Generalized and Extended Uncertainty Principles and their impact onto the Hawking radiation Mariusz P. Da¸browski Institute of Physics, University of Szczecin, Poland National Centre for Nuclear Research, Otwock, Poland Copernicus Center for Interdisciplinary Studies, Krakow,´ Poland ICNFP2019, Kolymbari 28 August 2019 Generalized and Extended Uncertainty Principles and their impact onto the Hawking radiation – p. 1/44 Plan: 1. Introduction. 2. Generalised Uncertainty Principle (GUP) and black hole thermodynamics 3. GUP influence onto Hawking radiation and its sparsity 4. Extended Uncertainty Principle (EUP) and GEUP duality. 5. Background geometry determined EUP (Rindler and Friedmann) and black hole thermodynamics 6. Conclusions. Generalized and Extended Uncertainty Principles and their impact onto the Hawking radiation – p. 2/44 References A. Alonso-Serrano, MPD, H. Gohar, GUP impact onto black holes information flux and the sparsity of Hawking radiation, Phys. Rev. D97, 044029 (2018) (arXiv: 1801.09660). A. Alonso-Serrano, MPD, H. Gohar, Minimal length and the flow of entropy from black holes, International Journal of Modern Physics D47, 028 (2018) (arXiv: 1805.07690). MPD, F. Wagner, Extended Uncertainty Principle for Rindler and cosmological horizons, EPJC to appear (2019), arXiv: 1905.09713 see also: MPD, H. Gohar, Phys. Lett. B748, 428 (2015). Generalized and Extended Uncertainty Principles and their impact onto the Hawking radiation – p. 3/44 1. Introduction It is believed that quantum gravity (QG) will add some new elements both into the relativity theory and into quantum mechanics (QM). One of the issues from relativistic side which is expected to emerge is Lorentz symmetry violation. From quantum side an issue is the modification of basic QM and, in particular, its uncertainty principle to include gravitational effects. The most suitable objects in which both relativistic and quantum effects show up are the black holes which are subject of black hole thermodynamics. In view of the recent detections of gravitational waves one may ask question of what are the effects of quantum gravity on the phenomenon of black hole mergers for example. In this talk I will concentrate on the effect of modified uncertainty principles onto the thermodynamics of black holes from both Planck and cosmological scales. Generalized and Extended Uncertainty Principles and their impact onto the Hawking radiation – p. 4/44 2. Generalized Uncertainty Principle (GUP) and black hole ther- modynamics Minimum length in quantum mechanics The minimum energy of a classical hydrogen atom p2 e2 E = (1) 2m − r at r = p = 0 is large and negative. This leads to a collapse of an atom. Quantum mechanics requires introduction of Heisenberg Uncertainty Principle (HUP) which makes the measurement ”fuzzy” ~ p (2) ≈ r and so the energy is ~2 e2 E = (3) 2mr2 − r and it has a minimum (Rydberg energy) E = me4/2~2 for the minimum min − length (Bohr radius) r = ~2/me2. min Generalized and Extended Uncertainty Principles and their impact onto the Hawking radiation – p. 5/44 GUP derivation Minimum length in quantum gravity While calculating the uncertainty for HUP one does not include the uncertainty due to gravitational interaction. Suppose we have an electron observed by a photon of momentum p so the HUP uncertainty of position is given by ~ ∆x . (4) ∼ ∆p This, however, should be appended with the uncertainty which comes from gravitational interaction of an electron and a photon which we can write down as ′ ∆(photon s energy) c∆p c∆p G∆p 2 ∆p ∆x = = 4 = = l , (5) 1 ∼ 4 maximum force 4F c c3 p ~ × max G 2 ~ 3 4 where lp = G /c is the Planck length, maximum force Fmax = c /4G. Generalized and Extended Uncertainty Principles and their impact onto the Hawking radiation – p. 6/44 Minimum length in quantum gravity regime This leads to the Generalized Uncertainty Principle ~ ∆p ~ α 2 ∆x = ∆x + ∆x + l2 = + ~ ∆p f(∆p), (6) GUP 1 ≥ ∆p p ~ ∆p α ≡ 0 where lpl α = α0 ~ −1 −1 is the constant with the dimension of inverse momentum kg m s, and α0 is a dimensionless constant which can be determined from data (e.g. Adler 2001). Assuming that the rhs of (6) is the function f(∆p) we can calculate its minimum which is reached for ∆p = ~ so that the minimum length uncertainty is now lp ∆x = f(∆p = ~/lp) = 2lp (7) which means that the Planck length plays the role of minimum or fundamental distance in quantum gravity regime. Generalized and Extended Uncertainty Principles and their impact onto the Hawking radiation – p. 7/44 Simple Newtonian derivation Gravitational interaction of an electron due to photon of mass E/c2 is (Adler & Santiago Mod. Phys. Lett. A14, 1371 (1999)) G(E/c2) ~r ~a = ~r¨ = (8) − r2 r and the interaction takes place in a characteristic region of length L r and a ∼ characteristic time t L/c, where r is the photon-electron distance. ∼ Then the velocity acquired by an electron and the distance it is moved are GE L GE L2 GE Gp ∆v , ∆x , (9) ∼ c2r2 c 1 ∼ c2r2 c2 ∼ c4 ∼ c3 which then leads to GUP as in (6). Alternative derivations are based on: string theory (e.g. Scardigli PLB452, 39 (1999)); LQG (Ashtekar et al CQG 20, 1031 (2003)); non-commutative spaces etc. Generalized and Extended Uncertainty Principles and their impact onto the Hawking radiation – p. 8/44 HUP minimum length and Hawking temperature Now assuming that near the horizon of a Schwarzshild black hole, the HUP position uncertainty has a minimum value (7) and the Planck is just the horizon 2 size lp = 2GM/c , we can recover Hawking temperature ~c ~c3 ∆pc = k T, (10) ≈ ∆x 4GM ≈ B which after including a “calibration factor” of 2π gives ~c3 c2 m2 T = = p , (11) 8πGkBM 8πkB M 2 ~ where, mp = c/G is the Planck mass, and kB is the Boltzmann constant. Generalized and Extended Uncertainty Principles and their impact onto the Hawking radiation – p. 9/44 GUP minimum length and Hawking temperature Similarly, using GUP, we can derive generalised Hawking temperature TGUP . To do this we first express ∆p in terms of ∆x using (6) ∆x ∆x 4~2α2 ∆p = 1 , (12) 2~α2 ∓ 2~α2 − (∆x)2 s and expand in series as follows ~ ~2α2 ~4α4 ∆p 1+ + 2 + ... (13) ≥ ∆x (∆x)2 (∆x)4 2 taking again ∆x = 2lp = 4GM/c and including the calibration factor into each term, we get (T is the Hawking temperature) 4α2π2k2 4α2π2k2 2 T = T 1+ B T 2 + 2 B T 4 + ... (14) GUP c2 c2 " # Generalized and Extended Uncertainty Principles and their impact onto the Hawking radiation – p. 10/44 GUP corrected Bekenstein entropy 2 Using 1st law of thermodynamics dSGUP = c dM/TGUP , after integration we obtain generalised Bekenstein entropy 2 2 2 4 4 4 2 2 α c mpkBπ S α c mpkBπ 1 SGUP = S ln + + ..., (15) − 4 S0 4 S where S is the Bekenstein entropy for a Schwarzschild black hole: A k c3 4πk GM 2 M 2 S = B = B = 4πk , (16) 4 ~G ~c B m p 3 with the integration constant S0 =(A0c kB)/4~G (A0 = const. with the unit of area) to keep logarithmic term dimensionless. Generalized and Extended Uncertainty Principles and their impact onto the Hawking radiation – p. 11/44 3. GUP influence onto Hawking radiation and its sparsity In the paper by Alonso-Serrano and Visser (PLB 57, 383 (2017)) it was calculated the entropy released during standard thermodynamic process of burning a lump of coal in a blackbody furnace and the reasoning was extended into the black hole evaporation. Firstly, they introduced the units of nats and bits Sˆ = S/kB Sˆ2 = S/(kB ln2) and calculated an average entropy flow in blackbody radiation π4 Sˆ = bits/photon 3.90 bits/photon, h 2i 30ζ(3)ln2 ≈ with the standard deviation to be (ζ(n) is the Riemann zeta function) 1 12ζ(5) π4 2 σ ˆ = bits/photon 2.52 bits/photon. (17) S2 ln2s ζ(3) − 30ζ(3) ≈ Generalized and Extended Uncertainty Principles and their impact onto the Hawking radiation – p. 12/44 Emitted information and the Hawking radiation It emerged that the Bekenstein entropy loss per emitted massless boson is equal to the entropy content per photon in blackbody radiation of a Schwarzshild black hole (Alonso-Serrano, Visser PLB 776, 10 (2018)). Information emitted by a black hole is perfectly compensated by the entropy gain of the radiation. What is mostly of our interest from these calculations is an estimate of the total number of emitted quanta in terms of the original Bekenstein entropy S which was found to be 30ζ(3) N = Sˆ 0.26 S.ˆ π4 ≈ We will extend this calculation onto the GUP case. Generalized and Extended Uncertainty Principles and their impact onto the Hawking radiation – p. 13/44 GUP corrected number of emitted Hawking quanta We start with the mass element E ~ ω dM = h idN = h idN, (18) c2 c2 where an average energy π4k E = ~ ω = B T . (19) h i h i 30ζ(3) GUP From these we can calculate the GUP modified Bekenstein entropy loss of a black hole 2 2 2 4 4 4 2 2 dSGUP dS/dt α c mpkBπ 1 α c mpkBπ 1 = 1 2 + ... , dN dN/dt × − 4 S − 4 S ! where standard (non-GUP) Bekenstein entropy loss is dS dS/dt 8πkB M Generalized and Extended Uncertainty~ Principles and their impact onto the Hawking radiation(20) – p. 14/44 = = 2 2 ω . dN dN/dt c mp h i GUP corrected number of emitted Hawking quanta Combining (14) (TGUP ), (19), and (20) we have (up to first order in GUP)) 2 dS k π4 αc 2 m2 = B 1+ p + ..