy conditions are predominately violated indicating a repulsive gravitational core. In addition, we show that the line element admits a wormhole solution which is not traversable, and the black hole at its vanishing horizon radius could be interpreted as a remnant.

Figures

Metric components for the GUP-modified black hole with $m = 1$, $Q_b = 0.1$, $Q_c = 10^{-6}$ (solid lines) and the classical black hole with $m = 1$ (dashed lines).

Metric components for the GUP-modified black hole with $m = 1$, $Q_b = 0.1$, $Q_c = 10^{-6}$ (solid lines) and the classical black hole with $m = 1$ (dashed lines).


: Radial geodesics with $E = 1$.

: Radial geodesics with $E = 1$.


: Expansion scalars.

: Expansion scalars.


: Exterior.

: Exterior.


: Interior.

: Interior.


Density and pressure combinations needed to determine the energy conditions for $m > \sqrt{Q_b}/2$ with $m = 1, Q_b = 0.1$, and $Q_c = 10^{-6}$.

Density and pressure combinations needed to determine the energy conditions for $m > \sqrt{Q_b}/2$ with $m = 1, Q_b = 0.1$, and $Q_c = 10^{-6}$.


: Radial geodesics with $E=4$.

: Radial geodesics with $E=4$.


: Expansion scalars.

: Expansion scalars.


: Carter-Penrose diagram.

: Carter-Penrose diagram.


: GUP wormhole with $m=1$, $Q_b = 5$, and $Q_c = 10^{-6}$.

: GUP wormhole with $m=1$, $Q_b = 5$, and $Q_c = 10^{-6}$.


Density and pressure combinations needed to determine the energy condition for $m < \sqrt{Q_b}/2$ with $m = 1, Q_b = 5$, and $Q_c = 10^{-6}$.

Density and pressure combinations needed to determine the energy condition for $m < \sqrt{Q_b}/2$ with $m = 1, Q_b = 5$, and $Q_c = 10^{-6}$.


: Temperature, $T$.

: Temperature, $T$.


: Heat capacity, $C_V$.

: Heat capacity, $C_V$.


Entropy and area versus mass for $Q_b = 0.1$ and $Q_c = 10^{-6}$. Also shown is the classical entropy.

Entropy and area versus mass for $Q_b = 0.1$ and $Q_c = 10^{-6}$. Also shown is the classical entropy.


Tortoise coordinate $r_*$ versus $r$. Discontinuities appear at $r = r_h$ and $r = 0$.

Tortoise coordinate $r_*$ versus $r$. Discontinuities appear at $r = r_h$ and $r = 0$.


References
  • [1] A. Kempf, G. Mangano and R.B. Mann, Hilbert space representation of the minimal length uncertainty relation, Phys. Rev. D 52 (1995) 1108 [hep-th/9412167].
  • [2] P. Bosso, G.G. Luciano, L. Petruzziello and F. Wagner, 30 years in: Quo vadis generalized uncertainty principle?, Class. Quant. Grav. 40 (2023) 195014 [2305.16193].
  • [3] F. Fragomeno, D.M. Gingrich, S. Hergott, S. Rastgoo and E. Vienneau, A generalized uncertainty-inspired quantum black hole, 2406.03909.
  • [4] A. Addazi et al., Quantum gravity phenomenology at the dawn of the multi-messenger era—A review, Prog. Part. Nucl. Phys. 125 (2022) 103948 [2111.05659].
  • [5] LISA collaboration, New horizons for fundamental physics with LISA, Living Rev. Rel. 25 (2022) 4 [2205.01597].
  • [6] LISA Cosmology Working Group collaboration, Cosmology with the Laser Interferometer Space Antenna, Living Rev. Rel. 26 (2023) 5 [2204.05434].
  • [7] R. Alves Batista et al., White Paper and Roadmap for Quantum Gravity Phenomenology in the Multi-Messenger Era, 2312.00409.
  • [8] L. Modesto, Semiclassical loop quantum black hole, Int. J. Theor. Phys. 49 (2010) 1649 [0811.2196].
  • [9] A. Ashtekar and J. Olmedo, Properties of a recent quantum extension of the Kruskal geometry, Int. J. Mod. Phys. D 29 (2020) 2050076 [2005.02309].
  • [10] Y.-C. Liu, J.-X. Feng, F.-W. Shu and A. Wang, Extended geometry of Gambini-Olmedo-Pullin polymer black hole and its quasinormal spectrum, Phys. Rev. D 104 (2021) 106001 [2109.02861].
  • [11] R. Gambini, J. Olmedo and J. Pullin, Loop Quantum Black Hole Extensions Within the Improved Dynamics, Front. Astron. Space Sci. 8 (2021) 74 [2012.14212].
  • [12] A. Alonso-Bardaji, D. Brizuela and R. Vera, Nonsingular spherically symmetric black-hole model with holonomy corrections, Phys. Rev. D 106 (2022) 024035 [2205.02098].
  • [13] A. Ashtekar, J. Olmedo and P. Singh, Quantum extension of the Kruskal spacetime, Phys. Rev. D 98 (2018) 126003 [1806.02406].
  • [14] A. Ashtekar and M. Bojowald, Quantum geometry and the Schwarzschild singularity, Class. Quant. Grav. 23 (2006) 391 [gr-qc/0509075].
  • [15] A. Ashtekar, T. Pawlowski and P. Singh, Quantum Nature of the Big Bang: Improved dynamics, Phys. Rev. D 74 (2006) 084003 [gr-qc/0607039].
  • [16] D.-W. Chiou, Phenomenological loop quantum geometry of the Schwarzschild black hole, Phys. Rev. D 78 (2008) 064040 [0807.0665].
  • [17] D.-W. Chiou, W.-T. Ni and A. Tang, Loop quantization of spherically symmetric midisuperspaces and loop quantum geometry of the maximally extended Schwarzschild spacetime, 1212.1265.
  • [18] P. Bosso, O. Obregón, S. Rastgoo and W. Yupanqui, Deformed algebra and the effective dynamics of the interior of black holes, Class. Quant. Grav. 38 (2021) 145006 [2012.04795].
  • [19] A. Ashtekar, J. Olmedo and P. Singh, Quantum Transfiguration of Kruskal Black Holes, Phys. Rev. Lett. 121 (2018) 241301 [1806.00648].
  • [20] E. Curiel, A Primer on Energy Conditions, Einstein Stud. 13 (2017) 43 [1405.0403].
  • [21] S.W. Hawking and G.F.R. Ellis, The large scale structure of space-time, Cambridge University Press (1973).
  • [22] I. Cho and H.-C. Kim, Simple black holes with anisotropic fluid, Chin. Phys. C 43 (2019) 025101 [1703.01103].
  • [23] M. Visser, Lorentizian Wormholes: from Einstein to Hawking, Springer-Verlag (1995).
  • [24] R. Carballo-Rubio, F. Di Filippo, S. Liberati and M. Visser, Geodesically complete black holes, Phys. Rev. D 101 (2020) 084047 [1911.11200].
  • [25] S. Rastgoo and S. Das, Probing the Interior of the Schwarzschild Black Hole Using Congruences: LQG vs. GUP, Universe 8 (2022) 349 [2205.03799].
  • [26] A. Simpson and M. Visser, Black-bounce to traversable wormhole, JCAP 02 (2019) 042 [1812.07114].
  • [27] A. Ashtekar, J. Olmedo and P. Singh, Regular black holes from Loop Quantum Gravity, 2301.01309.
  • [28] L. Modesto and I. Premont-Schwarz, Self-dual black holes in LQG: Theory and phenomenology, Phys. Rev. D 80 (2009) 064041 [0905.3170].
  • [29] R. Gambini, J. Olmedo and J. Pullin, Spherically symmetric loop quantum gravity: analysis of improved dynamics, Class. Quant. Grav. 37 (2020) 205012 [2006.01513].
  • [30] J.M. Bardeen, Non-singular general-relativisitc gravitational collaps, In Proc. Int. Conf. GR5, Tbilisi 174 (1968) .
  • [31] T.A. Roman and P.G. Bergmann, Stellar collapse without singularities?, Phys. Rev. D 28 (1983) 1265.
  • [32] V.P. Frolov, Notes on nonsingular models of black holes, Phys. Rev. D 94 (2016) 104056 [1609.01758].
  • [33] S.A. Hayward, Formation and evaporation of nonsingular black holes, Phys. Rev. Lett. 96 (2006) 031103.
  • [34] F. Di Filippo, R. Carballo-Rubio, S. Liberati, C. Pacilio and M. Visser, On the Inner Horizon Instability of Non-Singular Black Holes, Universe 8 (2022) 204 [2203.14516].
  • [35] M.R. Francis and A. Kosowsky, Geodesics in the generalized Schwarzschild solution, Am. J. Phys. 72 (2004) 1204 [gr-qc/0311038].
  • [36] R. Penrose, Conformal treatement of infinity, in Relativity, groups and topology, C. de Witt and B. de Witt, eds., (republised (2011) Gen Rel. Grav. 43 901-922), pp. 563–584, Gordon and Breach, New York (1964), DOI.