L. Parker, S. A. Fulling, PRD 9, (1974)
(
)
L.H. Ford, PRD 11, (1975)
J S D k R C i hl PRD ( 6)
J. S. Dowker, R. Critchley, PRD 13, (1976)
D. Hochberg,T. W.Kephart, PRD 49, (1994)
J G D
J.‐G. Demers, R.Lafrance, R.C.Myers, PRD 52, (1995) R L f
R C M
PRD (
) E.T. Akhmedov, T. Pilling, D. Singleton, JMPD 17. (2008) Concepts
y The spectrum of the Laplace‐Beltrami operator and p
p
p
μν
therefore also the zero‐point energy ‘ ‘ of a ω
[
g
]
∑
n
2
(scalar) quantum field depend on the background metric. (Casimir effect).
effect)
y The zero‐point energy diverges, but its change with certain variations in the background metric can (and g
physically ought to be) finite – for instance for an Extremal Reissner‐Nordström (E‐RN) background with conformally coupled scalars (classical BBMB‐solution).
coupled scalars (classical BBMB solution)
y Change in zero‐point energy of quantum fields enters the relation between mass and radius of the BH. We compute p
the leading semi‐classical (WKB) contribution to this zero‐point energy due to periodic classical orbits .
y To enhance
T h
th i
the importance of the zero‐point energy t
f th i t consider N identical conformally coupled scalars –
formally avoids complications with GR. The Setup
The Setup
I will consider the change in vacuum (zero‐point) energy due to the formation of a black hole. What should the black hole be made of? Dust? Quantum fields? We should expect to encounter terrible divergences unless we follow a “physically plausible “, if idealized l f ll “ h i ll l ibl “ if id li d scenario. I.e., if the BH forms from dust, we would have to address (renormalize) some of the properties of this material‐‐too complicated. Simplest consistent scenario: a black hole is formed by scalar fields Indeed: A self‐consistent BH –solution of classical GR was found in the 1970’s : Bochorova‐Bronnikov‐
f
d i h ’ B h
B
ik
Melnikov‐Bekenstein black hole
Reissner‐Nordström
e ss e o dst ö BH
The Reissner‐Nordström metric:
R+
R−
d = v(r )c dt
ds
d − ddr / v(r ) − r d Ω , v(r ) = (1 − )(1 − ),
)
r
r
For general is a static and spherically g
p
y
R+ > R−
2
2
2
2
2
2
symmetric vacuum solution of the Einstein‐Maxwell equations. It describes a BH of mass,
M BH c =
2
4π
κ
( R+ + R− ) = EBH ,
vanishing angular momentum and electric charge
Q =
2
8π
κ
R+ R− ; electric field
l
i fi ld
Q
E = rˆφ,r = rˆ 2
r
Classical BH Thermodynamics
B k
i and Hawking interpreted changes in the d H ki i
d h
i h Bekenstein
(classical) mass of a BH thermodynamically,
4π
dEBH =
(dR+ + dR− ) = TdS + ΦdQ
κ
2
π
k
4
π
k 2
With BH‐entropy: S =
N H
No
Hawking
ki
ABH =
R+
cκ
cκ
radiation from
c R+ − R−
E-RN
T=
Temperature: 4π k R+2
R =R =R
Potential:
Φ=
8π R−
Q
=
R+
κ R+
+
−
Note: Although T and S depend on , the “free energy” is entirely
classical ! Demers,
Demers Lafrance,
Lafrance Myers 1995: Free energy of E-RN
E RN
is not renormalized by scalars . Why?
GR coupled to N (massless) scalars
1 4
⎡ R μν
2⎤
Γ = ∫ d x −g ⎢ − g φ,μφ,ν − ξ Rφ ⎥
2
⎣κ
⎦
(0)
⇒ Gμν = Rμν − 12 g μν R = κ Tμν
Tμν = φ, μφ,ν − 12 g μν φ;α φ ;α + ξ [ g μν
∧
g
φ = ξ Rφ
2
2
2
φ
−
φ
+
G
φ
]
g
; μν
μν
Global SO(N)‐symmetry : consider classical solution for φ ,
which all but one, of the N scalars vanish:
ξ = 0 (minimal) ⇒ Schwarzschild: R− = 0,
φ =const.
R
ξ = (conformal) ⇒ Extremal RN: R− =R+ = R, φ =
κ r−R
Th E-RN
The
E RN solves
l
GR with
ith conformal
f
l coupled
l d SCALAR=
SCALAR
(Bochorova-Bronnikov-Melnikov-Bekenstein) black hole.(1970’s)
1
6
6
1‐loop effective action
Γ
(1)
=Γ
(0)
1/2
+ N ln Det [ g ] + ct's
with N = N + 2,
ct's
t' = ∫ d x − g ( Z Λ + Z R R + Z R2 ( R − Rμν R
4
5
2
2
μν
+ Rμνρσ R
μνρσ
⇒ No logarithmic 1-loop divergence for backgrounds with
∫d
4
x − g ( R − Rμν R
5
2
2
μν
+ Rμνρσ R
μνρσ
) = 0!
Makes sense? to compute (finite) Casimir
(
)
contributions to the mass of a BBMB black hole, e.g.
8π
N c
M E − RN c =
R+χ
R
κ
2
χ <> 0
Q: ?
))
Partial Wave Analysis
∞
1/2
ln Det [
R
0
] = Tr ln[
1
2
R
0
]=
−i
2
∫ d x ∫ d λG( x, x; λ )
4
0
G ( x, y; λ ) = i y
1
λ+
−
R
1
λ+
x = GR ( x, y; λ ) − G0 ( x, y; λ ) with
y x = δ 4 ( x − y)
0
PWA :
∞
d ε (l )
G( x, x ';' λ ) = ∑ ∑ Ylm (Ω ')Y (Ω) ∫
G (r , r ''; λ , ε ) exp[[iε c(t '− t )]
2π
l =0 m =− l
−∞
∞
l
*
lm
1/2
ln Det [
R
0
]=t
−i c
π
∞
∞
∞
∑ (−1) ∫ dε ∫ e π
2 inl
n
n =−∞
0
0
∞
∞
0
0
ldl ∫ r 2 dr ∫ d λ G (l ) (r , r; λ , ε )
Where the Poisson identity
∞
∑ (2l + 1) f (l (l + 1)) =
l =0
∞
∞
∑ (−1)
n =−∞
n
∫
e
2π in l 2 + 1 4
−1/4
Proceed with WKB-approximation
f (l 2 )dl 2
was used.
Semiclassics…. The radial‐Green’s functions satisfy:
ε2
l2 1
iδ (r − r ')
2
(l )
(λ −
+ 2 − 2 ∂ r r v(r )∂ r )G (r , r '; λ , ε ) =
2
v(r ) r
r
r
W riting: G ( l ) ( r , r '; λ , ε ) = A ( l ) ( r , r '; λ , ε ) exp[ iS ( l ) ( r , r '; λ , ε )]
ε2
l2
1
(l ) 2
1) λ =
− 2 − v ( r )( ∂ r S ) + ( l ) 2 ∂ r r 2 v ( r ) ∂ r A ( l )
v(r ) r
A r
pp
2) A ( l )δ ( r − r ')) = −∂∂ r ( A ( l ) ) 2 r 2 v ( r )( ∂ r S ( l ) )) WKB-approx:
=> HJ-equ.
G
(l )
⎡
⎤
1
dz
(r , r; λ , ε ) ~
exp ⎢ i ∫
k ( z; ε , l , λ ) ⎥
2
2r k (r; ε , l , λ )
⎣ r v( z)
⎦
k (r; ε , l , λ ) =
2
⎛
⎞
l
2
ε − ⎜ 2 + λ ⎟ v(r )
⎝r
⎠
2
2
⎛
⎞
l
l
R
k 2 (r ; ε , l , λ ) = ε 2 − ⎜ 2 + λ ⎟ v(r ) = 2 [α 2 − Veff ( x = ; β )]
R
r
⎝r
⎠
εR
λ R2
with α =
,β = 2
l
l
Veff ( x) = v( x)( x 2 + β )
β = β c = 18
=Unstable circular orbit
Horizon
E-RN
β =0
Schwarzschild β = 0
∞
ECas
∞
−i c ∞
n
~
(−1) ∫ d ε ∫ d
∑
2π n =−∞
0
0
n≠0
∞
∞
0
0
∫ dr ∫ dλ
Saddlepoint evaluation of integrals
about (unstable) critical trajectories
with λ = 0, r = 2 R, l = 2ε R
gives:
χ = +0.003
0 003
(preliminary)
The smallest E-RN black hole:
M min ∼ 0.05
0 05mP N ~ 1.2
1 2 N μg
Rmin ∼ 0.05
P
N ~ 8.9 × 10−35 N cm
r dz
2iπ n + 2i ∫
k ( z ;ε , ,λ )
v( z )
r
e
k (r; ε , , λ )
Conclusion and Speculation
Zero‐point energy tends to increase the mass (energy) of small p
gy
(
gy)
black holes and may prevent them from forming (much) below Planck scales.
o The smallest mass E
RN black hole, here estimated The smallest mass E‐RN black hole
here estimated semiclassically, compares well (qualitatively and quantitatively) with a Loop Quantum Gravity approach by L. Modesto (ArXive:0811 2196) who finds that a (ArXive:0811.2196) who finds that a ~ 0.1
0 1 mP Schwarzschild black hole is stabilized by graviton fluctuations ( ) .
N =2
o The accuracy of a semiclassical calculation could be doubted for such tiny BH. But dimensional analysis determines the form of h i BH B di
i
l l i d
i h f
f the quantum correction to the BH mass and the dimensionless coefficient may be estimated for large R (or N), where semiclassics
l
generally would be trusted.
ll ld b d
M μg
o Could BH “grow “ by emitting negative energy Hawking radiation thereby lowering the zero‐point energy of g
y
g
p
gy
quantum fields….? Is Dark Matter composed of stable primordial μ g ‐BHs ?