Temperature for the (2+1)-dimensional Black Hole with Non Linear Electrodynamics from the Generalized Uncertainty Principle

0.14 MB·English

by  Alexis Larranaga

(2+1)
Temperature for the -dimensional Bla
k Hole with Non Linear Ele
trodynami
s
from the Generalized Un
ertainty Prin
iple
Alexis Larrañaga
Universidad Na
ional de Colombia. Observatorio Astronómi
o Na
ional (OAN) and ∗
Universidad Distrital Fran
is
o José de Caldas. Fa
ultad de Ingeniería. Proye
to Curri
ular de Ingeniería Ele
tróni
a
He
tor J. Hortua †
Universidad Na
ional de Colombia. Observatorio Astronómi
o Na
ional (OAN)
(2+1)
In this paper, we study the thermodynami
al properties of the dimensional bla
k hole
with a non-linear ele
trodynami
s with a negative
osmologi
al
onstant, using the Generalized
Un
ertainty Prin
iple (GUP). This approa
h shows that there is a minimum mass or remnant for
9 the bla
k hole,
orresponding to the minimum radius of the event horizon that has a size of the
0 orderofthePlan
ks
ale. Wealsoshowthattheheat
apa
ityforthisbla
kholeisalwayspositive.
0
2
Keywords: Bla
k Holes Thermodynami
s, Generalized Un
ertanty Prin
iple.
n
PACS: 04.70.Dy; 04.20.-q; 11.10.Lm
a
J (2+1)
Working in dimensional gravity, and using as the sour
e of the Einstein equations the stress-energy tensor
3
2 of non-linear erle2
trodynami
s, Cataldo et. al. [1℄ found a solution with a Coulomb-like ele
tri
(cid:28)eld (proportional to
the inverse of ), that des
ribes
harged-AdS spa
e when
onsidering a negative
osmologi
al
onstant.

] As is well known, the thermodynami
al properties of bla
k holes are asso
iated with the presen
e of the event
c (2+1)
horizon. Re
ently[2℄,ithasbeen shownthatthe dimensional(cid:28)eldequationsforthebla
kholewithnonlinear
q
- ele
trodynami
s
an be interpreted as the di(cid:27)erential (cid:28)rst law with the usual form
r
g
dM =TdS+ΦdQ,
[
(1)

1 T κ
where is the Hawking temperature that
an be expressed in terms of the surfa
e gravity at the horizon by
v
7 κ
2 T = .
2π (2)
7
3
. On the other hand, in re
ent yearsthe un
ertainty relation that in
ludes gravity e(cid:27)e
ts, known as the Generalized
1
0 Un
ertainty Prin
iple (GUP), has shown interesting results in the
ontext of bla
k holelpev=ap1o.r6a1tion10[3−℄3,3ecxmtending
the relation between temperature and mass to s
ales of the order of the Plan
k lenght, × . This
9 lp
0 treatmentimplythmat=1is.2t2he s1m0a19llGesetVlengths
alein the theoryand it isrelatedto theexisten
eofanextreme mass
p
: (the Plan
k mass × , whi
h be
omes the bla
k hole remnant), that
orresponds to the maximum
v
possible temperature. This approa
h has been used re
ently [4℄ to extend the temperature-masss relation for the
i (2+1) (Λ=0)
X dimensionalbla
kholewithanonlinearele
tri
(cid:28)eldandwithout
osmologi
al
onstant . Thereisalso
shown that there is a maximum temperature permited for the bla
k hole that
orresponds to a horizon with size in
r
a the Plan
k s
ale.
(2+1)
In this paper, we investigate the thermodynami
s of the -dimensional bla
k hole with a nonlinear ele
tri
(cid:28)eld reported in [1℄, to show how the GUP extended to gravity in spa
es with
osmologi
al
onstant as proposed
T (M)
by Arraut et. al. [3℄
an be used to
al
ulate the Hawking temperature asso
iated with the bla
k hole. The
equation gives the usual relation for large masses but gets deformed when the mass be
omes small. We also show
how there is a minimum mass for the bla
k hole that
orresponds to a horizon with size in the Plan
k s
ale. Finally
we
al
ulate the heat
apa
ity for this bla
k hole, to show that it is always positive.
I. THE 3-DIMENSIONAL BLACK HOLE WITH NON-LINEAR ELECTRODYNAMICS
(2+1)
The metri
reported by Cataldo et. al. [1℄ is a solution of the dimensional Einstein's (cid:28)eld equations with a
Λ<0
negative
osmologi
al
onstant ,
∗
†Ele
troni
address: ealarranagaunal.edu.
o
Ele
troni
address: hjhortuaounal.edu.
o
2
G +Λg =8πT ,
µν µν µν
(3)
G= 1
where we have used units su
h that . To obtain a Coulomb-like ele
tri
(cid:28)eld, CataldoFet.=al1.FuseFdµaνnonlinear
ele
todynami
s, in whi
h the ele
tromagneti
a
tion does not depend only on the invariant 4 µν , but it is
F3/4
a fun
tion of ; and the energy-momentum tensor is restri
ted to be tra
eless. The stati

ir
ularly symmetri
solution obtained has the line element
dr2
ds2 = f(r)dt2+ +r2dϕ2,
− f(r) (4)
where
4Q2
f(r)= M Λr2+ .
− − 3r (5)
The ele
tri
(cid:28)eld for this solution is
Q
E(r)= ,
r2 (6)
Q
whi
h is the standard Coulomb (cid:28)eld for a point
harge. Note that the metri
depends on two parameters and
M
, that are identi(cid:28)ed as the ele
tri

hargeand the mass, respe
tively.
A. Horizons
The horizons of this solution are de(cid:28)ened by the
ondition
f(r)=0
(7)
or
4Q2
M Λr2+ =0.
− − 3r (8)
For a
osmologi
al
onstant in the range
M3
Λ<0,
− 12Q4 ≤ (9)
the roots of the third-order polynomial (8) are all real, and
an be written as [2℄
M 1 Q2 3Λ
r1 = 2 cos cos−1 2
− r−3Λ 3 M r−M!! (10)
M 1 Q2 3Λ 2π
r2 = 2 cos cos−1 2 +
− r−3Λ 3 M r−M! 3 ! (11)
M 1 Q2 3Λ 4π
r3 = 2 cos cos−1 2 + .
− r−3Λ 3 M r−M! 3 ! (12)
Note that the form of this solution impose the
ondition
Q2 3Λ
2 1.
M −M ≤ (13)
r
3
The equal sign de(cid:28)nes a extreme bla
k hole, with the maximum mass
Mmax = 3 12Q4Λ
− (14)
p
and with the horizons
Mmax
r1 = 2
− − 3Λ (15)
r
Mmax
r2 = r3 =2 .
− 3Λ (16)
r
r
1
Note that the radius is negative, so it does not represent a physi
al horizon be
ause for other values of the mass
r r
2 3
it is always negative. Therefore, this kind of bla
k holes have only two physi
al horizons, and , that, for the
extremal bla
k hole,
oin
ide,
2Q2 1/3
r2 =r3 =rext = .
− 3Λ (17)
(cid:18) (cid:19)
r r r
2 3 +
In the general
ase, the largestradius between and
orrespondsto the event horizon of the bla
k hole , while
r−
theUositnhgerth
eorerxesppaonnsidosntocotsh−e1ixnn≈erπ2h−orxizo−n16x3., the radii of the horizons r2 and r3 in equations (11) and (12)
an be
approximated up to the third order in
Q2 3Λ
x=2 ,
M −M (18)
r
as
M 1 √3 4
r2 =r+ √3 x+ x2+ x3 +O x4
≈ r−3Λ" − 3 18 81 # (19)
(cid:0) (cid:1)
M 2 8
r3 =r− x+ x3 +O x4 .
≈ −3Λ 3 81 (20)
r (cid:20) (cid:21)
(cid:0) (cid:1)
x
If we
onsider only the (cid:28)rst
ontribution in we get
M 2Q2
r2 =r+
≈ − Λ − 3 M (21)
r
4Q2
r3 =r− .
≈ 3 M (22)
II. HAWKING TEMPERATURE
In this se
tion we will dedu
e the Hawking temperature for this bla
k hole by using the usual de(cid:28)nition (2). The
surfa
e gravity
an be
al
ulated as
κ=χ(xµ)a,
(23)
a χ χ
where isthemagnitudeofthefour-a

elerationand isthered-shiftfa
tor. Inorderto
al
ulate ,wewill
onsider
aKsµtati
observer, for whomVµthe red-shift fa
tor is just the proportionality fa
tor between the timelike Killing ve
tor
and the four-velo
ity , i.e.
4
Kµ =χVµ.
(24)
The metri
(4) has the Killing ve
tor
Kµ =(1,0,0)
(25)
while the four-velo
ity is
al
ulated as
dxµ dt
Vµ = = ,0,0 .
dτ dτ (26)
(cid:18) (cid:19)
This gives
1
Vµ = f−1(r),0,0 = ,0,0 ,
 M Λr2+ 4Q2  (27)
(cid:0) (cid:1) − − 3r
q 
and therefore, the red-shift fa
tor is
4Q2
χ(r)= M Λr2+ .
− − 3r (28)
r
On the other hand, the four-a

elerationis given by
dVµ
aµ = ,
dτ (29)
that has
omponents
a0 =aϕ = 0
(30)
2Q2
ar = Λr .
− − 3 r2 (31)
and therefore, the magnitude of the four-a

eleration is
Λr 2Q2
a= g aµaν = − − 3 r2 .
µν
M Λr2+ 4Q2 (32)
p − − 3r
q
Then, the surfa
e gravity at the event horizon is given by the absolute value
2Q2
κ+ = −Λr− 3 r2 . (33)
(cid:12) (cid:12)r=r+
(cid:12) (cid:12)
(cid:12) (cid:12)
r = r κ((cid:12)r ) = 0 (cid:12)
ext max
Note that for , the surfa
e gravity is , i.e. that the extreme bla
k hole has zero Hawking
temperature,
T (Mmax)=0.
(34)
r+ r−
Usingtheapproximateformsfor and givenbyequations(19)and(20),we
anwritetheHawkingtemperature
x
as a fun
tion of the mass. For the event horizon we have, up to se
ond order in ,
5
κ+ 1 2Q2
T+(M)= 2π = 2π −Λr+− 3r2 (35)
(cid:12) +(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
−1 −2
1 M 1 2 M 1
T+(M) Λ √3 x Q2 √3 x
≈ 2π (cid:12)(cid:12)− r−3Λ(cid:20) − 3 (cid:21)− 3 (cid:18)−3Λ(cid:19) (cid:20) − 3 (cid:21) (cid:12)(cid:12) (36)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
1 MΛ1 2ΛQ2 2x −1
T+(M) √ MΛ x+ 3
≈ 2π (cid:12)(cid:12) − −r− 3 3 M (cid:18) − √3(cid:19) (cid:12)(cid:12) (37)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
1 MΛ1 2√3ΛQ2
T+(M) √ MΛ x+
≈ 2π (cid:12) − −r− 3 3 M 3√3 2x (cid:12) (38)
(cid:12) − (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:0) (cid:1)(cid:12)
(cid:12) (cid:12)
1 2ΛQ2 2√ MΛQ2
T+(M) √ MΛ + − .
≈ 2π (cid:12)(cid:12) − − 3 M 3 M3 4Q2(cid:12)(cid:12) (39)
(cid:12) − Λ − (cid:12)
(cid:12) q (cid:12)
(cid:12) (cid:12)
Q=0 (cid:12) (cid:12)
Note that, for , this expression be
omes
√ MΛ
TBTZ(M)= − ,
+ 2π (40)
that
orrespondsto the temperatureat the horizon of the stati
non-
hargedBTZ bla
k hole [5℄. Onthe otherhand,
r−
for the inner horizon we have
κ− 1 2Q2
T−(M)= 2π = 2π (cid:12)−Λr−− 3 r−2 (cid:12) (41)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
−2
1 M 2 2 M 2
T−(M) Λ x Q2 x
≈ 2π (cid:12)(cid:12)− r−3Λ3 − 3 "r−3Λ3 # (cid:12)(cid:12) (42)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
1 M 2 9ΛQ2
T−(M) Λ x+
≈ 2π (cid:12)− r−3Λ3 2Mx2(cid:12) (43)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
1 4ΛQ2 3M2
T−(M) .
≈ 2π −3 M − 8Q2 (44)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
III. HAWKING RADIATION AND THE GENERALIZED UNCERTAINTY PRINCIPLE
Nowwewill
onsidertheGUP andtheHawkingradiationderivedfromit. WewillshowhowtheGUP willprodu
e
a deformation in the temperature-mass relation when
onsidered
lose to the Plan
k lenght. As stacte=d~by=A1rraut et.
al. [3℄ the GUP
an be stated, in the
ase of a non-zero
osmologi
al
onstant and in units with , by the
relation
1 G γ 1
2
∆x∆p& + (∆p) ,
2 2 − 3Λ(∆p)2 (45)
6
G Λ γ
where is the gravitational
onstant, is
osmologi
al
onstant and is a
onstant fa
tor whi
h a

ounts for the
fa
t that the GUP equation is dealing with orders of magnitudes estimates. When
omparing the GUP results with
γ l
p
the aboveHawkingtemperature, wewill seethat isexpe
ted to be of the orderof1. Sin
e thePlan
k lenght
an
be written as
~2
l2 =G~= ,
p m2 (46)
p
m
p
where is the mass of Plan
k, the GUP
an be written as
1 l2 γ 1
∆x∆p& + p (∆p)2 ,
2 2 − 3Λ(∆p)2 (47)
or
2
1 1(∆p) γ 1
∆x∆p& + .
2 2 m2 − 3Λ(∆p)2 (48)
p
∆x
To apply this un
ertainty prin
iple to the bla
k hole evaporation pro
ess
onsist in identifying the with the
r+ ∆p 2π (∆P 2πT)
event horizon radius and the momentum with the Hawking temperature up to a fa
tor [? ℄ ∼ .
Therefore, we
an write the GUP as a fourth order equation for the temperature,
1 2π2T2 γ 1
r+2πT = 2 + m2 − 3Λ4π2T2 (49)
p
For high temperatures, the GUP equation
an be approximated by
1 2π2T2
r+2πT = 2 + m2 (50)
p
T
that
orrespondsto a quadrati
polynomial for ,
4π2
m2T2−4πr+T +1=0 (51)
p
From whi
h it follows that the temperature-mass relation is
T (M)= m2pr+ 1 r2 1 .
2π " ±s +− m2p# (52)
Note that the argument in the square root de(cid:28)nes a minimum radius for the event horizon,
1
rmin = =l ,
+ m p (53)
p
r
+
i.e. that
an not be smaller than the Plan
k lenght. This lenght
orresponds to a minimum mass or remnant for
the bla
k hole of the order of
Mmin =Λl2.
p (54)
Using equation (39), the emperature of the bla
k hole
orresponding to this mass is
7
1 2Q2 2 Λ l Q2
T+ Mmin ≈ 2π |Λ|lp− 3 l2 + 3 Λ|l3| p 4Q2 . (55)
(cid:12) p | | p− (cid:12)
(cid:0) (cid:1) (cid:12) (cid:12)
(cid:12) (cid:12)
Q=0 (cid:12) (cid:12)
Note that this expression gives, in the
ase, a minimum mass for the BTZ bla
k hole,
Λ l
TBTZ Mmin = | | p.
+ 2π (56)
(cid:0) (cid:1)
Λ=0
Finally, the heat
apa
ity for the bla
k hole
an be
al
ulated using equation (52),
∂M
C =
Q ∂T (57)
(cid:18) (cid:19)Q
= 1 r+2 − m12p 1 Λ + 2 Q2 −1.
2πm2pq2r+2 − m12p −2Λr−M 3M2! (58)
∂T <0
Note that the heat
apa
ity of this bla
k hole is negative, ∂M , when
1 l2
r2 < = p,
+ 2m2 2 (59)
p
rmin = l
+ p
but
ondition (53) impose that the minimum radius is . Therefore the heat
apa
ity of this bla
k hole is
always positive.
IV. CONCLUSION
(2+1)
We have studied the thermodynami
s of the dimensional bla
k hole with non-linear ele
trodynami
s and
with a negative
osmologi
al
onstant using the Generalized Un
ertainty Prin
iple. This gives a minimum mass or
remnant for the bla
k hole, that
orresponds to a maximum Hawking temperature depending only on the ele
tri
Q

harge . The solutionwith the maximumtemperatureis obtained when the bla
k hole hasasizeof the orderofthe
Plan
k s
ale (minimum horizon).
Equation (52) gives the temperature-mass relation, and asMismshinown, it gives the standard Hawking temperature
(39) for small masses, but gets deformed for masses
lose to . Finally, the heat
apa
ity of this bla
k hole is
(2+1)
always positive. This analysis
on(cid:28)rms that Plan
k lenght seems to be the smallest lenght in nature, even in
dimensions.
[1℄ M. Cataldo, N. Cruz, S. del Campo and A. Gar
ia. Phys. Lett. B 484, 154 (2000)
[2℄ A. Larranaga and L. A. Gar
ia. arXiv: 0811.3368 [gr-q
℄
[3℄ I. Arraut,D. Bati
and M. Nowakowski. arXiv: 0810.5156 [gr-q
℄
[4℄ A. Larranaga. arXiv: 0811.3368 [gr-q
℄
[5℄ M. Akbar,Chin. Phys.Lett. 24 (5) (2007) 1158