Black Hole Solutions for Scale Dependent Couplings: The de Sitter and the Reissner-Nordstr\"om Case

0.43 MB·English

by  Benjamin Koch

Black Hole Solutions for Scale Dependent Couplings:
The de Sitter and the Reissner-Nordstro¨m Case
Benjamin Koch and Paola Rioseco
Instituto de F´ısica,
Pontificia Universidad Cat´olica de Chile,
Av. Vicun˜a Mackenna 4860,
Santiago, Chile
(Dated: January 9, 2015)
Allowing for scale dependence of the gravitational couplings leads to a generalization of the
corresponding field equations. In this work, those equations are solved for the Einstein-Hilbert
and the Einstein-Maxwell case, leading to generalizations of the (Anti)-deSitter and the Reissner-
Nordstr¨omblack holes. Thosesolutions arediscussed andcompared totheirclassical counterparts.
5
1
0 PACSnumbers: 04.60.,04.70.
2

n
I. INTRODUCTION
a
J
8 Black holes are very fascinating objects. Apart from the fact that they are classical solutions of the equations
of general relativity, they also have additional “features”, such as the existence of an event horizon and an essential

] singularityattheorigin(r =0,typicallybehindtheeventhorizon). Theexistenceofthissingularitycanbeinterpreted
c
as the break down of the validity of the classicaltheory that predicts such solutions. In this sense the study of black
q
- holes gives an interesting opportunity of exploring general relativity in the transition between a regime where the
r classical solution is known to be valid to high precision, and a regime where corrections to the classical prediction
g
[ are to be expected. Those expected correctionsto the classicalregime are broadlyassumedto be of quantum nature.
A famous example for such quantum corrections is the predicted existence of thermal radiation, emitted by massive
2
black holes, and induced by quantum fluctuations [1].
v
When one takes general relativity as classical limit of a quantum theory seriously one has to deal with a lot of
4
conceptual and technical difficulties. Although, those difficulties are far from settled, there exist various promising
0
9 approaches aiming for a full (or partial) quantum formulation of generalrelativity [2–18] (for a review see [19]), even
0 without the necessity of going to a very different theory in higher dimensions. Although, the different approaches
0 are quite diverse, most of them still have a common feature: Like in most other quantum field theories, they show a
1. non-trivialscaledependence. This means thatthe effective couplingsofthe theories become actuallyscale dependent
0 quantities. For example, Newtons constant G0 acquires a scale dependence G0 G(k). This scaling behavior is
→
5 especially nice to se in the realizations of Weinbergs Asymptotic Safety program [20–27]. Such a scaling is expected
1 to modify classical observables, such as the black hole background [28–39]. Even though, the way how this scale de-
:
v pendenceiscalculatedandthefunctionalformofthescaledependenceitselfcanvaryamongthedifferentapproaches,
i the pure existence of such a scale dependence seems to be a solid statement.
X
In this paper the idea of scale dependent couplings will be implemented at the level of effective action and the
r corresponding improvedequations of motion. In order obtain more generic results, no particular functional form will
a
beassumedforthescaledependentcouplings(G(k),...). Instead,asymmetrycriterionwillbeimposedonthemetric
ansatz, which actually allows to solve the improved equations of motion without knowledge of the functional form
of the coupling constants (G(k),...). This technique will be used in order to obtain two new black hole solutions
that are on the one hand generalizations of the corresponding classical solution and on the other hand solutions
of the self-consistent field equations in the context of scale dependent couplings. The corrections with respect to
the classical solution will be parametrized in terms of an dimensionful integration constant ǫ, which for the case of
quantum-gravity-inducedscale dependence can be expected to be of the order of the Planck mass M .
Pl
The paper is organizedas follows. Inthe subsectionIA propertiesofthe classical(Anti)-de Sitter ((A)dS) solution
and the classical Reissner-Nordstro¨m (RN) will be shortly summarized. In the following subsection IB it will be
reviewed how a scale dependence of the gravitational couplings (G(k),...) can be implemented self-consistently at
the level of generalized gravitational field equations. In section II the generalized (A)dS black hole solution for the
generalized Einstein-Hilbert equations will be discussed. This solution is obtained and parametrized in subsection
IIA, then in subsection IIB the asymptotic behavior of this solution is calculated. A perturbative analysis of the
thermodynamicpropertiesofthis solutionis presentedinsubsectionIIC.InsectionIII the generalizedRNblackhole
solution for the generalized Einstein-Maxwell equations will be discussed. The generalized RN solution is presented
andparametrizedinsubsectionIIIA,theninsubsectionIIIBtheasymptoticbehaviorofthissolutioniscalculatedand
2
the conservedquantity correspondingto the electricalcharge is calculatedin subsection IIIC. The horizonstructure,
thethermodynamiccorrections,andthecosmiccensorshipofthesolutionarediscussedinsubsectionIIID.Theresults
of this work summarized in section IV and important features are highlighted.
A. The classical (A)dS and RN black hole solutions
In this subsection the key features such as line element, divergent behavior and location of the horizons of the
classical(A)dSandRNblackholesolutionswillbe listed. Thelineelementofbothofthoseblackholesolutionstakes
the form
ds2 =−f(r)dt2+f(r)−1dr2+r2dΩ22 , (1)
where dΩ2 is the volume element of the two-sphere.
2
For the case of the (A)dS solution, the function f(r) take the form
2G M 1
f(r) =1 0 0 Λ r2. (2)
|(A)dS − r − 3 0
HereG andΛ denotetheclassicalNewtonconstantandtheclassicalcosmologicalconstant. Theintegrationconstant
0 0
M is the classical mass of the black hole. The sign of Λ describes, a Schwarzschild-AdS (Λ < 0), Schwarzschild
0 0 0
(Λ =0), or a Schwarzschild-dS(Λ >0) black hole. The previouslymentioned space-likesingularity atr =0 can be
0 0
seen for the (A)dS solution by computing the invariant square of the Riemann tensor
48G2M2 8Λ2
R Rµνρσ = 0 0 + 0 . (3)
µνρσ |(A)dS r6 3
This singularity is hidden behind an event horizon. Horizons are found as zeros of the function f(r), which due to
the cubic nature of the function, allows for three solutions
1
r = U 1/3 U1/3, (4)
0|(A)dS − − − Λ
0
and
1 1 i√3
r±|(A)dS = 2 1±i√3 U−1/3+ ∓2Λ0 U1/3. (5)
(cid:16) (cid:17)
Where
U =3G M Λ2+ 9G2M2Λ4 Λ3 , (6)
0 0 0 0 0 0− 0
q
was defined as the typical scale of the solution. If the value of those roots is real and positive, they correspond to
a physical horizon. For the case of Λ 0 there is only a single horizon, given by (4). For the case of Λ > 0 and
0 0
≤
M >0 the solution has two physical horizons given by (5).
0
For the case of the classical RN solution, the function f(r) in the line element (1) takes the form
2G M 4πG Q2
f(r) =1 0 0 + 0 0 , (7)
|RN − r r2e 2
0
where the constant of integration Q is the classical electrical charge of the RN black hole and 1/e2 is the electro-
0 0
magnetic coupling constant. The classical solution for the electromagnetic stress energy tensor is
Q
0
F = F = . (8)
tr − rt r2
The leading singular behavior of the classicalRN solution at r =0 can be seen by computing the invariantsquare of
the Riemann tensor
14G2π2Q4
R Rαβγδ =82 0 0 . (9)
αβγδ |RN e4r8
0
3
Again,thissingularbehaviorcanbeshieldedbyahorizonwhichcanbefoundbysolvingtheconditionofvanishing(7)
4πQ 2
r =G M G 2M 2 0 . (10)
±|RN 0 0±s 0 0 − e02
Oneobservesthatthose twohorizons( )becomedegenerateifthe squarerootonthe righthandside vanishes. Even
±
more, beyond this point the square root turns negative and no physical horizon is present in the solution, which is
undesired since it would lead to a unshielded “naked” singularity. Thus, in order to not get in trouble due to the
appearance of a naked singularity one demands a minimal mass for the classical RN black hole.
Q
M 2√π 0 . (11)
0
≥ e G
0 0
This reasoning is known as the “cosmic censorship” argument [40].
B. Scale dependent couplings and scale setting
Thissubsectionsummarizesthe equationsofmotionforthe scaledependentEinstein-Hilbert-Maxwellsystem. The
notationfollowsclosely[42],wherealsoamoredetaileddescriptionofthissystemandtheproofofitsself-consistency
can be found. The three scale dependent couplings of the scale dependent Einstein-Hilbert-Maxwell system are, the
gravitationalcoupling G , the cosmologicalcoupling Λ , and the electromagnetic coupling 1/e . Further, the system
k k k
hasthreetypesofindependentfields,whicharethe metricfieldg (x), theelectromagneticfourpotentialA (x), and
µν µ
the scale field k(x). The equations of motion for the metric field g (x) are
µν
G
k
G = g Λ ∆t +8π T , (12)
µν − µν k− µν e2 µν
k
where the possible coordinate dependence of G induces an additional contribution to the stress-energy tensor [53]
k
1
∆t =G (g (cid:3) ) . (13)
µν k µν µ ν
−∇ ∇ G
k
The stress-energy tensor for the electromagnetic part is given by
1
T =F αF g F Fαβ , (14)
µν ν µα− 4 µν αβ
where F =D A D A is the antisymmetric electromagnetic field strength tensor. The equations of motion for
µν µ ν ν µ
−
the four potential A (x) are
µ
1
D Fµν =0 . (15)
µ e2
(cid:18) k (cid:19)
Finally the equations of motion for the scale-field k(x) are given by
1 Λ 4π
R 2 k F Fαβ (∂µk)=0 . (16)
∇µ G − ∇µ G − αβ ∇µ e2 ·
(cid:20) (cid:18) k(cid:19) (cid:18) k(cid:19) (cid:18) k(cid:19)(cid:21)
The above equations of motion are complemented by the relations correspondingto globalsymmetries of the system.
For the case of coordinate transformations one has
µG =0 (17)
µν
∇
and for the internal U(1) transformations the corresponding relations are
F =0 . (18)
[µ αβ]
∇
Please note that one has to work with (18) and not with e 2F = 0 [42]. In the following sections, two special
[µ − αβ]
∇
black hole solutions for this system will be presented and discussed. First, in section II a solution for the system
(12-16) will be presented, where the electromagnetic coupling is omitted (1/e2 = 0). Then, in section III, a solution
k
is found for the case of finite electromagnetic coupling and vanishing cosmologicalcoupling (Λ =0).
k
4
II. BLACK HOLE SOLUTION FOR THE EINSTEIN-HILBERT CASE
Inthe EinsteinHilberttruncationoneneglectsthe electromagneticcontributiontothe actionofthe system(12-16)
leading to simplified equations of motion for the metric field g
µν
G = g Λ ∆t , (19)
µν µν k µν
− −
and simplified equations of motion for the scale field k(x) [95]
1 Λ
k
R 2 =0 . (20)
µ µ
∇ G − ∇ G
(cid:18) k(cid:19) (cid:18) k(cid:19)
The most general line element consistent with spherical symmetry is
ds2 = f(r)dt2+h(r)dr2+r2dΩ2 . (21)
− 2
One notes that for this symmetry, the system (19, 20) has sufficient independent equations, in order to solve for the
three r-dependent functions f(r), h(r), and k(r). This is however, assuming that the functional form of the scale
dependent couplings G , and Λ is known, for example from a background independent integration of the functional
k k
renormalizationgroup [20–27].
Sincetheaimistogainsomeinformationonscaledependentblackholes,independentofthedetailsofthederivation
andintegrationofthe renormalizationgrouporthe particularapproachto quantumgravity,wewill use the following
reasoning:
Even if one does not know the functional form of G and Λ , one knows that both couplings will inherit some r-
k k
dependence from k(r) and therefore one might treat them as two independent fields G(r) and Λ(r). Thus, one has
encoded the ignorance(or ambiguity)on the scale dependent couplings in an additionalfield variable (G(r) andΛ(r)
instead of k(r)). Of course, now the system (19, 20) with three independent equations is in any case insufficient to
solveforthefourr-dependent fieldsf(r), h(r), G(r), andΛ(r) infullgenerality. Inordertoreduceagainthenumber
free fields one has to impose some condition on those functions. In the presented study we will restrict our search
to solutions that have only “standard” event horizons. By this we mean that on the one hand the signature of a
(t,r) line will change from minus to plus or vice versa when passing an event horizon (zero of f(r)), which suggests
that either f(r) h(r) or f(r) 1/h(r). On the other hand this means that we demand that the radial part of the
∼ ∼
line element diverges, when the time part of the line element vanishes. Those conditions are implemented straight
forwardly by imposing
1
f(r) . (22)
∼ h(r)
Thischoiceiscommonlyreferredtoas“Schwarzschildansatz”. Withtheexternalrestriction(22)thenumberoffields
is thus reduced to three: f(r), G(r), and Λ(r), which fits the number of independent equations in the system (19,
20).
A. Finding the solution
Based on the condition (22), the ansatz for the line element in the Einstein-Hilbert case will be
1
ds2 = f(r)c2dt2+ dr2+r2dΩ2 , (23)
− t f(r) 2
where the constant c implements explicitly the time-reparametrizationinvariance of the system. The equations (19,
t
20) have already been solved [43–45] by using the ansatz (23) for c = 1. However, in the parametrization found
t
in [43–45], the physical meaning of the integration constants and their relation to the classical (A)dS-Schwarzschild
metric remained unclear.
Here, a new parametrization (using the labels G ,Λ ,M , and ǫ, c , and c ) of the solution is presented, where
0 0 0 t 4
5
those problems were solved. The solutions for the three functions are
G
0
G(r) = (24)
ǫr+1
2G M Λ r2 c (ǫr+1)
f(r) = 1+3G M ǫ 0 0 (1+6ǫG M )ǫr 0 +r2ǫ2(6ǫG M +1)ln 4 (25)
0 0 0 0 0 0
− r − − 3 r
(cid:18) (cid:19)
72ǫ2r(ǫr+1) ǫr+ 1 G M ǫ+ 1 ln c4(ǫr+1) +4r3Λ ǫ2+ 12ǫ3+6Λ ǫ+72ǫ4G M r2
− 2 0 0 6 r 0 0 0 0
Λ(r) = (26)
(cid:0) (cid:1)(cid:0) (cid:1) (cid:16)2r(ǫr+(cid:17)1)2 (cid:0) (cid:1)
72ǫ3G M +11ǫ2+2Λ r+6ǫ2G M
0 0 0 0 0
+ .
2r(ǫr+1)2
(cid:0) (cid:1)
The new and intuitive feature of this non-trivial choice of constants of integration is given by the fact that it was
possible to isolate a combinationof the originalconstants ofintegrationsuch that the classical(A)dS solutioncan be
recoveredby sending a single constant (ǫ) to zero. For instance, the scale dependent Newton coupling reduces to the
classical Newton constant
limG(r)=G (27)
0
ǫ 0
→
and the scale dependent cosmological coupling reduces to the cosmologicalconstant
limΛ(r)=Λ . (28)
0
ǫ 0
→
Finally, the labeling of the constant M is justified by taking the same limit for the metric component
0
Λ r2 2G M
0 0 0
limf(r)= +1 , (29)
ǫ 0 − 3 − r
→
where the classical solution (2) and the mass M is recovered. Thus, the limits (27, 28, and 29) justify the choice of
0
constants of integration as G ,Λ ,M , and ǫ in contrast to the parametrizations found in [43–45]. Please note that
0 0 0
the problem with finding this new parametrization was that the limit ǫ 0 corresponds to sending two constants of
→
the parametrization [43–45] simultaneously and at a specific rate to infinity.
B. Asymptotic space-times
The asymptotic behavior of this solution for small scales (r 0) is closely linked to the singularity at the origin.
→
This singularity can be most clearly studied by evaluating geometrical invariants. For example the Ricci scalar for
the solution is
6G M ǫ 6ǫ+36G M ǫ2 c
R= 0 0 + 0 0 +7ǫ2+42G M ǫ3+4Λ 12ǫ2(1+6G M ǫ)log 4 + (r) . (30)
− r2 r 0 0 0− 0 0 r O
(cid:16) (cid:17)
One observes that the classical limit of 4Λ is modified by a new quadratic divergence of this quantity which is
0
proportional to ǫ. An other invariant quantity that is frequently studied is the higher curvature scalar
48G2M2 48G2M2ǫ
R Rµναβ = 0 0 0 0 + (r 4) . (31)
µναβ r6 − r5 O −
For this invariant one observes that to leading order the singular behavior of this quantity is the same as in the
classical case (3) and that modifications due to ǫ only appear at subleading orders in 1/r. From those two examples
one can already conclude that the elimination of this radial singularity is not possible for the given solution unless
one returns to trivial configurations - say with vanishing M and vanishing ǫ.
0
The other regime of asymptotic behavior can be studied in a large large radius expansion r . In order to get
→∞
a feeling for this behavior it is instructive to plot the radial function f(r) for varying values of ǫ. In figure 1 one
observesthatevenforthe caseΛ =0,the non-classicalscaledependence ǫ=0canmimictheeffectofacosmological
0
6
constant by generating an asymptotic (Anti)-de-Sitter space-time.
By studying the large r behavior of the metric function (25) one finds
Λ˜
f(r)= r2 + (r) , (32)
− 3 O
6
fHrL
1.0
0.8
0.6
0.4
0.2
r
0 5 10 15 20 25 30
FIG. 1: Radial function f(r) for G =1, M =1, Λ =0, and ǫ= 0,0.001,0.005,0.01,0.05
0 0 0
{ }
where the effective cosmologicalconstant Λ˜ is actually a shift of the classical value
Λ˜ =Λ 3ǫ2(6ǫG M +1)ln(c ǫ) . (33)
0 0 0 4
−
When it is referred to Λ˜ as “effective cosmological constant”, this is done in the sense that for any measurement,
say by observing trajectories, the result would be determined by the form of the metric function rather than by the
function Λ(r) in(26). One observesthat the ǫ induced shift inthe cosmologicalconstantΛ˜ Λ is determinedby the
0
sign of the logarithm in (25). In the asymptotic limit r one finds for example for ǫ>− 1 that for
→∞ −6G0M0
1
c < (Λ˜ Λ )>0 , (34)
4 0
ǫ ⇒ −
1
c = (Λ˜ Λ )=0 ,
4 0
ǫ ⇒ −
1
c > (Λ˜ Λ )<0 .
4 0
ǫ ⇒ −
For ǫ < 1/(6G M ) the relations (34) get inverted. A very interesting scenario turns out to be the case of c =
0 0 4
−
ǫ, where the effective cosmological constant agrees with the classical parameter Λ˜ = Λ . However, studying the
0
asymptotics of the metric function (25) is not the only way one might try to extract a notion of an asymptotic
cosmologicalconstant. Fora comparisononecantakethe limitoflarger forthe scale dependentquantity (26)which
gives
lim Λ(r)=2Λ 6ǫ2(6G M ǫ+1)ln(c ǫ) , (35)
0 0 0 4
r −
→∞
which is different from the “effective cosmologicalconstant” (33) extractedfrom the metric solution. This is however
not concerning, since one can argue that the asymptotic form of a space-time must be read from the metric and
not from a function appearing in the equation of motion. Based on this argument one sticks to (33) as the proper
definition of Λ˜. Still it is interesting to note that both possible notions of an “effective cosmological constant” (33
and 35) vanish for the same choice of parameters
ifΛ =3ǫ2(6ǫG M +1)ln(c ǫ) lim Λ(r)=Λ˜ =0 . (36)
0 0 0 4
⇒ r
→∞
Further one observes that the ǫ dependence of both notions vanishes for the particular choice c =1/ǫ.
4
C. Perturbative analysis for horizons and thermodynamics
Since scale dependence of coupling constants is generally assumed to be weak, it is reasonable to treat the dimen-
sionful parameter ǫ as small with respect to the other scales entering the problem such as 1/√G , or M . As it can
0 0
7
be seenfromtherelations(27-29),this constantencodesthe deviationfromtheclassicalsolution(2)andthereforeits
absolute value is also experimentally expected to be very small in comparison with other integration constant with
dimensions of energy. In principle ǫ could take positive or negative values. However, the following short discussion
will show that only small positive values give physically viable (real) solutions at the outside of the event horizon:
From the solution (25) one sees that the argument of logarithm in the metric function could become negative for
ǫ < 0 and c > 0, at very large values of r. Compensating this by making c < 0 is also not possible since in this
4 4
case the logarithm can become negative for somewhat smaller radii r < r < 1/ǫ, where r is the radius of the
S S
| |
horizon which for small ǫ can be approximated by the classical Schwarzschild radius. Thus, the parameter ǫ has to
| |
be positive and small right from the start.
In this context it is instructive to Taylor expand in this small parameter to see the leading corrections due to the
scale dependence of the couplings
G(r) = G ǫ G r+ (ǫ2) (37)
0 0
− · O
2G M r2Λ
f(r) = 1 0 0 0 +ǫ (3G M r)+ (ǫ2) (38)
0 0
− r − 3 · − O
Λ(r) = Λ +ǫ Λ r+ (ǫ2) . (39)
0 0
· O
Fromequation(38)oneseesnowmoreclearlywhypreviousattemptstoobtainameaningfulphysicalparametrization
failed. TheproblemwasthatitwasassumedthatforvanishingparameterM theflat(A)dS solutionf(r)=1 r2Λ0
0 − 3
wouldbe recovered. However,lookingat (38) one sees that this is actually not possible without completely returning
to the classicalsolution(ǫ=0). Apparently the deviationsfromthe classicalspace-timemetric in(38), couldbe used
inaphenomenologicalcontextinordertoconstrainthe valueofthe supposedlyverysmallparameterǫ. Suchastudy
is howeverbeyondthe scope ofthis work. The perturbativeanalysisis howeveruseful for a firstunderstanding of the
leading effects on the black hole horizons and the corresponding thermodynamics.
To first order in ǫ, the horizons are defined by the zeros of (38). For Λ > 0 and for 0 < M < 1 the two
0 0 3G0√Λ0
relevant real horizons are found to be
1 i√3 (1 i√3)P1/3 1 i√3 (1 i√3)(6G M P Λ ) 1
0 0 0
r = ± + ∓ +ǫ ∓ + ± − , (40)
± 2P1/3 2Λ0 4P2/3 4P4/3 − Λ0!
where r is the outer (cosmological) horizon and r is the inner Schwarzschild horizon. P is given by
+
−
1
P =3G M Λ2 1+ (1 ) . (41)
0 0 0 s − 9G20M02Λ0 !
In the classical case (ǫ=0) those two horizons become degenerate for the critical mass
1
M = , (42)
0,crit
3G √Λ
0 0
which corresponds to the Nariaiblack hole [41], which is the maximal allowedblack hole mass before the appearance
ofanakedsingularity. Fornon-vanishingǫthiscriticalmassvalueandthecorrespondingblackholeradiusgetslightly
shifted. A comparisonof the horizon structure as a function of the mass parameter M is shown in figure 2.
0
One observes that for a given M both horizons get shifted by positive ǫ towards smaller radii. Further, one sees
0
thatthesameholdstruefortheradiusofthecriticalNariaiblackholeandthatthecriticalmassparameterM which
0
is the value where the cosmological and the inner horizon merge r = r gets slightly increased with respect to the
+
classical value (42). −
Given the horizon structure and the functional form of (38) one can calculate the temperature of corresponding
black hole. At the inner horizon this temperature is given by
1 df 2G M 2r Λ
Tr− = 4π dr = r02 0 −ǫ− −3 0 +O(ǫ2) . (43)
(cid:12)(cid:12)r− −
(cid:12)
In figure 3 this temperature is shown as a fu(cid:12)nction of the mass parameter M0. One observes that for vast range
of parameters the modified temperature is indistinguishable from the classical value, and that only for the largest
masses, close to the M , a slight splitting of the curves occurs. This splitting shows a slightly increased temperature
c
for increasing ǫ values. Thus, in this region, the shift in the horizon radius r overcompensates the negative direct
contributionofǫtothe temperatureinequation(43). Thereforeittakesslightl−yhighervaluesofM inordertoreach
0
zero temperature, and thus the critical black hole state, as it can also be seen from the figures 2 and 3b .
8
r_S
11.0
10.5
10.0
9.5
9.0 M0
3.30 3.31 3.32 3.33 3.34
FIG. 2: Mass dependence of the black hole horizons r as a function of the mass parameter x=M /M , for G =1
c 0 0
and Λ =0.01. The black dashed curve is for ǫ=0 the±red curve is for ǫ=0.002 and the blue curve is for ǫ=0.004.
0
T_H
T_H
0.10
0.0014
0.08
0.0012
0.0010
0.06
0.0008
0.04 0.0006
0.0004
0.02
0.0002
0.00 M0 0.0000 M0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.30 3.31 3.32 3.33 3.34
(a) . (b) .
. MassrangeM0=0...4 . MassrangeM0=3.3...3.34
FIG. 3: M dependence of the temperature (43) for G =1 and Λ =0.01. The black dashed curve corresponds to
0 0 0
ǫ=0, the red curve is for ǫ=0.002 and the blue curve is for ǫ=0.004.
III. BLACK HOLE SOLUTION FOR THE EINSTEIN-MAXWELL CASE
In this section, a black hole solution for the Einstein-Maxwellcase will be constructedwithout taking into account
the cosmologicalterm (Λ =0). In this case the equations of motion for the metric field (12) simplify to
k
G
k
G = ∆t +8π T , (44)
µν − µν e 2 µν
k
while equations of motion (15) for the U(1) gauge field remain unchanged
1
D Fµν =0 . (45)
µ e2
(cid:18) k (cid:19)
The equations of motion for the scale-field k (16) simplify to
1 4π
R F Fαβ (∂µk)=0 . (46)
∇µ G −∇µ e2 αβ ·
(cid:20) (cid:18) k(cid:19) (cid:18) k(cid:19) (cid:21)
The invariance equations (17) and (18) remain unchanged.
9
A. Finding the solution
When searching a solution of the above equations we proceed by imposing spherical symmetry. For spherical
symmetry, the most general line element is again (21). Assuming electric and not magnetic charge, this symmetry
requirement reduces the degrees of freedom of the electromagnetic stress-energy tensor to
F = F =q(r) . (47)
tr rt
−
Under those assumptions and for a given scale dependence G and 1/e2 the system (44-46) contains four unknown
k k
functions f(r), h(r), q(r), and k(r).
Now,thereasoningofsection(II)willberepeatedandtheignorance(atleastmodeldependence)ofthecouplingflow
willbe encodedintradingthe radialscaledependence k=k(r) for radialcouplingdependence k(r) G(r), 1/e2(r).
→
The increase in unknown functions will be compensated by imposing the “standard black hole” condition (22). Due
to this, the system (44-46) will have to be solved for the four functions f(r), q(r), G(r), and 1/e2(r). The ansatz for
the line element will be
1
ds2 = f(r)dt2+ dr2+r2dΩ2 . (48)
− f(r) 2
Note that here,incontrastto (23), the constantc is set to one,since due to the F contributionone cannotexpect
t µν
to have time-rescaling invariance of the solution.
A good starting point for solving the system is to observe that f(r) actually decouples from the equations for the
radial electric field
1 ∂ q(r)
r2 =0 . (49)
r2∂r e2(r)
(cid:18) (cid:19)
This establishes a first relation between e(r) and q(r). With this, the remaining equations of motion (44 and 46) are
solved by
G
0
G(r) = (50)
ǫr+1
r4ǫ2e 2+4ǫr3e 2+4(1 G M ǫ)e 2r2 8rG M e 2+16πG Q 2
0 0 0 0 0 0 0 0 0 0
f(r) = − −
4r2(ǫr+1)2e 2
0
r6ǫ4e 2+3r5ǫ3e 2+ 3r4e 2 4r3e 2G M +48r2πG Q 2 ǫ2+48ǫrπG Q 2+16πG Q 2 e 2π
e2(r) = 0 0 0 − 0 0 0 0 0 0 0 0 0 0
Q 2G (ǫr+1)3
(cid:0) (cid:0) 0 0 (cid:1) (cid:1)
Q e2(r)
0
q(r) = .
4πe2 r2
0
Having learnedthe lessonfromsection II onthe subtleties of choosingthe constants of integration,the five constants
ofintegrationwerechosensuchthat, forthecasethatthefifth constantvanishes,the fourotherconstantscorrespond
to constants in the classical solution of f(r) . This allows to interpret this fifth constant, which again will be
RN
|
labeled ǫ, as deviation parameter which introduces corrections to the classicalsolution due to scale dependence. One
confirms for Newtons coupling
limG(r)=G , (51)
0
ǫ 0
→
and for the metric function
2G M 4πG Q2
limf(r)=1 0 0 + 0 0 , (52)
ǫ→0 − r r2e02
which reproduces the classical solution (7). Similarly one finds for the scale dependent electrical coupling
lime2(r)=(4π)2e2 (53)
0
ǫ 0
→
and the electrical field strength
4πQ
0
limq(r)= . (54)
ǫ 0 r2
→
10
One observes that this is also in agreement with the classical result. The factor (4π), that appears as different
normalization of q(r) is a convention which turns out to cancel for the ratios q2(r)/e2(r) that enter the equations
of motion (44 and 46). This convention is further justified when calculating the actual charge of the solution (50).
Takingagainǫsmallerthananydimensionfullscaleofthesystem,onecanexpandinthisparameter. Inthisexpansion
the lowest order corrections to the classical solution are
G(r) = G ǫG r+ ǫ2 , (55)
0 0
− O
2G M 4πG Q2 8G πQ2
f(r) = 1 0 0 + (cid:0) 0(cid:1) 0 +ǫ 3G M 0 0 r + ǫ2 ,
− r r2e 2 0 0− e2r − O
0 (cid:18) 0 (cid:19)
e4π(4G M r3 3r4) (cid:0) (cid:1)
e2(r) = e216π2 ǫ2 0 0 0 − + ǫ3 ,
0 − G Q2 O
0 0
Q 4π e2(4G M r 3r2) (cid:0) (cid:1)
q(r) = 0 ǫ2 0 0 0 − + ǫ3 .
r2 − 4G Q O
0 0
(cid:0) (cid:1)
Fortunately, due to the purely polynomial form of the solution, most of the following results can be discussed with
the complete solution (50), without the necessity to use the above expansion.
B. Asymptotic behavior of the solution
The asymptotic behavior of this generalized Reissner-Nordstro¨m solution for r 0 can be studied by evaluating
→
curvature invariants in this limit. For example, already the Ricci scalar shows a quadratic divergence for small radii
M ǫ+4πQ2ǫ2e 2
R= 6G 0 0 0− + (1/r) . (56)
− 0 r2 O
This is in contrast to the classical solution where the Ricci scalar vanishes in this limit.
Still the generalized solution incorporates the classical result, since in the classicallimit ǫ 0, the right hand side
→
of (56) vanishes accordingly. The invariant contraction of two Riemann tensors is also divergentin this limit, but for
this invariant, the leading divergence agrees with the classical behavior
G2π2Q4
R Rαβγδ =277 0 0 + 1/r7 . (57)
αβγδ e4r8 O
0
(cid:0) (cid:1)
This confirms that this solution is singular at the origin for the generalized solution, even in the classical limit.
Taking the opposite limit (for r ), the asymptotic behavior in brings a surprise since the metric function
→ ∞
approaches
1 1
f(r)= + + 1/r2 . (58)
4 2ǫr O
(cid:0) (cid:1)
This result does not resemble the classically expected value 1, not even by approaching a posteriori ǫ 0. The
→
supposed discrepancy can be explained by the fact all dimensionless terms ǫr are incompatible with first taking the
limit of large r and than the limit of small ǫ. Clearly, if one takes the limit of ǫ 0 first, the classical result is
→
recovered. The asymptotic line element corresponding to (58) is
1
ds2 = dt2+4dr2+r2dθ2+r2sin2θdφ2 . (59)
∞ −4
Onecantrytocastthisinamorefamiliarformbyintroducingarescaledtimeτ = 1tandarescaledradialcoordinate
2
R=2r, giving the line element
1
ds2 = dτ2+dR2+R2 dθ2+sin2θdφ2 . (60)
∞ − 4
(cid:0) (cid:1)
However, even though now the radial and temporal part of the line element take the familiar form, the angular part
suffers a non-trivial scaling, which corresponds to a deficit solid angle. For example, the area of a sphere with very
largeradiusRisnot4πR2 butratherπR2. Theremainingfactorof1/4mightbeabsorbedinarescalingoftheangles
θ and φ, but this clearly also implies the mentioned deficit angle of the asymptotic geometry.