Covariant $\overline{\mu }$-scheme effective dynamics, mimetic gravity, and nonsingular black holes: Applications to spherically symmetric quantum gravity

We propose a new $\overline{\mu }$-scheme Hamiltonian effective dynamics in the spherical symmetric spacetime which is generally covariant as derived from a covariant Lagrangian. The Lagrangian belongs to the class of extended mimetic gravity Lagrangians in four dimensions. We apply the effective dynamics to both cosmology and black hole. The effective dynamics reproduces the nonsingular loop-quantum-cosmology effective dynamics. From the effective dynamics, we obtain the nonsingular black hole solution, which has a killing symmetry in addition to the spherical symmetry and reduces to the Schwarzschild geometry asymptotically near the infinity. The black hole spacetime resolves the classical singularity and approaches asymptotically the Nariai geometry ${\mathrm{dS}}_2\times S^2$ at the future infinity in the interior of the black hole. The resulting black hole spacetime has the complete future null infinity ${ℐ}^{+}$. Thanks to the general covariance, the effective dynamics can be reformulated in the light cone gauge.

Figure 1

The figures plot the solution (orange curves) of (99)–(101), with $k=0$ (and $\beta =1$), and compare to the classical FRW cosmology (blue curves). The bouncing time is at $t_c\simeq 0$.

Figure 2

Illustration of constant $t$ slices and constant $z$ slices in Schwarzschild spacetime.

Figure 3

The absolute (blue line) and relative error (orange line) of the numerical solution with $z_0=3\times {10}^8$, $\mathrm{\Delta }=0.01$, $\beta =1$, $R_s={10}^5$. The absolute error is defined as ${\epsilon }_a=\frac{1}{4}{\sum }_i|\frac{\mathrm{d}N[f_i](z)}{\mathrm{d}z}-{\mathcal{F}}_{f_i}(N[f](z))|,f_i\in f=\{{\zeta }_1,{\zeta }_2,\xi ,\psi \}$, where $N[f]$ represents the numerical solution and ${\mathcal{F}}_f(z)$ is given by the equation of motion (122). Similarly, the relative error is defined as ${\epsilon }_r=\frac{1}{4}{\sum }_i|\frac{\frac{\mathrm{d}N[f_i](z)}{z}-{\mathcal{F}}_{f_i}(N[f](z))}{{\mathcal{F}}_{f_i}(N[f](z)}|$.

Figure 4

Plots of $\mathrm{sgn}({\mathrm{\Theta }}_k)$ (orange) and $\mathrm{sgn}({\mathrm{\Theta }}_l)$ (blue) of the numerical solution with $z_0=3\times {10}^8$, $\mathrm{\Delta }=0.01$, $\beta =1$, $R_s={10}^5$. ${\mathrm{\Theta }}_k=0$ is at $z_H\simeq 6.67\times {10}^4\simeq \frac{2}{3}{R}_s$. The correction $|z_H-2R_s/3|\sim {10}^{-9}$.

Figure 5

Plot of Kretschmann invariant $\mathcal{K}$ of the numerical solutions with $z_0=3\times {10}^8$, $\mathrm{\Delta }=0.01$, $R_s={10}^5$.

Figure 6

(a) $\mathrm{\Delta }$ versus the maximum ${\mathcal{K}}_{\max ,0}$, for $z$ inside the neighborhood $N_0$. (b) $\mathrm{\Delta }$ versus the maximum of ${\mathcal{K}}_{\max ,<}$, for $z$ inside the neighborhood $N_{<}$.

Figure 7

The distance $|z|$ between the locations of two $\mathcal{K}$ maxima in $N_0$ and $N_{<}$ and the relations with $\mathrm{\Delta }$ and $R_s$.

Figure 8

Plots of $E^x=e^{2\psi },\log (E^{\phi })=\psi +\xi ,{\zeta }_1\sqrt{\mathrm{\Delta }},{\zeta }_2\sqrt{\mathrm{\Delta }}$ of the numerical solutions in the regime $z<1$ with $z_0=3\times {10}^8$, $\mathrm{\Delta }=0.01$, $R_s={10}^5$.

Figure 9

The blue dots are the numerical values of $r_0$, $a_0$ from the solutions at different values of $\mathrm{\Delta }\in [{10}^{-3},{10}^{-1}]$. The orange lines plot the best fit functions. Other parameters used in the numerics are $z_0=3\times {10}^8$, $R_s={10}^5$.

Figure 10

The conformal diagram of the nonsingular black hole spacetime reduced to 2D covered by $(t,x)$ coordinate. $\mathcal{S}$ (black curve) is a typical spatial slice with constant $t$. Dashed curves are another spatial slice in the far past. The green line illustrate the killing horizon. Near the future infinity, the 4D asymptotic geometry is ${\mathrm{dS}}_2\times S^2$ with Planckian radii.

Figure 11

The conformal diagram of the 2D maximal extension.

Figure 12

This figure plots the trajectory of the $z$ evolution in the $({\zeta }_1,{\zeta }_2)$ space. The arrow indicates the direction of the evolution from $z>0$ to $z<0$. The orange dot indicates the initial value of ${\zeta }_1$, ${\zeta }_2$ at $z_0$. The attractor is the ${\mathrm{dS}}_2\times S^2$ geometry.

Figure 13

(a) The plots of expansions ${\mathrm{\Theta }}_k$ and ${\mathrm{\Theta }}_l$ for $z\le 10$. (b) The plots of the difference ${\mathrm{\Theta }}_k-{\mathrm{\Theta }}_l$.

Figure 14

(a) The plot of the effective tension on $S^2$. (b) The plot of the effective energy density. (c) The effective energy density. (d) The plot of the norm of the effective energy flow.

Figure 15

The subfigures (a) and (b) plot, respectively, the regimes where $E^x$ and $E^{\phi }$ reach their minima in the evolution. The parameters used in the figures are the same as in Fig. 8.

Figure 16

Log-log plot of the conformal factor ${\mathrm{\Omega }}^2$ as a function of $z$.

Figure 17

The conformal diagram of the 2D spacetime from the nonsingular black hole solution. The dashed lines are the $\tau$, $z$ coordinate lines, where constant $z$ lines are timelike outside (spacelike inside) the killing horizon.