GUP Effective Metric Without GUP: Implications for the Sign of GUP Parameter and Quantum Bounce

The standard form of generalized uncertainty principle (GUP) predicts that the Hawking temperature is modified near the Planck scale and that the Bekenstein-Hawking entropy receives a logarithmic correction, consistent with other approaches to quantum gravity. However, due to the heuristic arguments in most GUP literature, it is not clear how to obtain the Schwarzschild metric that incorporates GUP correction. In this work, we try a different approach. We will start with the entropy expression with the standard logarithmic correction term, and use the recently proposed "generalized entropy and varying-G correspondence" (GEVAG) to obtain the associated metric. We show that the Hawking temperature obtained from this metric matches the GUP version. In this sense, we have derived in a consistent and reliable manner, a metric tensor that can describe the standard GUP physics, and use it to clarify some shortcomings in the heuristic GUP approach itself. In particular, if the strict Bekenstein bound is imposed, then the GUP parameter is negative. We also speculate on the possibility that instead of a stable remnant, the final stage of black hole evaporation could be a "bounce" due to an effective gravitational repulsion, once higher order corrections are included.