generalized and extended uncertainty principles and their
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Generalized and Extended
Uncertainty Principles and their
impact onto the Hawking radiation
Mariusz P. Dabrowski
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
E =p2
2m− e2
r(1)
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 ≈ ~
r(2)
and so the energy is
E =~2
2mr2− e2
r(3)
and it has a minimum (Rydberg energy) Emin = −me4/2~2 for the minimum
length (Bohr radius) rmin = ~2/me2.
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 ∼ ~
∆p. (4)
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
∆x1 ∼ ∆(photon′s energy)
4×maximum force=
c∆p
4Fmax=c∆pc4
G
=G∆p
c3= l2p
∆p
~, (5)
where l2p = G~/c3 is the Planck length, maximum force Fmax = c4/4G.
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
∆xGUP = ∆x+∆x1 ≥ ~
∆p+ l2p
∆p
~=
~
∆p+ ~
(
α
α0
)2
∆p ≡ f(∆p), (6)
where
α = α0lpl~
is the constant with the dimension of inverse momentum kg−1m−1s, 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 = ~
lpso that the minimum length uncertainty is now
∆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))
~a = ~r = −G(E/c2)
r2~r
r(8)
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
∆v ∼ GE
c2r2L
c, ∆x1 ∼ GE
c2r2L2
c2∼ GE
c4∼ Gp
c3, (9)
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
size lp = 2GM/c2, we can recover Hawking temperature
∆pc ≈ ~c
∆x=
~c3
4GM≈ kBT, (10)
which after including a “calibration factor” of 2π gives
T =~c3
8πGkBM=
c2
8πkB
m2p
M, (11)
where, m2p = ~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)
∆p =
(
∆x
2~α2
)
∓ ∆x
2~α2
√
1− 4~2α2
(∆x)2, (12)
and expand in series as follows
∆p ≥ ~
∆x
[
1 +~2α2
(∆x)2+ 2
~4α4
(∆x)4+ . . .
]
. (13)
taking again ∆x = 2lp = 4GM/c2 and including the calibration factor into each
term, we get (T is the Hawking temperature)
TGUP = T
[
1 +4α2π2k2B
c2T 2 + 2
(
4α2π2k2Bc2
)2
T 4 + . . .
]
. (14)
Generalized and Extended Uncertainty Principles and their impact onto the Hawking radiation – p. 10/44
GUP corrected Bekenstein entropy
Using 1st law of thermodynamics dSGUP = c2dM/TGUP , after integration we
obtain generalised Bekenstein entropy
SGUP = S −α2c2m2
pkBπ
4ln
S
S0+α4c4m4
pk2Bπ
2
4
1
S+ . . . , (15)
where S is the Bekenstein entropy for a Schwarzschild black hole:
S =A
4
kBc3
~G=
4πkBGM2
~c= 4πkB
(
M
mp
)2
, (16)
with the integration constant S0 = (A0c3kB)/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
S = S/kB S2 = S/(kB ln 2)
and calculated an average entropy flow in blackbody radiation
〈S2〉 =π4
30ζ(3)ln2bits/photon ≈ 3.90 bits/photon,
with the standard deviation to be (ζ(n) is the Riemann zeta function)
σS2=
1
ln2
√
12ζ(5)
ζ(3)−(
π4
30ζ(3)
)2
bits/photon ≈ 2.52 bits/photon. (17)
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
N =30ζ(3)
π4S ≈ 0.26 S.
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
dM =〈E〉c2
dN =~〈ω〉c2
dN, (18)
where an average energy
〈E〉 = ~〈ω〉 = π4kB30ζ(3)
TGUP . (19)
From these we can calculate the GUP modified Bekenstein entropy loss of a
black hole
dSGUP
dN=dS/dt
dN/dt×(
1−α2c2m2
pkBπ
4
1
S−α4c4m4
pk2Bπ
2
4
1
S2+ . . .
)
,
where standard (non-GUP) Bekenstein entropy loss is
dS
dN=dS/dt
dN/dt=
8πkBc2
M
m2p
~〈ω〉. (20)Generalized and Extended Uncertainty Principles and their impact onto the Hawking radiation – p. 14/44
GUP corrected number of emitted Hawking quanta
Combining (14) (TGUP ), (19), and (20) we have (up to first order in GUP))
dS
dN=
kBπ4
30ζ(3)
1 +(αc
4
)2(
m2p
M
)2
+ ...
.
and using (15) (SGUP ), we obtain
dSGUP
dN=
kBπ4
30ζ(3)
1−(αc
4
)4(
m2p
M
)4
+ ...
, (21)
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
α=0
α=0.2
α=0.4
α=0.6
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2.6985
2.6990
2.6995
2.7000
2.7005
2.7010
M
dS
GU
P/
dN
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
dNGUP
dM=
30c2ζ(3)
π4kBTGUP=
30c2ζ(3)
π4kBT
(
1 +4π2α2k2B
c2T 2
)−1
, (22)
which can be integrated to give the total number of emitted Hawking quanta
NGUP , when GUP corrections are present as
NGUP =30ζ(3)
π4
[
4π
m2p
M2 −α2c2m2
pπ
4ln
(
M2
M20
)
]
, (23)
where M is the initial mass of a black hole and M0 = (A0c4)/(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
η = Cλ2thermal
gA, (24)
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
λ2thermal
Aeff=
64π3
27∼ 73.5...≫ 1, (26)
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
Aeff |GUP =27
4AGUP =
27
4
[
A− ~2α2π ln
A
A0
]
(27)
with A0 an integration constant with the unit of area, and the GUP corrected
thermal wavelength is
λthermal|GUP =2π~c
kBTGUP=
2π~c
kBT[
1 +4π2α2k2
B
c2 T 2] . (28)
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
η =λ2thermal
Aeff|GUP =
64π3
27× M6
[
M2 − (αc4 )2m4p ln
(
M2
M2
0
)]
[
M2 + (αc4 )2m4p
]2,
(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
α=0
α=0.2
α=0.4
α=0.6
0.0 0.5 1.0 1.5 2.0 2.5
0
20
40
60
80
M
η
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 6= 0 and M → mp one has η ≪ 1.
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
σ2x = 〈x2〉 − 〈x〉2 σ2
p = 〈p2〉 − 〈p〉2
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 followsGeneralized and Extended Uncertainty Principles and their impact onto the Hawking radiation – p. 23/44
Generalised Extended Uncertainty Principle (GEUP) and duality
σxσp ≥ ~
2
(
1 +α0l
2p
~2σ2p +
β0r2hor
σ2x
)
, (30)
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~
σp ↔√β0
rhorσx (31)
as well as both GUP sector (β0 = 0) and EUP sector (α0 = 0) exhibit dualities as
follows √α0lp~
σp ↔ ~√α0lp
σ−1p ,
√β0lH
σx ↔ rhor√β0σ−1x , (32)
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
~a = ~r =
(
−G(E/c2)
r2+
Λc2
3r
)
~r
r, (33)
where Λ is the cosmological constant associated to the cosmological de Sitter
space with the Hubble horizon
rhor =c
H=
(
3
Λ
)1/2
. (34)
Then, a new contribution to the uncertainty of a measuring particle momentum is
added into the scheme (Bambi & Urban CQG 25, 095006 (2008)).
Then the particle is moved by (Λ/3 = r−2hor)
∆x2 ∼ rc2
r2hor
L2
c2∼ (∆x)3
r2hor, ∆xEUP∆p ∼ ~
(
1 +(∆x)2
r2hor
)
. (35)Generalized and Extended Uncertainty Principles and their impact 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:
∆ψ + λψ = 0
inside D with the requirement that ψ = 0 on the boundary, λ denotes the
eigenvalue, and ∆ 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
〈p〉 = 0, and so one can obtain the uncertainty of a momentum
p = −i~∂i measurement as
σp =√
〈p2〉 = ~
√
−〈ψ|∆|ψ〉 ≥ ~
√
λ1 (36)
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
σp∆x ≥ π~
√
1− K
π2(∆x)2. (37)
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
ds2 =c2dl2
2αl+ d~y2
⊥, (38)
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
l0 = 2c2/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 = [l0 −∆x, l0 +∆x].
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
λn = n2π2 α
2c2δ2, (39)
which after inserting into (36) produces an exact formula of EUP for Rindler
spacetime (Fig. 30).
σp∆x ≥ π~α∆x2c2
√
1 + α∆x2c2 −
√
1− α∆x2c2
, (40)
or Taylor expanded formula for the sake of comparison with the common form
of the EUP (for small values of α∆x/(2c2))
σp∆x & π~
(
1− α2(∆x)2
32c4+O
[
(
α2(∆x)2
2c4
)2])
. (41)
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
ds2 =dr2
A(r, t0)+ r2dΩ2, A(r, t0) = 1− r2
r2H(r, t0), (42)
where the apparent horizon
r2H =c2
H2 + Kc2
a2
, (43)
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)
σp∆x ≥ ~∆x
rH
√
(
π
2 arctan f(∆x)− π/2
)2
− 1, f(∆x) =
√
1−∆x/rH1 + ∆x/rH
,
(44)
which can be Taylor expanded for small values of ∆x/rH giving the standard
form of such an EUP
σp∆x & π~
(
1− 3 + π2
6π2
(∆x)2
r2H+O
[
(∆x/rH)4]
)
. (45)
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)
Tσp=σpc
kB(46)
which for the horizons under study takes the form
Tσp,min = TH lim∆x→1
g(∆x) (47)
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
σp ∼ π~
∆x
(
1 + β0∆x2
r2hor+O[(rs/rhor)
4]
)
, (48)
which leads to the EUP corrected black hole Hawking temperature
TH,as = T(0)H
(
1 + β0r2sr2hor
+O[(rs/rhor)4]
)
(49)
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
SH,as =πkBr
2hor
β0l2plog
(
1 + β0r2sr2hor
+O[(rs/rhor)4]
)
(50)
≃ S(0)BH
(
1− β02
r2sr2hor
+O[(rs/rhor)4]
)
(51)
≃ S(0)BH
1− β02
S(0)BH
Shor+O
(
S(0)BH
Shor
)2
, (52)
where the horizon entropy of the background spacetime is equal to
Shor =πkBr
2hor
l2p. (53)
and T(0)H and S
(0)BH 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
TH,R =~α
8πckB
(√
1 +αrs2c2
−√
1− αrs2c2
)−1
(54)
which leads to the entropy
SBH,R =16πkB3l2p
c4
α2
[
(
1 +αrs2c2
)3/2
+(
1− αrs2c2
)3/2
− 2
]
. (55)
For small black holes (αrs/2c2 ≪ 1) this result can be expanded to yield
SBH,R ≃ S(0)BH
(
1 +S(0)BH
16SR+O
[
(
S0BH/SR
)2]
)
(56)
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
Rindler horizon distance αrs/2c2 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
TH,F =c~
kB
1
4π2rH
√
(
π
2 arctan f(rs)− π/2
)2
− 1
. (57)
Unfortunately, the integration of the entropy cannot be done analytically.
Therefore it will be given in its integral form
SBH,F =2π2kBrH
l2p
∫
drs√
(
π2 arctan f(rs)−π/2
)2
− 1
+ S0 (58)
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
SBH,F ≃ S(0)BH
(
1 +3 + π2
12π2
S(0)BH
SH+O
[
(
S0BH/SH
)2]
)
, (59)
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 ∝ 1r gets infinite in the limit r → 0. In
relativity there exists a maximum force due to the phenomenon of
gravitational collapse and black hole formation (Gibbons (2002), Schiller
(2005))
Fmax = c4/4G
The nicest derivation of the principle comes from the application of the
cosmic string deficit angle
φ = (8πG/c4)F ≤ 2π.
In fact, the factor c4
G = 1.3× 1044 Newtons appears in the Einstein field
equations Tµν = 18π
c4
GGµν 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
power for GWs is Pmax = cFmax = c5/4G
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
Fr = −T dS
drh= −γ c
4
G= −4γFmax,
where T is the Hawking temperature (rh - horizon radius, γ - a parameter)
T =γℏc
2πkBrh,
and S is the Bekenstein entropy
S =kBc
3A
4ℏG=πkBc
3
Gℏr2h,
(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
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