Fermion's Tunnelling with Effects of Quantum Gravity

Abstract

In this paper, using Hamilton-Jacobi method, we address the tunnelling of fermions in a 4-dimensional Schwarzschild spacetime. Based on the generalized uncertainty principle, we introduce the influence of quantum gravity. After solving the equation of motion of the spin-1/2 field, we derive the corrected Hawking temperature. It turns out that the correction depends not only on the black hole&#x2019;s mass but also on the mass (energy) of emitted fermions. It is of interest that, in our calculation, the quantum gravity correction decelerates the temperature increase during the radiation explicitly. This observation then naturally leads to the remnants in black hole evaporation. Our calculation shows that the residue mass is <svg style="vertical-align:-4.74141pt;width:63.075001px;" id="M1" height="19.362499" version="1.1" viewBox="0 0 63.075001 19.362499" width="63.075001" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.162)"><path id="x2273" d="M531 285l-474 -214v56l416 183l-416 184v56l474 -215v-50zM538 -11q-54 -70 -120 -70q-47 0 -137 43q-78 38 -108 38q-55 0 -91 -57l-31 33q51 74 122 74q45 0 137 -44q76 -37 108 -37q55 0 91 57z" /></g><g transform="matrix(.017,-0,0,-.017,14.749,12.162)"><path id="x1D440" d="M998 650l-8 -28q-71 -4 -86 -16t-22 -69l-50 -397q-3 -28 -4.5 -44t2 -29t6.5 -18.5t17 -10.5t24.5 -6.5t37.5 -3.5l-8 -28h-271l7 28q63 6 78 22t25 90l60 415h-2l-353 -552h-23l-130 536h-2l-70 -254q-44 -158 -47 -188q-5 -38 9 -51t71 -18l-6 -28h-241l8 28&#xA;q45 4 67 18.5t35 45.5q16 38 74 233l52 173q24 79 11.5 98t-89.5 26l6 28h177l136 -508l337 508h172z" /></g> <g transform="matrix(.012,-0,0,-.012,32.112,16.25)"><path id="x1D45D" d="M570 304q0 -108 -87 -199q-40 -42 -94.5 -74t-105.5 -43q-41 0 -65 11l-29 -141q-9 -45 -1.5 -58t45.5 -16l26 -2l-5 -29l-241 -10l4 26q51 10 67.5 24t26.5 60l113 520q-54 -20 -89 -41l-7 26q38 28 105 53l11 49q20 25 77 58l8 -7l-17 -77q39 14 102 14q82 0 119 -36&#xA;t37 -108zM482 289q0 114 -113 114q-26 0 -66 -7l-70 -327q12 -14 32 -25t39 -11q59 0 118.5 81.5t59.5 174.5z" /></g> <g transform="matrix(.017,-0,0,-.017,39.8,12.162)"><path id="x2F" d="M368 703l-264 -866h-60l265 866h59z" /></g><g transform="matrix(.017,-0,0,-.017,46.803,12.162)"><path id="x1D6FD" d="M558 587q0 -32 -14 -61t-40 -53.5t-48.5 -41t-54.5 -36.5q144 -51 144 -174q0 -55 -43.5 -108t-104.5 -87q-77 -42 -131 -42q-31 0 -54 20t-31 47l11 18q48 -29 108 -29q79 0 119.5 43t40.5 109t-44.5 107.5t-119.5 50.5l22 47q34 1 65 21q96 61 96 157q0 42 -24 67.5&#xA;t-62 25.5q-24 0 -43.5 -9t-35 -29.5t-27 -44t-22.5 -63t-19.5 -75.5t-18.5 -91q-57 -294 -68 -380q-26 -190 -35 -200q-26 -31 -97 -37l-4 26q19 9 48 170l77 413q23 121 52.5 187.5t83.5 114.5q70 62 148 62q51 0 88.5 -34t37.5 -91z" /></g> <g transform="matrix(.012,-0,0,-.012,56.675,17.2)"><path id="x30" d="M241 635q53 0 94 -28.5t63.5 -76t33.5 -102.5t11 -116q0 -58 -11 -112.5t-34 -103.5t-63.5 -78.5t-94.5 -29.5t-95 28t-64.5 75t-34.5 102.5t-11 118.5q0 58 11.5 112.5t34.5 103t64.5 78t95.5 29.5zM238 602q-32 0 -55.5 -25t-35.5 -68t-17.5 -91t-5.5 -105&#xA;q0 -76 10 -138.5t37 -107.5t69 -45q32 0 55.5 25t35.5 68.5t17.5 91.5t5.5 105t-5.5 105.5t-18 92t-36 68t-56.5 24.5z" /></g> </svg>, where <svg style="vertical-align:-4.74141pt;width:25.1625px;" id="M2" height="18.3125" version="1.1" viewBox="0 0 25.1625 18.3125" width="25.1625" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D440"/></g> <g transform="matrix(.012,-0,0,-.012,17.412,15.188)"><use xlink:href="#x1D45D"/></g> </svg> is the Planck mass and <svg style="vertical-align:-3.25793pt;width:16.3375px;" id="M3" height="17.4" version="1.1" viewBox="0 0 16.3375 17.4" width="16.3375" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.162)"><use xlink:href="#x1D6FD"/></g> <g transform="matrix(.012,-0,0,-.012,9.938,17.2)"><use xlink:href="#x30"/></g> </svg> is a dimensionless parameter accounting for quantum gravity effects. The evaporation singularity is then avoided.