Dedicated to the memory of Iaonnis Bakas QFT Corrections to Black Holes Hessamaddin Arfaei In collaboratin with J. Abedi, A. Bedroya, M. N. Kuhani, M. A. Rasulian and K. S. Vaziri Sharif University of Technology arfaei@ipm.ir July 9, 2017 H. Arfaei (SUT) QFT Corrections to Black Holes July 9, 2017 1 / 29
Targets Understanding of QFT fluctuations on background geometry.- by Applying it to back-reaction effects to the BH solutions, and studying its consequences to geometric and thermal of BH s, collapse process, Hawking Radiation. H. Arfaei (SUT) QFT Corrections to Black Holes July 9, 2017 2 / 29
Setup Semi classical approximation Classical Gravity + Quantum Field Theory + Minimal coupling Lagrangian. Semi classical approximation G µν = T µν + T µν µ F µν = µ F µν = 0 where T µν is the the expectation value of T µν resulting from the quantum field fluctuations. In a general curved background it is a quadratic function of the curvature, trace of which has been exactly calculated. H. Arfaei (SUT) QFT Corrections to Black Holes July 9, 2017 3 / 29
History Semiclassical backreaction and black hole formation: A. Paranjape and T. Padmanabhan, Phys.Rev. D80, 044011 (2009), arxiv:0906.1768 [gr-qc]. V. P. Frolov, in Proceedings, 18th International Seminar on High Energy Physics (Quarks 2014) (2014) arxiv:1411.6981 [hep-th]. J. M. Bardeen, (2014), arxiv:1406.4098 [gr-qc]. L. Mersini-Houghton, (2014), 10.1016/j.physletb.2014.09.018, arxiv:1406.1525 [hep-th]. L. Mersini-Houghton and H. P. Pfeiffer, (2014), arxiv:1409.1837 [hep-th]. S. Chakraborty, S. Singh, and T. Padmanabhan, (2015), arxiv:1503.01774 [gr-qc]. H. Kawai, Y. Matsuo, and Y. Yokokura, Int. J. Mod. Phys. A28, 1350050 (2013), arxiv:1302.4733 [hep-th]. H. Kawai and Y. Yokokura, Int. J. Mod. Phys. A30, 1550091 (2015), arxiv:1409.5784 [hep-th]. Quantum effects derived through conformal anomaly lead to an inflationary model that can be either stable or unstable: H. Arfaei (SUT) QFT Corrections to Black Holes July 9, 2017 4 / 29
T µν The backreaction of QFT fluctuations corresponds to an effective energy momentum tensor T µν ; Effective backreaction The trace of this effective tensor is known as trace(conformal) anomaly: T ρ ρ = 32π {(c A + c A )(F + 2 3 R)c A E + c A R} c A = 1 90π (n 0 + 7 4 nm 1 2 c A = 1 90π (1 2 n 0 + 11 4 nm 1 2 + 7 2 nd 1 13n 1 + 212n 2 ) 2 + 11 2 nd 1 2 + 31n 1 + 243n 2 ) n s is the number of spin s particles and M, D respectively stand for Majorana and Dirac spinors. H. Arfaei (SUT) QFT Corrections to Black Holes July 9, 2017 5 / 29
Symmetries + trace anomaly T µν Going beyond trace is not a straight forward, In certain highly symmetric conditions it becomes possible. We are looking for corrections to RN and Schwarzschild solutions; Step 1: We limit our study to stationary spherical charged BHs. Step 2: T ν µ is diagonal T 2 2 = T 3 3 Step 3: Invariance of the Riemann tensor under radial boost T µ ν, would also remain unchanged. Which further implies T r r = T t t. H. Arfaei (SUT) QFT Corrections to Black Holes July 9, 2017 6 / 29
Symmetries + trace anomaly T µν Step 4: Another useful constraint is that T ν µ is divergence free. To implement it we expand T ν µ in terms of inverse power of r. Each term must be divergent free and for the stationary Rotationally invariant background, it implies that each term has to satisfy the constarint. Hence T µ ν = p I T p ( I2 0 0 ( p 2 + 1)I 2 ) r p This is a powerful constraint imposed on the form of T ν µ uniquely find T ν µ from the r-expansion of its trace. such that we can H. Arfaei (SUT) QFT Corrections to Black Holes July 9, 2017 7 / 29
Modified perturbative equations for RN background R µν 1 2 g µνr = 8π T µν + T µν (1) Substituting the zeroth expansion component of metric into the right hand side of equation (1) one can find the next order of the LHS. Up to first order in terms of l p. The smallest perturbative component of T ν µ takes the following form: T ν µ = 3c A l 2 p 4π ( I2 0 0 2I 2 (6c A + c A )l 2 p 16π ) M 2 r 6 + c Alp 2 ( 2I2 0 2π 0 5I 2 ( ) I2 0 Q 4 0 3I 2 and so the modified metric takes the following form: (α = 3 5 + c A f (r) = 1 2M r + Q2 r 2 + c Al 2 p ( 2M 2 r 4 r 8 2MQ2 r 5 + αq4 r 6 10c A ) ) ) MQ 2 r 7 H. Arfaei (SUT) QFT Corrections to Black Holes July 9, 2017 8 / 29
Two different modes of modified solution, small QM lp 2 large QM lp 2 and Modified horizons ( l P M Q l P 1): r ± = M ± M 2 Q 2 c Alp 2 ( Mβ± 2 1 + α 2 where β ± = 1 ± 1 (Q/M) 2. Modified horizons ( Q l P l P M 1): (β ± 2) 2 ) β ± 1 r + = 2M c Al 2 P 4M Q2 2M r = c 1 3 A l 2 3 P M 1 3 6Q 2 M H. Arfaei (SUT) QFT Corrections to Black Holes July 9, 2017 9 / 29
Extremal black holes (QM l 2 P ) For classical RN solution without quantum corrections f approaches f 0 (r, M, Q) = 1 2M r + Q2, resulting in the extremal values of Q and r r 2 ±, r ± = Q = M Quamtum correction modifies extremal solutions as following: Q = M c A l 2 p r ± = M + c A l 2 p α 2M 2α 1 M H. Arfaei (SUT) QFT Corrections to Black Holes July 9, 2017 10 / 29
Extremal black holes (QM l 2 P ) For QM lp 2, we should expand in terms of Q l P changes the extremal solutions as following: rather than l P M. This 32 3 M ext = 27 c Q 2 Al p + 32 ca l p 8 3 r ± = 3 c Q 2 Al p + 512 ca l p H. Arfaei (SUT) QFT Corrections to Black Holes July 9, 2017 11 / 29
Energy density of EM field vs Quamtum fluctuations 1.5 1 Inner horizon Outer horizon T 00 /m p 0.5 0-0.5 0 5 10 15 20 r/l P H. Arfaei (SUT) QFT Corrections to Black Holes July 9, 2017 12 / 29
Energy density of EM field vs Quamtum fluctuations 1.5 1 Inner horizon Outer horizon T 00 /m p 0.5 0-0.5 0 5 10 15 20 r/l P H. Arfaei (SUT) QFT Corrections to Black Holes July 9, 2017 13 / 29
Modification on thermal properties, temperature The Hawking temperature takes quantum correction as well, ( T H = TH RN c Alp 2 β 1 πm 3 β 5 1 + 3α (β 2) 2 ) (β 2/3) 4 (β 1) 2 H. Arfaei (SUT) QFT Corrections to Black Holes July 9, 2017 14 / 29
Modification on thermal properties, temperature Where TH RN = (β 1) /( 2πMβ 2) is the classical temperature. Note that the line Q = M, or β = 1 + 1 (Q/M) 2 = 1 no longer represents the T = 0 state; the first term of the correction vanishes at this line but the second term diverges negatively. This is another indication that the classically extremal case has moved to a nonphysical region with naked singularity as the result of quantum correction. H. Arfaei (SUT) QFT Corrections to Black Holes July 9, 2017 15 / 29
Modification on thermal properties, temperature The T = 0 curve deviates from its asymptote as the mass moves down from large values. Hence the deviation from the classical extremality condition becomes larger for small mass and charge, i.e. the charge to mass ratio becomes smaller than the classical case to the extent that the zero charge limit acquires a finite mass. The finite mass of the zero charge limit of the extremal case is of the order of Planck s mass. H. Arfaei (SUT) QFT Corrections to Black Holes July 9, 2017 16 / 29
Modification on thermal properties, phase transition 0.04 Classical Temperature Corrected Temperature Critical point 0.03 T/T P 0.02 0.01 0 0 2 4 6 8 10 M/m P 1000 800 600 CQ=0 = TH S TH 400 200 0 200 400 600 800 1000 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 M H. Arfaei (SUT) QFT Corrections to Black Holes July 9, 2017 17 / 29
Modification on thermal properties, phase transition Phase transition maximum temperature H. Arfaei (SUT) QFT Corrections to Black Holes July 9, 2017 18 / 29
Modification on thermal properties, maximum temperature 0.05 Increasing the Q decreasing of T Max 0.045 Max Temperature (T/T P ) 0.04 0.035 0.03 0.025 0.02 0.015 0.01 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Electric charge (Q/q P ) H. Arfaei (SUT) QFT Corrections to Black Holes July 9, 2017 19 / 29
Modification on thermal properties, entropy On the outer horizon, g 00 = f (Q, M, r) vanishes so we have, ) f dm + f ) dq + f ) d r + = 0 M r +,Q Q r +,M r + M,Q dm = f Q f M ) ) r +,M r +,Q dq T H ) f M r +,Q 4πd r + Comparison with the standard thermodynamic relation de = TdS + φdq gives the expression for ds, ( ) f 1 ds = 4πdr + = M 2πr + dr ( ( + 1 c A lp 2 2M r+ 3 Q2 r 4 + )) H. Arfaei (SUT) QFT Corrections to Black Holes July 9, 2017 20 / 29
Modification on thermal properties, entropy Modified entropy First order quantum corrections gives logarithmic correction to the entropy. S Classic = A 4 S Quantum = A 4 + πc Alp 2 ln(a) 10 Classical entropy Corrected entropy 8 6 S/S P 4 2 0 0 2 4 6 8 10 2 A/l P H. Arfaei (SUT) QFT Corrections to Black Holes July 9, 2017 21 / 29
Modification on thermal properties, chemical potential The chemical potential is defined to be the work needed to adiabatically increase the ADM mass, M, per unit of eletric charge, Q. In the classical case the µ is given by Q r +. After applying quantum correction, entropy gets modifed and so the adiapatic paths get correction. Figure: M = 1m P (Left) and M = 10m P (Right) 0.7 0.6 Classical chemical potential Corrected chemical potenital 0.7 0.6 Classical chemical potential Corrected chemical potenital 0.5 0.5 0.4 0.4 Q/q P Q/q P 0.3 0.3 0.2 0.2 0.1 0.1 0 0 0.2 0.4 0.6 0.8 µ/m P 0 0 2 4 6 8 µ/m P H. Arfaei (SUT) QFT Corrections to Black Holes July 9, 2017 22 / 29
Modification on thermal properties, chemical potential Physics behind this correction After applying quantum correction, entropy gets modifed and so the adiapatic paths get correction. Hence in order to keep the entropy unchanged under falling of a charged particle with electric charge dq and rest mass dm, dm cannot be zero anymore, like the classical case. This nonzero restmass results in a gravitational work appearing in the chemical potental. µ = Q r + [1 c A l 2 p ( 2 r 2 + + )] (1 2α)Q2 r+ 4 H. Arfaei (SUT) QFT Corrections to Black Holes July 9, 2017 23 / 29
Hawking radiation, third law - Third law of Thermodynamics states that reaching zero temperature in finite time should be impossible. - Before studying the quantum corrections, blackholes were belived to evaporate in finite time. But now we know the fate of any black hole will be extremal, even if it s not electrically charged! -So the Third Law states that the lifetime of all black holes must be infinite. Any small quantum correction results in such a strong change H. Arfaei (SUT) QFT Corrections to Black Holes July 9, 2017 24 / 29
Hawking radiation, third law Here is the consistency check Let us take the black body radiation for black hole, dm dt = σa HT 4 H = σ(4πr 2 +) ( f r r=r+ ) 4 (4π) 4 (2) Using the above equation, we shall find lifetime of black hole τ: rext τ = (4π) 3 σ r+ 0 dm dr + dr + f 4 (r + )r 2 + (3) This integral diverges and so the lifetime of black holes is infinite, as it was supposed to be. H. Arfaei (SUT) QFT Corrections to Black Holes July 9, 2017 25 / 29
Remnant, extremal Schwarzschild black hole, Candidate for dark matter H. Arfaei (SUT) QFT Corrections to Black Holes July 9, 2017 26 / 29
Conclusion Conclusions: 1)Change in Extermality Condition 2) Change in the radii of the Horizons 3) Minimum mass for black holes 4)Logarithmic correction to Entropy 5)Correction to thermal quantities such as Chemical potential and specific heat 6)Maximum temperature for Black Holes 7) evasion of singularity in the process of collapse.. 8)Existence of a remnant after the evaporation with zero temperature which can be a candidate for dark matter H. Arfaei (SUT) QFT Corrections to Black Holes July 9, 2017 27 / 29
Thank you H. Arfaei (SUT) QFT Corrections to Black Holes July 9, 2017 28 / 29