generalized and extended uncertainty principles and their

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.


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).


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.


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.


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.


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.


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)


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.


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)


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.


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.


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).


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.


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”.


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.


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)



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.



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.


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).


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.


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)


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].


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)


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.


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).



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)



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.


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).


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.


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.


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.



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.



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


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).


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

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