Extended Uncertainty Principles and Their Impact Onto the Hawking Radiation

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 thermodynamics

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 + ... . dN 30ζ(3)  4 M !    and using (15) (SGUP ), we obtain

4 dS k π4 αc 4 m2 GUP = B 1 p + ... , (21) dN 30ζ(3)  − 4 M !    from which we conclude that the Bekenstein entropy loss is no longer a constant as it happens in a non-GUP case, but it depends on the mass of a black hole - i.e. the information does not escape at the same rate when GUP is applied.

Generalized and Extended Uncertainty Principles and their impact onto the Hawking radiation – p. 15/44 GUP modiﬁed entropy loss

2.7010

2.7005 α=0 0.2 dN 2.7000 α= / α=0.4

GUP 0.6 2.6995 α= dS

2.6990

2.6985

0.0 0.5 1.0 1.5 2.0 2.5 3.0 M

The GUP modiﬁed Bekenstein entropy loss per emitted photons, dSGUP /dN, as given by (21) as the function of M for different values of the GUP parameter α (its zero value shows that the rate of loss is constant).

Generalized and Extended Uncertainty Principles and their impact onto the Hawking radiation – p. 16/44 Total number of emitted quanta and a black hole remnant.

Applying (14) we obtain the number of particles per emitted mass

− dN 30c2ζ(3) 30c2ζ(3) 4π2α2k2 1 GUP = = 1+ B T 2 , (22) dM π4k T π4k T c2 B GUP B which can be integrated to give the total number of emitted Hawking quanta

NGUP , when GUP corrections are present as

30ζ(3) 4π α2c2m2π M 2 N = M 2 p ln , (23) GUP π4 m2 − 4 M 2 " p 0 #

4 where M is the initial mass of a black hole and M0 =(A0c )/(16πG) is an integration constant. This shows that the introduction of GUP results in decreasing the total number of emitted particles. It makes sense since the ﬁnal state of evaporation is a remnant of the Planck size.

Generalized and Extended Uncertainty Principles and their impact onto the Hawking radiation – p. 17/44 Sparsity of Hawking radiation

Hawking radiation is very sparse while emitted. Sparsity is measured studying the ratio between an average time between the emission of two consecutive quanta and the natural timescale (Gray et al. CQG 33, 115003 (2016)). In the ﬁrst approximation one assumes the exact Planck spectrum and it results in a general expression for the Minkowski spacetime that should be speciﬁed depending on a dimensionless parameter η

λ2 η = C thermal , (24) gA

where the constant C is dimensionless and depends on the speciﬁc parameter (η) we are choosing, g is the spin degeneracy factor, A is the area and

λthermal = 2π~c/(kBT ) (25)

is the “thermal wavelength”.

Generalized and Extended Uncertainty Principles and their impact onto the Hawking radiation – p. 18/44 Sparsity of Hawking radiation

Schwarzschild black hole: - the temperature in the thermal wavelength is given by the Hawking temperature - the area is replaced by an effective area (which corresponds to the universal

cross section at high frequencies) equal to Aeff = (27/4)A. For massless bosons the ratio

λ2 64π3 thermal = 73.5... 1, (26) Aeff 27 ∼ ≫

which means that for massless bosons the gap between successive Hawking quanta is on average much larger than the natural timescale associated with each individual emitted quantum, so the ﬂux is very sparse (note that the mass M of a black hole is not present in the formula). In fact, in normal laboratory conditions emitters have η 1. ≪

Generalized and Extended Uncertainty Principles and their impact onto the Hawking radiation – p. 19/44 GUP modiﬁed sparsity of Hawking radiation

When GUP is applied then both the area and the “thermal wavelength” are modiﬁed when the system approaches the Planck scale. This results in modifying the frequency of emitted quanta from a black hole. We obtain that the new generalised by GUP effective area is

27 27 A A = A = A ~2α2π ln (27) eff |GUP 4 GUP 4 − A 0

with A0 an integration constant with the unit of area, and the GUP corrected thermal wavelength is

2π~c 2π~c λthermal = = 2 2 2 . (28) GUP 4π α k | kBTGUP B 2 kBT 1+ c2 T h i

Generalized and Extended Uncertainty Principles and their impact onto the Hawking radiation – p. 20/44 GUP modiﬁed sparsity of Hawking radiation

Finally, the GUP corrected parameter η that determines the sparsity of the ﬂux, is given by

λ2 64π3 M 6 η = thermal = , GUP 2 2 Aeff | 27 × 2 αc 2 4 M 2 αc 2 4 M ( ) mp ln 2 M +( ) mp − 4 M0 4 h i (29) which now depends on the mass M of a black hole, and on the GUP parameter α. In fact, radiation ceases to be sparse (η 1) when the process of ≫ evaporation reaches its last stages near the Planck scale since close to this scale the parameter becomes less than one and behaves as standard laboratory radiation with η 1. ≪

Generalized and Extended Uncertainty Principles and their impact onto the Hawking radiation – p. 21/44 GUP modiﬁed sparsity of Hawking radiation

60 α=0 α=0.2 η 40 α=0.4 α=0.6

0 0.0 0.5 1.0 1.5 2.0 2.5 M

GUP-corrected sparsity of the Hawking ﬂux η as given by (29) versus M for different values of GUP parameter α (its zero value shows that sparsity is constant and large (η 1), while for α = 0 and M m one has η 1. ≫ 6 → p ≪ Generalized and Extended Uncertainty Principles and their impact onto the Hawking radiation – p. 22/44 4. Extended Uncertainty Principle (EUP) and GEUP duality.

GUP takes into account gravitational uncertainty of position related to the minimum fundamental scale in physics (photon-electron gravitational interaction) while still there is a problem of taking into account the geometrical aspects of curvature on large fundamental scales of the order of Hubble horizon. This is what is the matter of EUP which takes into account also the uncertainty related to the background spacetime manifested by external horizons. Both components can be related to the standard deviations of position x and momentum p

σ2 = xˆ2 xˆ 2 σ2 = pˆ2 pˆ 2 x h i−h i p h i−h i and they lead to the most general asymptotic Generalised Extended Uncertainty Principle (GEUP) which includes both GUP and EUP (Adler, Santiago 1999); Bambi, Urban CQG 25, 095006 (2008)) as follows Generalized and Extended Uncertainty Principles and their impact onto the Hawking radiation – p. 23/44 Generalised Extended Uncertainty Principle (GEUP) and duality

2 ~ α0l β σ σ 1+ p σ2 + 0 σ2 , (30) x p ~2 p 2 x ≥ 2 rhor !

where rhor is the radius of the horizon which is introduced by the background

space-time, α0 was introduced in (6) and β0 is a new dimensionless parameter. GEUP (30) possesses the invariance under the duality transformation

√α0lp √β0 σp σx (31) ~ ↔ rhor

as well as both GUP sector (β0 = 0) and EUP sector (α0 = 0) exhibit dualities as follows ~ √α0lp −1 √β0 rhor −1 ~ σp σp , σx σx , (32) ↔ √α0lp lH ↔ √β0 They reﬂect some general relations between black hole and cosmological horizons (e.g. Artymowski, Mielczarek 2018).

Generalized and Extended Uncertainty Principles and their impact onto the Hawking radiation – p. 24/44 EUP simple Newtonian derivation

It is known that in the Newtonian limit of relativistic background an acceleration of a particle of mass m due to the particle of mass M is

G(E/c2) Λc2 ~r ~a = ~r¨ = + r , (33) − r2 3 r where Λ is the cosmological constant associated to the cosmological de Sitter space with the Hubble horizon

c 3 1/2 r = = . (34) hor H Λ Then, a new contribution to the uncertainty of a measuring particle momentum is added into the scheme (Bambi & Urban CQG 25, 095006 (2008)). −2 Then the particle is moved by (Λ/3= rhor)

rc2 L2 (∆x)3 (∆x)2 ∆x , ∆x ∆p ~ 1+ . (35) 2 ∼ r2 c2 ∼ r2 EUP ∼ r2 hor horGeneralized and Extended Uncertainty Principles and their impacthor onto the Hawking radiation – p. 25/44 Background geometry determined EUP

We follow an idea of Schurmann¨ (2017, 2018) that the measurement of momentum depends on a given space-time background. To measure the momentum one needs to consider a compact domain D with boundary ∂D characterised by the geodesic length ∆x around the location of the measurement with Dirichlet boundary conditions. Thus the wavefunction is conﬁned to D (which lies on a spacelike hypersurface). The method then reduces to the solution of an eigenvalue problem for the wave function ψ:

∆ˆ ψ + λψ = 0

inside D with the requirement that ψ = 0 on the boundary, λ denotes the eigenvalue, and ∆ˆ is the Laplace-Beltrami operator.

Generalized and Extended Uncertainty Principles and their impact onto the Hawking radiation – p. 26/44 Background geometry determined EUP

As we can choose ψ to be real (the eigenvalue problem is the same for the real and the imaginary part), the Dirichlet boundary conditions assure that pˆ = 0, and so one can obtain the uncertainty of a momentum h i pˆ = i~∂ measurement as − i

σ = pˆ2 = ~ ψ ∆ˆ ψ ~ λ (36) p h i −h | | i≥ 1 p q p where λ1 denotes the ﬁrst eigenvalue. Multiplying by ∆x, the uncertainty relation corresponding to this momentum measurement is obtained. A formula found by Schürmann (2018) applied for Riemannian 3-manifolds of constant curvature K reads

K ~ 2 (37) σp∆x π 1 2 (∆x) . ≥ r − π

Generalized and Extended Uncertainty Principles and their impact onto the Hawking radiation – p. 27/44 5. Background geometry determined EUP (Rindler and Fried- mann) and black hole thermodynamics

The method requires a foliation of spacetime and we consider only the spatial part of the Rindler metric

2 2 2 c dl 2 ds = + d~y⊥, (38) 2αl

α – the acceleration describing a boost in the l-direction, and ~y⊥ – components of the metric perpendicular to l direction. − An observer/a particle moving with the acceleration α is located at 2 l0 = 2c /α and sees a horizon at a distance l0 at l = 0. For simplicity the directions transversal to the acceleration will not play any role in this treatment. Thus, the obtained uncertainty will account for the effect on measurements done along the direction of acceleration. As we basically describe one-dimensional problem, the domain can most conveniently be taken to be the interval I =[l ∆x, l + ∆x]. 0 − 0

Generalized and Extended Uncertainty Principles and their impact onto the Hawking radiation – p. 28/44 EUP for Rindler spacetime

Solution to the eigenvalue problem gives eigenvalues as α λ = n2π2 , (39) n 2c2δ2 which after inserting into (36) produces an exact formula of EUP for Rindler spacetime (Fig. 30).

α∆x 2c2 σp∆x π~ , (40) ≥ α∆x α∆x 1+ 2 1 2 2c − − 2c q q or Taylor expanded formula for the sake of comparison with the common form of the EUP (for small values of α∆x/(2c2))

α2(∆x)2 α2(∆x)2 2 σ ∆x & π~ 1 + O . (41) p − 32c4 2c4 " #!

Generalized and Extended Uncertainty Principles and their impact onto the Hawking radiation – p. 29/44 EUP for Rindler spacetime

Conclusion: uncertainty never reaches zero although it is monotonically decreasing with increasing ∆x and it features a minimum value of 1/√2 in units

of ~/2 where ∆x = l0.

Generalized and Extended Uncertainty Principles and their impact onto the Hawking radiation – p. 30/44 EUP for Friedmann spacetime

Next we consider Friedmann universe with hypersurfaces of constant Schwarzschild-like time (in deSitter/ anti-deSitter space this slicing corresponds to static coordinates) with spatial metric

2 2 2 dr 2 2 r ds = + r dΩ , A(r, t0) = 1 2 , (42) A(r, t0) − rH (r, t0)

where the apparent horizon

2 2 c r = 2 , (43) H 2 Kc H + a2

with the scale factor a, Hubble-parameter H =a/a ˙ , the curvature index K, and the metric of the two sphere dΩ. Subtlety: in this approach the homogeneity of the universe is broken, putting an observer at the center of symmetry (isotropy w.r.t. just one point).

Generalized and Extended Uncertainty Principles and their impact onto the Hawking radiation – p. 31/44 EUP for Friedmann spacetime

After ﬁnding the eigenvalues we get an exact EUP formula for Friedmann spacetime (Fig. 33)

2 ∆x π 1 ∆x/rH σp∆x ~ 1, f(∆x)= − , ≥ rH s 2 arctan f(∆x) π/2 − 1 + ∆x/rH − s (44)

which can be Taylor expanded for small values of ∆x/rH giving the standard form of such an EUP

3+ π2 (∆x)2 ~ 4 (45) σp∆x & π 1 2 2 + O (∆x/rH ) . − 6π rH h i

Generalized and Extended Uncertainty Principles and their impact onto the Hawking radiation – p. 32/44 EUP for Friedmann spacetime

The EUP (44) for Friedmann background with manifest horizon again never reaches zero. Here given in terms of the rescaled position uncertainty in units of π~. In these units the uncertainty approaches a minimum value of √3/π.

Generalized and Extended Uncertainty Principles and their impact onto the Hawking radiation – p. 33/44 EUP relation to Hawking temperature

Minimum of momentum uncertainty σp allows to deﬁne the temperature (of spacetime)

σpc Tσp = (46) kB

which for the horizons under study takes the form

Tσp,min = TH lim g(∆˜x) (47) ∆˜x→1

where TH , is the Hawking temperature of the respective horizons,

∆˜x = ∆x/l0 for Rindler and ∆˜x = ∆x/rH for Friedmann space-time, respectively. The function g(∆˜x) possesses a limit of the order of √2π2 for Rindler and 2π/√3 for Friedmann for horizon size uncertainties (∆˜x = 1).

Generalized and Extended Uncertainty Principles and their impact onto the Hawking radiation – p. 34/44 Black hole thermodynamics in background spacetimes with horizons

The spacetime horizons of radius rhor inﬂuence black holes immersed into them and we can calculate the (EUP) uncertainty related to the background

geometry (parameter β0) as

π~ ∆x2 σ 1+ β + O[(r /r )4] , (48) p ∼ ∆x 0 r2 s hor hor which leads to the EUP corrected black hole Hawking temperature

r2 T = T (0) 1+ β s + O[(r /r )4] (49) H,as H 0 r2 s hor hor Here ”as” means that we use the asymptotic Taylor expanded form of EUP.

Generalized and Extended Uncertainty Principles and their impact onto the Hawking radiation – p. 35/44 Black hole thermodynamics in background spacetimes with horizons

and the EUP corrected Bekenstein entropy

πk r2 r2 S = B hor log 1+ β s + O[(r /r )4] (50) H,as β l2 0 r2 s hor 0 p hor β r2 S(0) 1 0 s + O[(r /r )4] (51) ≃ BH − 2 r2 s hor hor (0) (0) 2 (0) β0 SBH SBH SBH 1 + O , (52) ≃  − 2 Shor  Shor !     where the horizon entropy of the background spacetime is equal to

πk r2 B hor (53) Shor = 2 . lp

(0) (0) and TH and SBH are the standard (non-GUP) Hawking temperature and Bekenstein entropy. Generalized and Extended Uncertainty Principles and their impact onto the Hawking radiation – p. 36/44 EUP corrected accelerated black holes in Rindler spacetime

Applying the exact relation (40), the Hawking temperature of an accelerated black hole reads

−1 ~α αr αr T = 1+ s 1 s (54) H,R 8πck 2c2 − − 2c2 B r r which leads to the entropy

16πk c4 αr 3/2 αr 3/2 B s s (55) SBH,R = 2 2 1+ 2 + 1 2 2 . 3lp α 2c − 2c − For small black holes (αr /2c2 1) this result can be expanded to yield s ≪ (0) (0) SBH 0 2 SBH,R SBH 1+ + O SBH /SR (56) ≃ 16SR ! h i

with the entropy of the Rindler horizon SR which is the result for the calculation

in the asymptotic form. Generalized and Extended Uncertainty Principles and their impact onto the Hawking radiation – p. 37/44 EUP corrected accelerated black holes in Rindler spacetime

The temperature (left) and the entropy (right) of an accelerated black hole of an accelerated black hole as a function of the Schwarzschild horizon in units of the 2 Rindler horizon distance αrs/2c for ﬁxed acceleration α in comparison to the asymptotic result. The presence of a Rindler horizon decreases the temperature of a black hole thus increasing its entropy. This effect is maximal when one uses the exact formulas.

Generalized and Extended Uncertainty Principles and their impact onto the Hawking radiation – p. 38/44 EUP corrected accelerated black holes in Friedmann spacetime

Analogously, the entropy of a black hole surrounded by a Friedmann horizon can be obtained. Correspondingly, the Hawking temperature becomes

c~ 1 π 2 (57) TH,F = 2 1 . kB 4π rH s 2 arctan f(rs) π/2 −  −   Unfortunately, the integration of the entropy cannot be done analytically. Therefore it will be given in its integral form

2 2π kBrH drs SBH,F = + S0 (58) l2 2 p π Z − 1 2 arctan f(rs) π/2 − r with the integration constant S0, again, chosen in a way that SBH,F (rs = 0) = 0.

Generalized and Extended Uncertainty Principles and their impact onto the Hawking radiation – p. 39/44 EUP corrected accelerated black holes in Friedmann spacetime

The expansion for small rs/rH reads

2 (0) 3+ π S 2 (0) BH 0 (59) SBH,F SBH 1+ 2 + O SBH /SH , ≃ 12π SH ! h i

where the Hubble-horizon entropy SH equals to the asymptotic result.

Generalized and Extended Uncertainty Principles and their impact onto the Hawking radiation – p. 40/44 EUP corrected accelerated black holes in Friedmann spacetime

The Hawking temperature (left) and the Bekenstein entropy (left) of a black hole surrounded by a cosmological horizon as a function of the Schwarzschild horizon

in units of the cosmological horizon distance rs/rH for a ﬁxed horizon distance

rH in comparison to the asymptotic result. The presence of the horizon decreases the temperature and increases the entropy. The application of the exact relation results in a considerable ampliﬁcation of this effect. Generalized and Extended Uncertainty Principles and their impact onto the Hawking radiation – p. 41/44 Brief comment on Principle of Maximum Tension in relativity

Force in the Newton’s theory F 1 gets inﬁnite in the limit r 0. In ∝ r → relativity there exists a maximum force due to the phenomenon of gravitational collapse and black hole formation (Gibbons (2002), Schiller (2005)) 4 Fmax = c /4G The nicest derivation of the principle comes from the application of the cosmic string deﬁcit angle

φ = (8πG/c4)F 2π. ≤ 4 c 44 In fact, the factor G = 1.3 10 Newtons appears in the Einstein ﬁeld 4 1 c × equations Tµν = 8π G Gµν which can be considered in analogy with the elastic force equation F = kx (k = c4/G - an elastic constant, x - the displacement) which relates it to gravitational waves. Besides, maximum 5 power for GWs is Pmax = cFmax = c /4G

Generalized and Extended Uncertainty Principles and their impact onto the Hawking radiation – p. 42/44 Maximum Tension Principle and the entropic force

We make an observation that similar ratio c4/G appears in the expression for the entropic force within the framework of entropic cosmology (Easson et al.; 2011). The entropic force is deﬁned as

dS c4 Fr = T = γ = 4γFmax, − drh − G −

where T is the Hawking temperature (rh - horizon radius, γ - a parameter)

γℏc T = , 2πkBrh

and S is the Bekenstein entropy

k c3A πk c3 S = B = B r2, 4ℏG Gℏ h (minus sign – the force points in the direction of increasing entropy).

Generalized and Extended Uncertainty Principles and their impact onto the Hawking radiation – p. 43/44 6. Conclusions

Hawking temperature and Bekenstein entropy are essentially modified while applying GUP and EUP. The information does not escape at the same rate from black holes when GUP is applied. Introduction of GUP results in decreasing the total number of emitted particles which is obvious since the final state of evaporation when GUP is applied is a remnant of the Planck size. GUP corrected Hawking radiation ceases to be sparse (sparsity parameter η 1) when the process of evaporation reaches its last ≫ stages near the Planck scale and despite HUP radiation which is sparse it behaves as standard laboratory radiation with η 1. ≪ The existence of spacetime horizons influence black holes immersed into them – it decreases their Hawking temperature and increases their Bekenstein entropy.

Generalized and Extended Uncertainty Principles and their impact onto the Hawking radiation – p. 44/44