Covariant <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mover accent="true"><mml:mi>μ</mml:mi><mml:mo stretchy="false">¯</mml:mo></mml:mover></mml:math>-scheme effective dynamics, mimetic gravity, and nonsingular black holes: Applications to spherically symmetric quantum gravity

We propose a new $\overline{\ensuremath{\mu}}$-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 ${\mathrm{dS}}_{2}\ifmmode\times\else\texttimes\fi{}{S}^{2}$ at the future infinity in the interior of the black hole. The resulting black hole spacetime has the complete future null infinity ${\ensuremath{\mathcal{I}}}^{+}$. Thanks to the general covariance, the effective dynamics can be reformulated in the light cone gauge.

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