Quantum black holes in scalar–tensor gravity: a Green function approach

Abstract It is well-known that a Lagrangian resembling scalar–tensor gravity naturally emerges from string theory in the low-energy limit. This suggests that a scalar–tensor theory intrinsically inhabits precious information about quantum gravitational effects. With this motivation, we quantize the scalar–tensor theory $$R+ \lambda R\phi ^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>R</mml:mi> <mml:mo>+</mml:mo> <mml:mi>λ</mml:mi> <mml:mi>R</mml:mi> <mml:msup> <mml:mi>ϕ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> in order to unfold the black hole physics in quantum gravity. We are interested in the behavior of the wave function near the interior singularity of a black hole, which would give a pathway to resolving the singularity. Consequently, we employ canonical quantization in a Kantowski–Sachs minisuperspace representation for the black hole interior. The ensuing Wheeler–DeWitt equation turns out to be intricately coupled among the three minisuperspace variables. Upon a suitable transformation, one of the variables could be easily separated – however, the scalar field $$\phi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ϕ</mml:mi> </mml:math> remains coupled intricately and cannot be separated by any obvious transformation. Consequently, we employ a perturbative approach in powers of $$\lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> </mml:math> . As a result, the zeroth order equation is immediately separable, and the first order equation still remains coupled in two variables. We adopt a Green function approach to solve this first order (inhomogeneous) equation. Our detailed calculation reveals that the Green function is well-defined in the vicinity of the black hole singularity, an important result for admitting the DeWitt criterion for singularity resolution. Furthermore, our calculations illustrate that both the zeroth order and first order wave functions are well-defined at the singularity as they are consistent with the DeWitt criterion, paving the way to black hole singularity resolution in scalar–tensor quantum gravity.